Tabella de parve gruppos

La prime tabella lista le gruppos finite con un ordine minor o equal a 20 excepte isomorphitate.

Ordine	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Quantitate de gruppos abelian	1	1	1	2	1	1	1	3	2	1	1	2	1	1	1	5	1	2	1	2
Quantitate de gruppos non-abelian	0	0	0	0	0	1	0	2	0	1	0	3	0	1	0	9	0	3	0	3
Quantitate de gruppos in toto	1	1	1	2	1	2	1	5	2	2	1	5	1	2	1	14	1	5	1	5

Abbreviaturas

• $A_2\cong S_1\cong {\mathbb{Z}}_1$ es le gruppo trivial.
• $A_n$ es le gruppo alternante del grado $n$, con $n!/2$ permutationes de $n$ elementos pro $n\ge 2$.
• $C_n$ = ${\mathbb{Z}}_n$ = $\mathbb{Z}/n\mathbb{Z}$ es le gruppo cyclic del ordine $n$.
• $D_n$ es le gruppo dihedre del ordine $2n$.
• ${\mathrm{D}\mathrm{i}\mathrm{c}}_n$ es le gruppo dicyclic del ordine $4n$.
• $Q_8$ es le gruppo de quaterniones del ordine $8$.
• $S_n$ es le gruppo symmetric del grado $n$, con $n!$ permutationes de $n$ elementos.
• $V_4$ es le gruppo de Klein del ordine $4$.

Tabella

Ordine	Gruppo	Subgruppos non-trivial	Proprietates	Graphico cyclo
1	${\mathbb{Z}}_1\cong S_1\cong A_2$		abelian, cyclic
2	${\mathbb{Z}}_2\cong S_2\cong D_1$		abelian, finite, simple, cyclic, le minus grande gruppo non-trivial
3	${\mathbb{Z}}_3\cong A_3$		abelian, simple, cyclic
4	${\mathbb{Z}}_4\cong {\mathrm{D}\mathrm{i}\mathrm{c}}_1$	${\mathbb{Z}}_2$	abelian, cyclic
	$V_4\cong {\mathbb{Z}}_2^2\cong D_2$	$3\cdot {\mathbb{Z}}_2$	abelian, le minus grande gruppo non-cyclic
5	${\mathbb{Z}}_5$		abelian, simple, cyclic
6	${\mathbb{Z}}_6\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_3$	${\mathbb{Z}}_3$, ${\mathbb{Z}}_2$	abelian, cyclic
	$S_3\cong D_3$   gruppo symetric $S_3$	${\mathbb{Z}}_3$, $3\cdot {\mathbb{Z}}_2$	le minus grande gruppo non-ablian
7	${\mathbb{Z}}_7$		abelian, simple, cyclic
8	${\mathbb{Z}}_8$	${\mathbb{Z}}_4$, ${\mathbb{Z}}_2$	abelian, cyclic
	${\mathbb{Z}}_2\times {\mathbb{Z}}_4$	$2\cdot {\mathbb{Z}}_4$, $3\cdot {\mathbb{Z}}_2$, $D_2$	abelian
	${\mathbb{Z}}_2^3\cong D_2\times {\mathbb{Z}}_2$	$7\cdot {\mathbb{Z}}_2$, $7\cdot D_2$	abelian
	$D_4$	${\mathbb{Z}}_4$, $2\cdot D_2$, $5\cdot {\mathbb{Z}}_2$	non-abelian
	$Q_8\cong {\mathrm{D}\mathrm{i}\mathrm{c}}_2$	$3\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_2$	non-abelian; le minus grande gruppo hamiltonian
9	${\mathbb{Z}}_9$	${\mathbb{Z}}_3$	abelian, cyclic
	${\mathbb{Z}}_3^2$	$4\cdot {\mathbb{Z}}_3$	abelian
10	${\mathbb{Z}}_{10}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_5$	${\mathbb{Z}}_5$, ${\mathbb{Z}}_2$	abelian, cyclic
	$D_5$	${\mathbb{Z}}_5$, $5\cdot {\mathbb{Z}}_2$	non-abelian
11	${\mathbb{Z}}_{11}$		abelian, simple, cyclic
12	${\mathbb{Z}}_{12}\cong {\mathbb{Z}}_4\times {\mathbb{Z}}_3$	${\mathbb{Z}}_6$, ${\mathbb{Z}}_4$, ${\mathbb{Z}}_3$, ${\mathbb{Z}}_2$	abelian, cyclic
	${\mathbb{Z}}_2\times {\mathbb{Z}}_6\cong {\mathbb{Z}}_2^2\times {\mathbb{Z}}_3\cong D_2\times {\mathbb{Z}}_3$	$3\cdot {\mathbb{Z}}_6$, ${\mathbb{Z}}_3$, $D_2$, $3\cdot {\mathbb{Z}}_2$	abelian
	$D_6\cong D_3\times {\mathbb{Z}}_2$	${\mathbb{Z}}_6$, $2\cdot D_3$, $3\cdot D_2$, ${\mathbb{Z}}_3$, $7\cdot {\mathbb{Z}}_2$	non-abelian
	$A_4$	$D_2$, $4\cdot {\mathbb{Z}}_3$, $3\cdot {\mathbb{Z}}_2$	non-abelian; nulle subgruppo de ordine 6
	${\mathrm{D}\mathrm{i}\mathrm{c}}_3$	${\mathbb{Z}}_6$, $3\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_3$, ${\mathbb{Z}}_2$	non-abelian
13	${\mathbb{Z}}_{13}$		abelian, simple, cyclic
14	${\mathbb{Z}}_{14}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_7$	${\mathbb{Z}}_7$, ${\mathbb{Z}}_2$	abelian, cyclic
	$D_7$	${\mathbb{Z}}_7$, $7\cdot {\mathbb{Z}}_2$	non-abelian
15	${\mathbb{Z}}_{15}\cong {\mathbb{Z}}_3\times {\mathbb{Z}}_5$	${\mathbb{Z}}_5$, ${\mathbb{Z}}_3$	abelian, cyclic
16	${\mathbb{Z}}_{16}$	${\mathbb{Z}}_8$, ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2$	abelian, cyclic
	${\mathbb{Z}}_2^4$	$15\cdot {\mathbb{Z}}_2$, $35\cdot D_2$, $15\cdot {\mathbb{Z}}_2^3$	abelian
	${\mathbb{Z}}_4\times {\mathbb{Z}}_2^2$	$7\cdot {\mathbb{Z}}_2$, $4\cdot {\mathbb{Z}}_4$, $7\cdot D_2$, ${\mathbb{Z}}_2^3$, $6\cdot {\mathbb{Z}}_4\times {\mathbb{Z}}_2$	abelian
	${\mathbb{Z}}_8\times {\mathbb{Z}}_2$	$3\cdot {\mathbb{Z}}_2$, $2\cdot {\mathbb{Z}}_4$, $D_2$, $2\cdot {\mathbb{Z}}_8$, ${\mathbb{Z}}_4\times {\mathbb{Z}}_2$	abelian
	${\mathbb{Z}}_4^2$	$3\cdot {\mathbb{Z}}_2$, $6\cdot {\mathbb{Z}}_4$, $D_2$,$3\cdot {\mathbb{Z}}_4\times {\mathbb{Z}}_2$	abelian
	$D_8$	${\mathbb{Z}}_8$, $2\cdot D_4$, $4\cdot D_2$, ${\mathbb{Z}}_4$, $9\cdot {\mathbb{Z}}_2$	non-abelian
	$D_4\times {\mathbb{Z}}_2$	$4\cdot D_4$, ${\mathbb{Z}}_4\times {\mathbb{Z}}_2$, $2\cdot {\mathbb{Z}}_2^3$, $13\cdot {\mathbb{Z}}_2^2$, $2\cdot {\mathbb{Z}}_4$, $11\cdot {\mathbb{Z}}_2$	non-abelian
	$Q_{16}\cong \mathrm{D}\mathrm{i}{\mathrm{c}}_{\mathrm{4}}$	${\mathbb{Z}}_8$, $2\cdot Q_8$, $5\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_2$	non-abelian
	$Q_8\times {\mathbb{Z}}_2$	$3\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_4$, $4\cdot Q_8$, $6\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $3\cdot {\mathbb{Z}}_2$	non-abelian, gruppo hamiltonian
	gruppo quasi-dihedre	${\mathbb{Z}}_8$, $Q_8$, $D_4$, $3\cdot {\mathbb{Z}}_4$, $2\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $5\cdot {\mathbb{Z}}_2$	non-abelian
	M-gruppo (gruppo non-abelian, non-hamiltonian, modular)	$2\cdot {\mathbb{Z}}_8$, ${\mathbb{Z}}_4\times {\mathbb{Z}}_2$, $2\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $3\cdot {\mathbb{Z}}_2$	non-abelian
	producto semidirecte ${\mathbb{Z}}_4⋊{\mathbb{Z}}_4$	$3\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_4$, $6\cdot {\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $3\cdot {\mathbb{Z}}_2$	non-abelian
	le gruppo create per matrices de Pauli	$3\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_4$, $3\cdot D_4$, $Q_8$, $4\cdot {\mathbb{Z}}_4$, $3\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $7\cdot {\mathbb{Z}}_2$	non-abelian
	$G_{4,4}=V_4⋊{\mathbb{Z}}_4$	$2\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $4\cdot {\mathbb{Z}}_4$, $7\cdot {\mathbb{Z}}_2\times {\mathbb{Z}}_2$, $7\cdot {\mathbb{Z}}_2$	non-abelian
17	${\mathbb{Z}}_{17}$		abelian, simple, cyclic
18	${\mathbb{Z}}_{18}\cong {\mathbb{Z}}_9\times {\mathbb{Z}}_2$	${\mathbb{Z}}_9,{\mathbb{Z}}_6,{\mathbb{Z}}_3,{\mathbb{Z}}_2$	abelian, cyclic
	${\mathbb{Z}}_6\times {\mathbb{Z}}_3$	${\mathbb{Z}}_6,{\mathbb{Z}}_3,{\mathbb{Z}}_2$	abelian
	$D_9$		non-abelian
	$S_3\times {\mathbb{Z}}_3$		non-abelian
	$({\mathbb{Z}}_3\times {\mathbb{Z}}_3){⋊}_{\alpha }{\mathbb{Z}}_2$ con $\alpha (1)=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)$		non-abelian
19	${\mathbb{Z}}_{19}$		abelian, simple, cyclic
20	${\mathbb{Z}}_{20}\cong {\mathbb{Z}}_5\times {\mathbb{Z}}_4$	${\mathbb{Z}}_{10},{\mathbb{Z}}_5,{\mathbb{Z}}_4,{\mathbb{Z}}_2$	abelian, cyclic
	${\mathbb{Z}}_{10}\times {\mathbb{Z}}_2\cong {\mathbb{Z}}_5\times {\mathbb{Z}}_2\times {\mathbb{Z}}_2$	${\mathbb{Z}}_5,{\mathbb{Z}}_2$	abelian
	$Q_{20}\cong {\mathrm{D}\mathrm{i}\mathrm{c}}_5$		non-abelian
	${\mathbb{Z}}_5⋊{\mathbb{Z}}_4\cong$ gruppo affine ${\mathrm{A}\mathrm{G}\mathrm{L}}_1$(5)		non-abelian
	$D_{10}\cong D_5\times {\mathbb{Z}}_2$	${\mathbb{Z}}_{10},D_5,{\mathbb{Z}}_5,5\cdot V_4,6\cdot {\mathbb{Z}}_2$	non-abelian