4.6: Classification of Groups

 Classification of finite groups

The following table gives the number of distinct groups of finite order. Here $C_n$ stands for a cyclic group of order $n$. 

Group order	Abelian	Non-abelian	Comments
1	{e}
2	$C_2=D_1$		Prime p
3	$C_3$		Prime p
4	$C_4$ or  $C_2 \times C_2\cong D_2 \cong K_4$		Order of the form $p^2$
5	$C_5$
6	$C_6$	$S_3\cong D_3$	Order of the form $pq$, p, q are prime with $q \equiv 1 mod p$
7	$C_7$
8	$C_8$ or    $C_2 \times C_4$ or $C_2 \times C_2 \times C_2$	$D_4$  or $Q_8$	$p^3$
9	$C_9$ or  $C_3 \times C_3$		$p^2$
10	$C_{10}$	$D_5$
11	$C_{11}$
12	$C_{12}$	$D_6$ or $A_4$
13	$C_{13}$
14	$C_{14}$	$D_7$
15	$C_{15}$
16	$C_{16}$ or
17	$C_{17}$