# Subgroup structure of symmetric group:S7

## Contents

View subgroup structure of particular groups | View other specific information about symmetric group:S7

This article discusses the subgroup structure of symmetric group:S7, which is the symmetric group on the set $\{ 1, 2,3,4,5,6,7\}$. The group has order 5040.

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

### Quick summary

Item Value
Number of subgroups 11300
Compared with $S_n, n=1,2,\dots$: 1,2,6,30,156,1455,11300,151221
Number of conjugacy classes of subgroups 96
Compared with $S_n, n=1,2,\dots$: 1,2,4,11,19,56,96,296,554,1593,...
Number of automorphism classes of subgroups 96
Compared with $S_n, n=1,2,\dots$: 1,2,4,11,19,37,96,296,554,1593,...
Isomorphism classes of Sylow subgroups 2-Sylow: direct product of D8 and Z2 (order 16)
3-Sylow: elementary abelian group:E9 (order 9)
5-Sylow: cyclic group:Z5 (order 5)
7-Sylow: cyclic group:Z7 (order 7)
Hall subgroups Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{ 2,3 \}$-Hall subgroups (of order 144) and $\{ 2,3,5 \}$-Hall subgroups (of order 720), the latter being S6 in S7. Note that the $\{ 2,3 \}$-Hall subgroups are not contained in $\{ 2,3,5 \}$-Hall subgroups.
maximal subgroups maximal subgroups have orders 42, 144, 240, 720, 2520
normal subgroups the whole group, trivial subgroup, and alternating group:A7 (embedded as A7 in S7)
subgroups that are simple non-abelian groups alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size $2^2 + 2 + 1 = 7$), alternating group:A6 (order 360), alternating group:A7 (order 2520)