# Subgroup structure of alternating group:A7

## Contents

View subgroup structure of particular groups | View other specific information about alternating group:A7

This article discusses the subgroup structure of alternating group:A7, which is the alternating group on the set $\{ 1, 2,3,4,5,6,7\}$. The group has order 2520.

## Family contexts

Family name Parameter values General discussion of subgroup structure of family
alternating group degree $n = 7$, i.e., the group $A_7$ subgroup structure of alternating groups

## 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 3786
Compared with $A_n, n = 3,4,5,\dots$: 2, 10, 59, 501, 3786, 48337, ...
Number of conjugacy classes of subgroups 40
Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 22, 40, 137, ...
Number of automorphism classes of subgroups 37
Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 16, 37, 112, ...
Isomorphism classes of Sylow subgroups and the corresponding fusion systems 2-Sylow: dihedral group:D8 (order 8) as D8 in A7 (with its non-inner fusion system -- see fusion systems for dihedral group:D8). Sylow number is 315.
3-Sylow: elementary abelian group:E9 (order 9) as E9 in A7. Sylow number is 70.
5-Sylow: cyclic group:Z5 (order 5) as Z5 in A7. Sylow number is 126.
7-Sylow: cyclic group:Z7 (order 7) as Z7 in A7. Sylow number is 120.
Hall subgroups Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{ 2,3 \}$-Hall subgroups (of order 72) and $\{ 2,3,5 \}$-Hall subgroups (of order 360), the latter being A6 in A7. Note that the $\{ 2,3 \}$-Hall subgroups are not contained in $\{ 2,3,5 \}$-Hall subgroups.
maximal subgroups maximal subgroups have orders 72, 120, 168, 360.
normal subgroups only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple.
subgroups that are simple non-abelian groups (apart from the whole group itself) 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)