Subgroup structure of symmetric group:S6

This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S6.
View subgroup structure of particular groups | View other specific information about symmetric group:S6

This article discusses the subgroup structure of symmetric group:S6, which is the symmetric group on the set $\{1,2,3,4,5,6\}$. The group has order 720.

Family contexts

Family name	Parameter values	General discussion of subgroup structure of family
symmetric group	degree $n=6$, i.e., the group $S_{6}$	subgroup structure of symmetric groups
symplectic group of degree four	field:F2	subgroup structure of symplectic group of degree four over a finite field

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	1455 Compared with $S_{n},n=1,2,3,\dots$: 1, 2, 6, 30, 156, 1455, 11300, 151221, ...
Number of conjugacy classes of subgroups	56 Compared with $S_{n},n=1,2,3,\dots$: 1, 2, 4, 11, 19, 56, 96, 296, ...
Number of automorphism classes of subgroups	37 Compared with $S_{n},n=1,2,3,\dots$: 1, 2, 4, 11, 19, 37, 96, 296, ...
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: direct product of D8 and Z2 (order 16), Sylow number is 45 3-Sylow: elementary abelian group:E9 (order 9), Sylow number is 10 5-Sylow: cyclic group:Z5 (order 5), Sylow number is 36
Hall subgroups	No Hall subgroups other than the Sylow subgroups, whole group, and trivial subgroup. In particular, there is no $\{2,3\}$-Hall subgroup, $\{2,5\}$-Hall subgroup, and $\{3,5\}$-Hall subgroup.
maximal subgroups	maximal subgroups have order 48, 72, 120, and 360
normal subgroups	The only normal subgroups are the whole group, the trivial subgroup, and alternating group:A6 as A6 in S6.

Table classifying subgroups up to conjugacy

The below lists subgroups up to conjugacy, i.e., up to automorphisms arising from conjugation in symmetric group:S6. This is not the same as the classification up to automorphisms because of the presence of other automorphisms, a phenomenon unique to degree six (see symmetric groups on finite sets are complete).

Conjugacy class of subgroups	Representative subgroup (full list if small, generating set if large)	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes	Size of each conjugacy class	Total number of subgroups	Note
trivial subgroup	$()$	trivial group	1	720	1	1	1	trivial
subgroup generated by transposition in S6	$\{(),(1,2)\}$	cyclic group:Z2	2	360	1	15	15
subgroup generated by triple transposition in S6	$\{(),(1,2)(3,4)(5,6)\}$	cyclic group:Z2	2	360	1	15	15
subgroup generated by double transposition in S6	$\{(),(1,2)(3,4)\}$	cyclic group:Z2	2	360	1	45	45

The table needs to be completed.