Subgroup structure of symmetric group:S6
From Groupprops
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 . The group has order 720.
Family contexts
Family name | Parameter values | General discussion of subgroup structure of family |
---|---|---|
symmetric group | degree , i.e., the group | 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 : 1, 2, 6, 30, 156, 1455, 11300, 151221, ... |
Number of conjugacy classes of subgroups | 56 Compared with : 1, 2, 4, 11, 19, 56, 96, 296, ... |
Number of automorphism classes of subgroups | 37 Compared with : 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 -Hall subgroup, -Hall subgroup, and -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 | cyclic group:Z2 | 2 | 360 | 1 | 15 | 15 | ||
subgroup generated by triple transposition in S6 | cyclic group:Z2 | 2 | 360 | 1 | 15 | 15 | ||
subgroup generated by double transposition in S6 | cyclic group:Z2 | 2 | 360 | 1 | 45 | 45 |
The table needs to be completed.