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
|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
|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.