Finite Group DatabasebetaFinite Group DatabasebetaOrder 7533.
110 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14578892 | C3 | 1 | 8 | yes | maximal, complemented |
| 2 | C3×C31⋊C27 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C31×#737 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | 3−1+2×C31 | C9 | 1 | 8 | yes | — |
| 5 | H(3,𝔽3)×C31 | C9 | 1 | 1 | yes | characteristic |
| 6 | C3×C9×C31 | C32 | 1 | 4 | yes | — |
| 7 | C31⋊C27 | C32 | 1 | 1 | yes | characteristic |
| 8 | C31⋊C27 | C32 | 1 | 8 | yes | — |
| 9 | C32×C31 | C3×C9 | 1 | 4 | yes | — |
| 10 | C9×C31 | C3×C9 | 1 | 8 | yes | — |
| 11 | C9×C31 | C33 | 1 | 1 | yes | characteristic |
| 12 | #6903 | — | 31 | 1 | no | maximal, Sylow |
| 13 | C3×C31 | — | 3 | 4 | no | — |
| 14 | C3×C31 | C32×C9 | 1 | 1 | yes | characteristic |
| 15 | #737 | C31⋊C3 | 1 | 1 | yes | characteristic |
| 16 | C27⋊C3 | — | 31 | 8 | no | — |
| 17 | C3×C27 | — | 31 | 4 | no | — |
| 18 | C31 | #6903 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | 3−1+2 | C31⋊C9 | 1 | 8 | yes | — |
| 20 | H(3,𝔽3) | C31⋊C9 | 1 | 1 | yes | characteristic |
| 21 | C3×C9 | C3×C31⋊C3 | 1 | 4 | yes | — |
| 22 | C27 | — | 31 | 1 | no | — |
| 23 | C27 | — | 31 | 8 | no | — |
| 24 | C32 | C3×C31⋊C9 | 1 | 4 | yes | — |
| 25 | C9 | C3×C31⋊C9 | 1 | 8 | yes | — |
| 26 | C9 | C32×C31⋊C3 | 1 | 1 | yes | characteristic |
| 27 | C3 | — | 3 | 4 | no | — |
| 28 | C3 | C32×C31⋊C9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C32×C31⋊C3 |
| commutator, socle | C3×C31 | C32×C9 |
| Fitting | C31×#737 | C3 |
| hypercenter | #737 | C31⋊C3 |
| hypocenter | C31 | #6903 |
| LCS | G ▷ C3×C31 ▷ C31 | |
| UCS | C1 ◁ C9 ◁ #737 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14578892 | C3 | semidirect |
| 2. C3×C31⋊C27 | C3 | semidirect |
| 18. C31 | #6903 | semidirect |