Finite Group DatabasebetaFinite Group DatabasebetaOrder 10731.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×C73 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #13095276 | — | 7 | 1 | no | maximal |
| 3 | #13095275 | — | 7 | 1 | no | maximal |
| 4 | C7×C73 | — | 3 | 2 | no | — |
| 5 | C7×C73 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 6 | C7×C73 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 7 | C73⋊C3 | — | 49 | 1 | no | Hall |
| 8 | #3862 | — | 73 | 1 | no | maximal, Hall |
| 9 | C73 | #3862 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C72 | C73⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | ΓL(1,𝔽8) | — | 511 | 1 | no | — |
| 12 | ΓL(1,𝔽8) | — | 511 | 1 | no | — |
| 13 | C7 | — | 3 | 2 | no | — |
| 14 | C7 | #13095275 | 1 | 1 | yes | characteristic, complemented |
| 15 | C7 | #13095276 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3 | — | 3577 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C73 | C3 |
| LCS | G ▷ C72×C73 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C72×C73 | C3 | semidirect |
| 5. C7×C73 | ΓL(1,𝔽8) | semidirect |
| 6. C7×C73 | ΓL(1,𝔽8) | semidirect |
| 9. C73 | #3862 | semidirect |
| 10. C72 | C73⋊C3 | semidirect |
| 14. C7 | #13095275 | semidirect |
| 15. C7 | #13095276 | semidirect |