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 | C3×C7×C73 | — | 7 | 1 | no | maximal |
| 3 | ΓL(1,𝔽8)×C73 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 4 | C7×C73 | — | 3 | 2 | no | — |
| 5 | C7×C73 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 6 | C7×C73 | C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×C73 | — | 7 | 1 | no | Hall |
| 8 | C7×ΓL(1,𝔽8) | C73 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 9 | C73 | C7×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C72 | C3×C73 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C3×C7 | — | 7 | 1 | no | — |
| 12 | ΓL(1,𝔽8) | C7×C73 | 1 | 1 | yes | characteristic, direct product factor |
| 13 | C7 | — | 3 | 2 | no | — |
| 14 | C7 | ΓL(1,𝔽8)×C73 | 1 | 1 | yes | characteristic, direct product factor |
| 15 | C7 | C3×C7×C73 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3 | — | 7 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C73 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C7×C73 |
| Frattini | C1 | G |
| Fitting, socle | C72×C73 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C7×C73 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C72×C73 | C3 | semidirect |
| 3. ΓL(1,𝔽8)×C73 | C7 | direct (N × H) |
| 5. C7×C73 | ΓL(1,𝔽8) | direct (N × H) |
| 6. C7×C73 | C3×C7 | semidirect |
| 8. C7×ΓL(1,𝔽8) | C73 | direct (N × H) |
| 9. C73 | C7×ΓL(1,𝔽8) | direct (N × H) |
| 10. C72 | C3×C73 | semidirect |
| 12. ΓL(1,𝔽8) | C7×C73 | direct (N × H) |
| 14. C7 | ΓL(1,𝔽8)×C73 | direct (N × H) |
| 15. C7 | C3×C7×C73 | semidirect |