Finite Group DatabasebetaFinite Group DatabasebetaOrder 10731.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C49×C73 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | ΓL(1,𝔽8)×C73 | — | 7 | 1 | no | maximal |
| 3 | C7×C73 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 4 | C3×C73 | — | 49 | 1 | no | Hall |
| 5 | C49⋊C3 | C73 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C73 | C49⋊C3 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 7 | C49 | C3×C73 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | ΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 9 | C7 | ΓL(1,𝔽8)×C73 | 1 | 1 | yes | characteristic |
| 10 | C3 | — | 49 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C73 | C49⋊C3 |
| commutator, cosocle, hypocenter | C49 | C3×C73 |
| Frattini | C7 | ΓL(1,𝔽8)×C73 |
| Fitting | C49×C73 | C3 |
| socle | C7×C73 | ΓL(1,𝔽8) |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C73 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C49×C73 | C3 | semidirect |
| 5. C49⋊C3 | C73 | direct (N × H) |
| 6. C73 | C49⋊C3 | direct (N × H) |
| 7. C49 | C3×C73 | semidirect |