Finite Group DatabasebetaFinite Group DatabasebetaOrder 8162.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C11×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C11×D53 | — | 7 | 1 | no | maximal, Hall |
| 3 | D371 | C11 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C11×C53 | D7 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C7×C53 | C2×C11 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C11×D7 | — | 53 | 1 | no | maximal, Hall |
| 7 | D53 | — | 7 | 1 | no | Hall |
| 8 | C7×C11 | D53 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C53 | C11×D7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×C11 | — | 371 | 1 | no | Hall |
| 11 | D7 | — | 53 | 1 | no | Hall |
| 12 | C11 | D371 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C7 | C11×D53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 371 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | D371 |
| commutator, hypocenter | C7×C53 | C2×C11 |
| Frattini | C1 | G |
| Fitting, socle | C7×C11×C53 | C2 |
| LCS | G ▷ C7×C53 | |
| UCS | C1 ◁ C11 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C11×C53 | C2 | semidirect |
| 3. D371 | C11 | direct (N × H) |
| 4. C11×C53 | D7 | semidirect |
| 5. C7×C53 | C2×C11 | semidirect |
| 8. C7×C11 | D53 | semidirect |
| 9. C53 | C11×D7 | semidirect |
| 12. C11 | D371 | direct (N × H) |
| 13. C7 | C11×D53 | semidirect |