Finite Group DatabasebetaFinite Group DatabasebetaOrder 30849.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C13×C113 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C13⋊C3×C113 | — | 7 | 1 | no | maximal, Hall |
| 3 | ΓL(1,𝔽8)×C113 | — | 13 | 1 | no | maximal, Hall |
| 4 | C13×C113 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C7×C113 | C13⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C3×C113 | — | 91 | 1 | no | Hall |
| 7 | #63270 | C113 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C113 | #63270 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C7×C13 | C3×C113 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C13⋊C3 | — | 7 | 1 | no | Hall |
| 11 | ΓL(1,𝔽8) | — | 13 | 1 | no | Hall |
| 12 | C13 | ΓL(1,𝔽8)×C113 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C7 | C13⋊C3×C113 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C3 | — | 91 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C113 | #63270 |
| commutator, cosocle, hypocenter | C7×C13 | C3×C113 |
| Frattini | C1 | G |
| Fitting, socle | C7×C13×C113 | C3 |
| LCS | G ▷ C7×C13 | |
| UCS | C1 ◁ C113 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C13×C113 | C3 | semidirect |
| 4. C13×C113 | ΓL(1,𝔽8) | semidirect |
| 5. C7×C113 | C13⋊C3 | semidirect |
| 7. #63270 | C113 | direct (N × H) |
| 8. C113 | #63270 | direct (N × H) |
| 9. C7×C13 | C3×C113 | semidirect |
| 12. C13 | ΓL(1,𝔽8)×C113 | semidirect |
| 13. C7 | C13⋊C3×C113 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C13×C113 | 3 |
| C13⋊C3×C113 | 7 |
| ΓL(1,𝔽8)×C113 | 13 |
| #63270 | 113 |