Finite Group DatabasebetaFinite Group DatabasebetaOrder 31424.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D491 |
| commutator, socle, cosocle | C22×C491 | C24 |
| Fitting | #126×C491 | C2 |
| hypercenter | #126 | D491 |
| hypocenter | C491 | #560 |
| LCS | G ▷ C22×C491 ▷ C491 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30628334 | C2 | semidirect |
| 2. #30628331 | C2 | semidirect |
| 3. #30628322 | C2 | semidirect |
| 4. #30628312 | C2 | semidirect |
| 5. #30628310 | C2 | semidirect |
| 6. #30628307 | C2 | semidirect |
| 7. #126×C491 | C2 | semidirect |
| 11. C2×D1964 | C22 | semidirect |
| 13. C2×C4×D491 | C22 | semidirect |
| 14. Dic491⋊C4 | C22 | semidirect |
| 17. (C2×D491)⋊C4 | C22 | semidirect |
| 18. C4⋊C4×C491 | C22 | semidirect |
| 20. C42×C491 | C22 | semidirect |
| 50. C491 | #560 | semidirect |
| 54. #126 | D491 | semidirect |
| 66. C4⋊C4 | C2×D491 | semidirect |
| 71. C42 | C2×D491 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30628334 | 2 |
| #30628331 | 2 |
| #30628322 | 2 |
| #30628312 | 2 |
| #30628310 | 2 |
| #30628307 | 2 |
| #126×C491 | 2 |
| #560 | 491 |