Finite Group DatabasebetaFinite Group DatabasebetaOrder 31424.
234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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 |
| prime radical | #30628310 | C2 |
| hypercenter | #126 | D491 |
| hypocenter | C491 | #529 |
| 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. #30628331 | C2 | semidirect |
| 2. #30628327 | C2 | semidirect |
| 3. #30628323 | C2 | semidirect |
| 4. #30628322 | C2 | semidirect |
| 5. #30628319 | C2 | semidirect |
| 6. #30628315 | C2 | semidirect |
| 8. #30628302 | C2 | semidirect |
| 9. #30628299 | C2 | semidirect |
| 10. (C22⋊D491)⋊C4 | C2 | semidirect |
| 11. (C22⋊D491)⋊C4 | C2 | semidirect |
| 12. #30628295 | C2 | semidirect |
| 13. #30628293 | C2 | semidirect |
| 14. C42×D491 | C2 | semidirect |
| 15. #126×C491 | C2 | semidirect |
| 103. C491 | #529 | semidirect |
| 106. #126 | D491 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30628331 | 2 |
| #30628327 | 2 |
| #30628323 | 2 |
| #30628322 | 2 |
| #30628319 | 2 |
| #30628315 | 2 |
| #30628310 | 2 |
| #30628302 | 2 |
| #30628299 | 2 |
| (C22⋊D491)⋊C4 | 2 |
| #30628295 | 2 |
| #30628293 | 2 |
| C42×D491 | 2 |
| #126×C491 | 2 |
| #529 | 491 |