Finite Group DatabasebetaFinite Group DatabasebetaOrder 27584.
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×D431 |
| commutator, socle, cosocle | C22×C431 | C24 |
| Fitting | #126×C431 | C2 |
| hypercenter | #126 | D431 |
| hypocenter | C431 | #560 |
| LCS | G ▷ C22×C431 ▷ C431 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28352186 | C2 | semidirect |
| 2. #28352183 | C2 | semidirect |
| 3. #28352174 | C2 | semidirect |
| 4. #28352164 | C2 | semidirect |
| 5. #28352162 | C2 | semidirect |
| 6. #28352159 | C2 | semidirect |
| 7. #126×C431 | C2 | semidirect |
| 11. C2×D1724 | C22 | semidirect |
| 13. C2×C4×D431 | C22 | semidirect |
| 14. Dic431⋊C4 | C22 | semidirect |
| 17. (C2×D431)⋊C4 | C22 | semidirect |
| 18. C4⋊C4×C431 | C22 | semidirect |
| 20. C42×C431 | C22 | semidirect |
| 50. C431 | #560 | semidirect |
| 54. #126 | D431 | semidirect |
| 66. C4⋊C4 | C2×D431 | semidirect |
| 71. C42 | C2×D431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28352186 | 2 |
| #28352183 | 2 |
| #28352174 | 2 |
| #28352164 | 2 |
| #28352162 | 2 |
| #28352159 | 2 |
| #126×C431 | 2 |
| #560 | 431 |