Finite Group DatabasebetaFinite Group DatabasebetaOrder 27584.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 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 |
| prime radical | C2×C22⋊D431 | C22 |
| hypercenter | #126 | D431 |
| hypocenter | C431 | #563 |
| 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. #28352189 | C2 | semidirect |
| 2. #28352188 | C2 | semidirect |
| 3. #28352184 | C2 | semidirect |
| 4. #28352183 | C2 | semidirect |
| 5. #28352182 | C2 | semidirect |
| 6. #28352180 | C2 | semidirect |
| 7. #28352179 | C2 | semidirect |
| 11. #28352169 | C2 | semidirect |
| 12. #28352167 | C2 | semidirect |
| 13. #28352166 | C2 | semidirect |
| 14. #28352166 | C2 | semidirect |
| 15. #126×C431 | C2 | semidirect |
| 97. C431 | #563 | semidirect |
| 104. #126 | D431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28352189 | 2 |
| #28352188 | 2 |
| #28352184 | 2 |
| #28352183 | 2 |
| #28352182 | 2 |
| #28352180 | 2 |
| #28352179 | 2 |
| #28352175 | 2 |
| #28352173 | 2 |
| #28352171 | 2 |
| #28352169 | 2 |
| #28352167 | 2 |
| #28352166 | 2 |
| #126×C431 | 2 |
| #563 | 431 |