Finite Group DatabasebetaFinite Group DatabasebetaOrder 27712.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×D433 |
| commutator, socle, cosocle | C22×C433 | C24 |
| Fitting | #123×C433 | C2 |
| prime radical | #28357192 | C2 |
| hypercenter | #123 | D433 |
| hypocenter | C433 | #529 |
| LCS | G ▷ C22×C433 ▷ C433 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28357215 | C2 | semidirect |
| 2. #28357213 | C2 | semidirect |
| 3. #28357210 | C2 | semidirect |
| 4. #28357206 | C2 | semidirect |
| 5. #28357203 | C2 | semidirect |
| 6. #28357197 | C2 | semidirect |
| 8. #28357185 | C2 | semidirect |
| 9. D4×Dic433 | C2 | semidirect |
| 10. #28357180 | C2 | semidirect |
| 11. #28357180 | C2 | semidirect |
| 12. #28357178 | C2 | semidirect |
| 13. #28357176 | C2 | semidirect |
| 14. C2×C4×Dic433 | C2 | semidirect |
| 15. #123×C433 | C2 | semidirect |
| 103. C433 | #529 | semidirect |
| 110. #123 | D433 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28357215 | 2 |
| #28357213 | 2 |
| #28357210 | 2 |
| #28357206 | 2 |
| #28357203 | 2 |
| #28357197 | 2 |
| #28357192 | 2 |
| #28357185 | 2 |
| D4×Dic433 | 2 |
| #28357180 | 2 |
| #28357178 | 2 |
| #28357176 | 2 |
| C2×C4×Dic433 | 2 |
| #123×C433 | 2 |
| #529 | 433 |