Finite Group DatabasebetaFinite Group DatabasebetaOrder 16832.
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×D263 |
| commutator, socle, cosocle | C22×C263 | C24 |
| Fitting | #126×C263 | C2 |
| prime radical | C2×C22⋊D263 | C22 |
| hypercenter | #126 | D263 |
| hypocenter | C263 | #563 |
| LCS | G ▷ C22×C263 ▷ C263 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22554470 | C2 | semidirect |
| 2. #22554469 | C2 | semidirect |
| 3. #22554465 | C2 | semidirect |
| 4. #22554464 | C2 | semidirect |
| 5. #22554463 | C2 | semidirect |
| 6. #22554461 | C2 | semidirect |
| 7. #22554460 | C2 | semidirect |
| 11. #22554450 | C2 | semidirect |
| 12. #22554448 | C2 | semidirect |
| 13. #22554447 | C2 | semidirect |
| 14. #22554447 | C2 | semidirect |
| 15. #126×C263 | C2 | semidirect |
| 97. C263 | #563 | semidirect |
| 104. #126 | D263 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22554470 | 2 |
| #22554469 | 2 |
| #22554465 | 2 |
| #22554464 | 2 |
| #22554463 | 2 |
| #22554461 | 2 |
| #22554460 | 2 |
| #22554456 | 2 |
| #22554454 | 2 |
| #22554452 | 2 |
| #22554450 | 2 |
| #22554448 | 2 |
| #22554447 | 2 |
| #126×C263 | 2 |
| #563 | 263 |