Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
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×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | #126×C419 | C2 |
| prime radical | C2×C22⋊D419 | C22 |
| hypercenter | #126 | D419 |
| hypocenter | C419 | #563 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27211718 | C2 | semidirect |
| 2. #27211717 | C2 | semidirect |
| 3. #27211713 | C2 | semidirect |
| 4. #27211712 | C2 | semidirect |
| 5. #27211711 | C2 | semidirect |
| 6. #27211709 | C2 | semidirect |
| 7. #27211708 | C2 | semidirect |
| 11. #27211698 | C2 | semidirect |
| 12. #27211696 | C2 | semidirect |
| 13. #27211695 | C2 | semidirect |
| 14. #27211695 | C2 | semidirect |
| 15. #126×C419 | C2 | semidirect |
| 97. C419 | #563 | semidirect |
| 104. #126 | D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27211718 | 2 |
| #27211717 | 2 |
| #27211713 | 2 |
| #27211712 | 2 |
| #27211711 | 2 |
| #27211709 | 2 |
| #27211708 | 2 |
| #27211704 | 2 |
| #27211702 | 2 |
| #27211700 | 2 |
| #27211698 | 2 |
| #27211696 | 2 |
| #27211695 | 2 |
| #126×C419 | 2 |
| #563 | 419 |