Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #123×C419 | C2 |
| hypercenter | #123 | D419 |
| hypocenter | C419 | #553 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27211713 | C2 | semidirect |
| 2. #27211711 | C2 | semidirect |
| 3. #27211708 | C2 | semidirect |
| 4. #27211700 | C2 | semidirect |
| 5. #27211691 | C2 | semidirect |
| 6. #27211691 | C2 | semidirect |
| 7. #27211690 | C2 | semidirect |
| 8. #27211689 | C2 | semidirect |
| 9. #27211680 | C2 | semidirect |
| 10. C4×C22⋊D419 | C2 | semidirect |
| 11. C4×C22⋊D419 | C2 | semidirect |
| 12. (C22⋊D419)⋊C4 | C2 | semidirect |
| 13. #27211674 | C2 | semidirect |
| 14. C4⋊C4×D419 | C2 | semidirect |
| 15. #123×C419 | C2 | semidirect |
| 109. C419 | #553 | semidirect |
| 115. #123 | D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27211713 | 2 |
| #27211711 | 2 |
| #27211708 | 2 |
| #27211700 | 2 |
| #27211691 | 2 |
| #27211690 | 2 |
| #27211689 | 2 |
| #27211680 | 2 |
| C4×C22⋊D419 | 2 |
| (C22⋊D419)⋊C4 | 2 |
| #27211674 | 2 |
| C4⋊C4×D419 | 2 |
| #123×C419 | 2 |
| #553 | 419 |