Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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 | #121×C419 | C2 |
| hypercenter | #121 | D419 |
| hypocenter | C419 | #560 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27211714 | C2 | semidirect |
| 2. #27211713 | C2 | semidirect |
| 3. #27211711 | C2 | semidirect |
| 4. #27211703 | C2 | semidirect |
| 5. #27211702 | C2 | semidirect |
| 6. #27211701 | C2 | semidirect |
| 7. #27211692 | C2 | semidirect |
| 8. #27211691 | C2 | semidirect |
| 9. #27211690 | C2 | semidirect |
| 10. #27211690 | C2 | semidirect |
| 11. #27211689 | C2 | semidirect |
| 12. #27211688 | C2 | semidirect |
| 13. #27211686 | C2 | semidirect |
| 14. #27211686 | C2 | semidirect |
| 15. #121×C419 | C2 | semidirect |
| 24. C2×C22⋊D419 | C22 | semidirect |
| 25. C2×C22⋊D419 | C22 | semidirect |
| 26. C22×Dic419 | C22 | semidirect |
| 27. C2×C4×D419 | C22 | semidirect |
| 28. C4⋊Dic419 | C22 | semidirect |
| 29. C4⋊Dic419 | C22 | semidirect |
| 30. Dic419⋊C4 | C22 | semidirect |
| 31. Dic419⋊C4 | C22 | semidirect |
| 32. Dic419⋊C4 | C22 | semidirect |
| 35. C22⋊Dic419 | C22 | semidirect |
| 42. (C2×D419)⋊C4 | C22 | semidirect |
| 43. C4×Dic419 | C22 | semidirect |
| 46. C2×D4×C419 | C22 | semidirect |
| 47. C22×C4×C419 | C22 | semidirect |
| 48. C4⋊C4×C419 | C22 | semidirect |
| 50. C22⋊C4×C419 | C22 | semidirect |
| 122. C419 | #560 | semidirect |
| 135. #121 | D419 | semidirect |
| 151. C2×D4 | C2×D419 | semidirect |
| 154. C22×C4 | C2×D419 | semidirect |
| 160. C4⋊C4 | C2×D419 | semidirect |
| 172. C22⋊C4 | C2×D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27211714 | 2 |
| #27211713 | 2 |
| #27211711 | 2 |
| #27211703 | 2 |
| #27211702 | 2 |
| #27211701 | 2 |
| #27211692 | 2 |
| #27211691 | 2 |
| #27211690 | 2 |
| #27211689 | 2 |
| #27211688 | 2 |
| #27211686 | 2 |
| #121×C419 | 2 |
| #560 | 419 |