Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#1141 |
| commutator, cosocle | C7×C8 | C24 |
| Frattini | C8 | C23×D7 |
| Fitting | C7×#576 | C2 |
| socle | C2×C7 | C22×D8 |
| hypercenter | #576 | D7 |
| hypocenter | C7 | #3416 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #576 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#576 | C2 | semidirect |
| 2. #90444 | C2 | semidirect |
| 3. #89967 | C2 | semidirect |
| 4. #89966 | C2 | semidirect |
| 5. C2×#6401 | C2 | semidirect |
| 6. #89919 | C2 | semidirect |
| 7. #89912 | C2 | semidirect |
| 15. C2×Dic28 | C22 | semidirect |
| 16. #6467 | C22 | semidirect |
| 17. #6403 | C22 | semidirect |
| 18. #6402 | C22 | semidirect |
| 19. #6401 | C22 | semidirect |
| 20. #6401 | C22 | semidirect |
| 21. #6400 | C22 | semidirect |
| 22. #6386 | C22 | semidirect |
| 23. C2×C7⋊C16 | C22 | semidirect |
| 47. #576 | D7 | semidirect |
| 136. C7 | #3416 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#576 | 2 |
| #90444 | 2 |
| #89967 | 2 |
| #89966 | 2 |
| C2×#6401 | 2 |
| #89919 | 2 |
| #89912 | 2 |
| #3416 | 7 |