Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D28 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#443 | C2 |
| socle | C2×C7 | D42 |
| hypercenter | #443 | D7 |
| hypocenter | C7 | #3293 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90462 | C2 | semidirect |
| 2. #90444 | C2 | semidirect |
| 3. C7×#443 | C2 | semidirect |
| 4. #89915 | C2 | semidirect |
| 5. #89668 | C2 | semidirect |
| 6. #89598 | C2 | semidirect |
| 7. #89472 | C2 | semidirect |
| 8. #89471 | C2 | semidirect |
| 71. #443 | D7 | semidirect |
| 185. C7 | #3293 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90462 | 2 |
| #90444 | 2 |
| C7×#443 | 2 |
| #89915 | 2 |
| #89668 | 2 |
| #89598 | 2 |
| #89472 | 2 |
| #89471 | 2 |
| #3293 | 7 |