Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#477 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #477 | D7 |
| hypocenter | C7 | #3305 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #477 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90425 | C2 | semidirect |
| 2. #90268 | C2 | semidirect |
| 3. C7×#477 | C2 | semidirect |
| 4. #89860 | C2 | semidirect |
| 5. C4×#1143 | C2 | semidirect |
| 6. #89657 | C2 | semidirect |
| 7. #89655 | C2 | semidirect |
| 8. #89615 | C2 | semidirect |
| 9. #89606 | C2 | semidirect |
| 10. #89583 | C2 | semidirect |
| 11. #89576 | C2 | semidirect |
| 81. #477 | D7 | semidirect |
| 239. C7 | #3305 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90425 | 2 |
| #90268 | 2 |
| C7×#477 | 2 |
| #89860 | 2 |
| C4×#1143 | 2 |
| #89657 | 2 |
| #89655 | 2 |
| #89615 | 2 |
| #89606 | 2 |
| #89583 | 2 |
| #89576 | 2 |
| #3305 | 7 |