Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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×#427 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #427 | D7 |
| hypocenter | C7 | #3339 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90221 | C2 | semidirect |
| 2. C7×#427 | C2 | semidirect |
| 3. #89887 | C2 | semidirect |
| 4. C8×C22⋊D7 | C2 | semidirect |
| 5. #89775 | C2 | semidirect |
| 6. #89655 | C2 | semidirect |
| 7. D7×#107 | C2 | semidirect |
| 8. #89650 | C2 | semidirect |
| 9. #89644 | C2 | semidirect |
| 10. #89643 | C2 | semidirect |
| 11. #89635 | C2 | semidirect |
| 79. #427 | D7 | semidirect |
| 215. C7 | #3339 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90221 | 2 |
| C7×#427 | 2 |
| #89887 | 2 |
| C8×C22⋊D7 | 2 |
| #89775 | 2 |
| #89655 | 2 |
| D7×#107 | 2 |
| #89650 | 2 |
| #89644 | 2 |
| #89643 | 2 |
| #89635 | 2 |
| #3339 | 7 |