Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
484 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#384 | C2 |
| prime radical | C23×D7 | C23 |
| hypercenter | #384 | D7 |
| hypocenter | C7 | #2499 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#384 | C2 | semidirect |
| 5. #89753 | C2 | semidirect |
| 6. #89749 | C2 | semidirect |
| 7. #89746 | C2 | semidirect |
| 9. #89701 | C2 | semidirect |
| 62. #384 | D7 | semidirect |
| 192. C7 | #2499 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6460 | 2 |
| C2×#6455 | 2 |
| C7×#384 | 2 |
| #89765 | 2 |
| #89753 | 2 |
| #89749 | 2 |
| #89746 | 2 |
| C4×(C2×D7)⋊C4 | 2 |
| #89701 | 2 |
| #2499 | 7 |