Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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 | C2×C7×#125 | C2 |
| prime radical | C22×D28 | C22 |
| hypercenter | C2×#125 | D7 |
| hypocenter | C7 | #2633 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#125 | C2 | semidirect |
| 3. #89765 | C2 | semidirect |
| 4. #89764 | C2 | semidirect |
| 6. #89761 | C2 | semidirect |
| 7. #89755 | C2 | semidirect |
| 8. #89709 | C2 | semidirect |
| 50. C2×#125 | D7 | semidirect |
| 167. C7 | #2633 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#125 | 2 |
| C2×C4×D28 | 2 |
| #89765 | 2 |
| #89764 | 2 |
| #89763 | 2 |
| #89761 | 2 |
| #89755 | 2 |
| #89709 | 2 |
| #2633 | 7 |