Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 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, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#390 | C2 |
| prime radical | C22×C7×D4 | C22 |
| hypercenter | #390 | D7 |
| hypocenter | C7 | #2808 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #89750 | C2 | semidirect |
| 4. #89725 | C2 | semidirect |
| 5. #89723 | C2 | semidirect |
| 6. #89723 | C2 | semidirect |
| 7. #89719 | C2 | semidirect |
| 8. #89701 | C2 | semidirect |
| 9. #89700 | C2 | semidirect |
| 216. C7 | #2808 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#390 | 2 |
| #89994 | 2 |
| #89750 | 2 |
| #89725 | 2 |
| #89723 | 2 |
| #89719 | 2 |
| #89701 | 2 |
| #89700 | 2 |
| #2808 | 7 |