Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 484 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, prime radical | C7×#392 | C2 |
| hypercenter | #392 | D7 |
| hypocenter | C7 | #2752 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #89994 | C2 | semidirect |
| 3. #89993 | C2 | semidirect |
| 4. #89992 | C2 | semidirect |
| 5. #89725 | C2 | semidirect |
| 6. #89723 | C2 | semidirect |
| 7. #89721 | C2 | semidirect |
| 8. #89715 | C2 | semidirect |
| 9. #89426 | C2 | semidirect |
| 221. C7 | #2752 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#392 | 2 |
| #89994 | 2 |
| #89993 | 2 |
| #89992 | 2 |
| #89725 | 2 |
| #89723 | 2 |
| #89721 | 2 |
| #89715 | 2 |
| #89426 | 2 |
| #2752 | 7 |