Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
520 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 119 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×#401 | C2 |
| prime radical | #89443 | C2 |
| hypercenter | #401 | D7 |
| hypocenter | C7 | #2760 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #401 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#401 | C2 | semidirect |
| 2. #89445 | C2 | semidirect |
| 4. #89437 | C2 | semidirect |
| 5. #89437 | C2 | semidirect |
| 6. #89436 | C2 | semidirect |
| 7. C7⋊#401 | C2 | semidirect |
| 41. #401 | D7 | semidirect |
| 144. C7 | #2760 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#401 | 2 |
| #89445 | 2 |
| #89443 | 2 |
| #89437 | 2 |
| #89436 | 2 |
| C7⋊#401 | 2 |
| #2760 | 7 |