Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 386 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 | C22 | C22×D28 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#405 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #405 | D7 |
| hypocenter | C7 | #3176 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90226 | C2 | semidirect |
| 2. C7×#405 | C2 | semidirect |
| 3. #89907 | C2 | semidirect |
| 4. #89887 | C2 | semidirect |
| 5. #89485 | C2 | semidirect |
| 6. #89483 | C2 | semidirect |
| 7. #89467 | C2 | semidirect |
| 8. C4×D56 | C2 | semidirect |
| 9. #89455 | C2 | semidirect |
| 59. #405 | D7 | semidirect |
| 152. C7 | #3176 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90226 | 2 |
| C7×#405 | 2 |
| #89907 | 2 |
| #89887 | 2 |
| #89485 | 2 |
| #89483 | 2 |
| #89467 | 2 |
| C4×D56 | 2 |
| #89455 | 2 |
| #3176 | 7 |