Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#500 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #500 | D7 |
| hypocenter | C7 | #3169 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90416 | C2 | semidirect |
| 2. C7×#500 | C2 | semidirect |
| 3. #89860 | C2 | semidirect |
| 4. #89655 | C2 | semidirect |
| 5. #89644 | C2 | semidirect |
| 6. #89467 | C2 | semidirect |
| 7. C4×C8×D7 | C2 | semidirect |
| 48. #500 | D7 | semidirect |
| 71. C4×D7 | D8 | semidirect |
| 72. C7×C8 | #41 | semidirect |
| 150. C8 | #1161 | semidirect |
| 151. C7 | #3169 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90416 | 2 |
| C7×#500 | 2 |
| #89860 | 2 |
| #89655 | 2 |
| #89644 | 2 |
| #89467 | 2 |
| C4×C8×D7 | 2 |
| #3169 | 7 |