Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 418 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×#446 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #446 | D7 |
| hypocenter | C7 | #3399 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ #446 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90276 | C2 | semidirect |
| 2. #90268 | C2 | semidirect |
| 3. C7×#446 | C2 | semidirect |
| 4. #89620 | C2 | semidirect |
| 5. #89616 | C2 | semidirect |
| 6. #89615 | C2 | semidirect |
| 7. #89615 | C2 | semidirect |
| 8. #89484 | C2 | semidirect |
| 9. #89483 | C2 | semidirect |
| 10. #89467 | C2 | semidirect |
| 11. C4×#1132 | C2 | semidirect |
| 86. #446 | D7 | semidirect |
| 120. C7×C8 | #41 | semidirect |
| 242. C8 | #1161 | semidirect |
| 243. C7 | #3399 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90276 | 2 |
| #90268 | 2 |
| C7×#446 | 2 |
| #89620 | 2 |
| #89616 | 2 |
| #89615 | 2 |
| #89484 | 2 |
| #89483 | 2 |
| #89467 | 2 |
| C4×#1132 | 2 |
| #3399 | 7 |