Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
532 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 141 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×#431 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #431 | D7 |
| hypocenter | C7 | #3205 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90168 | C2 | semidirect |
| 2. C2×#6437 | C2 | semidirect |
| 3. C7×#431 | C2 | semidirect |
| 4. #89615 | C2 | semidirect |
| 5. #89504 | C2 | semidirect |
| 6. #89467 | C2 | semidirect |
| 7. #89466 | C2 | semidirect |
| 15. C2×D56 | C22 | semidirect |
| 21. #6437 | C22 | semidirect |
| 23. C4×D28 | C22 | semidirect |
| 24. #6432 | C22 | semidirect |
| 25. C7×C4⋊C8 | C22 | semidirect |
| 27. C4×C7×C8 | C22 | semidirect |
| 28. #6395 | C22 | semidirect |
| 66. #431 | D7 | semidirect |
| 75. C2×C4×C7 | D8 | semidirect |
| 113. C4⋊C8 | C2×D7 | semidirect |
| 116. C4×C8 | C2×D7 | semidirect |
| 192. C2×C4 | D56 | semidirect |
| 200. C7 | #3205 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90168 | 2 |
| C2×#6437 | 2 |
| C7×#431 | 2 |
| #89615 | 2 |
| #89504 | 2 |
| #89467 | 2 |
| #89466 | 2 |
| #3205 | 7 |