Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 524 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 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#427 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #427 | D7 |
| hypocenter | C7 | #3058 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D56 | C2 | semidirect |
| 2. C2×#6464 | C2 | semidirect |
| 3. #90216 | C2 | semidirect |
| 4. C7×#427 | C2 | semidirect |
| 5. #89773 | C2 | semidirect |
| 6. #89655 | C2 | semidirect |
| 7. #89637 | C2 | semidirect |
| 17. C2×#1131 | C22 | semidirect |
| 18. C2×C8×D7 | C22 | semidirect |
| 19. #6458 | C22 | semidirect |
| 23. C7×#107 | C22 | semidirect |
| 24. C7×#106 | C22 | semidirect |
| 25. #6383 | C22 | semidirect |
| 40. C2×C7×C8 | D4 | semidirect |
| 58. #427 | D7 | semidirect |
| 98. #107 | C2×D7 | semidirect |
| 99. #106 | C2×D7 | semidirect |
| 133. C2×C8 | D28 | semidirect |
| 168. C7 | #3058 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D56 | 2 |
| C2×#6464 | 2 |
| #90216 | 2 |
| C7×#427 | 2 |
| #89773 | 2 |
| #89655 | 2 |
| #89637 | 2 |
| #3058 | 7 |