Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
452 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D7 |
| commutator, socle | C22×C7 | C23×C4 |
| Fitting | C7×#431 | C2 |
| prime radical | C22×D28 | C4 |
| cosocle | C2×C4×C7 | C24 |
| hypercenter | #431 | D7 |
| hypocenter | C7 | #2975 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#431 | C2 | semidirect |
| 3. #89697 | C2 | semidirect |
| 4. #89610 | C2 | semidirect |
| 5. #89498 | C2 | semidirect |
| 6. #89460 | C2 | semidirect |
| 12. C2×#1131 | C22 | semidirect |
| 17. C7×C4⋊C8 | C22 | semidirect |
| 19. C4×C7×C8 | C22 | semidirect |
| 21. (C2×D7)⋊C8 | C22 | semidirect |
| 22. C4⋊(C7⋊C8) | C22 | semidirect |
| 23. C4×C7⋊C8 | C22 | semidirect |
| 51. #431 | D7 | semidirect |
| 60. C2×C4×C7 | M16 | semidirect |
| 86. C4⋊C8 | C2×D7 | semidirect |
| 91. C4×C8 | C2×D7 | semidirect |
| 147. C2×C4 | #1131 | semidirect |
| 156. C7 | #2975 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D28 | 2 |
| C7×#431 | 2 |
| #89697 | 2 |
| #89610 | 2 |
| #89498 | 2 |
| #89460 | 2 |
| #2975 | 7 |