Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
788 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#384 | C2 |
| hypercenter | #384 | D7 |
| hypocenter | C7 | #2537 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6458 | C2 | semidirect |
| 2. C2×C4×D28 | C2 | semidirect |
| 3. C7×#384 | C2 | semidirect |
| 4. #89763 | C2 | semidirect |
| 5. #89759 | C2 | semidirect |
| 28. C2×D28 | D4 | semidirect |
| 34. C42×C7 | D4 | semidirect |
| 41. #384 | D7 | semidirect |
| 91. C4×C7 | #121 | semidirect |
| 111. C42 | D28 | semidirect |
| 140. C7 | #2537 | semidirect |
| 156. C4 | #6458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6458 | 2 |
| C2×C4×D28 | 2 |
| C7×#384 | 2 |
| #89763 | 2 |
| #89759 | 2 |
| #2537 | 7 |