Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
644 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C22×C42×C7 | C2 |
| prime radical | C24×C7 | C23 |
| hypercenter | C22×C42 | D7 |
| hypocenter | C7 | #2857 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#6432 | C2 | semidirect |
| 5. #89707 | C2 | semidirect |
| 6. #89701 | C2 | semidirect |
| 7. #89700 | C2 | semidirect |
| 30. C22×C4×C7 | Q8 | semidirect |
| 100. C22×C7 | C4⋊Q8 | semidirect |
| 109. C4×C7 | C22⋊Q8 | semidirect |
| 122. C22×C4 | Dic14 | semidirect |
| 168. C7 | #2857 | semidirect |
| 173. C22 | #6432 | semidirect |
| 183. C4 | #6487 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C7 | 2 |
| C2×#6487 | 2 |
| C2×#6432 | 2 |
| #89984 | 2 |
| #89707 | 2 |
| #89701 | 2 |
| #89700 | 2 |
| #2857 | 7 |