Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 564 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 | C2×C4×C7×D4 | C2 |
| prime radical | C22×C7×D4 | C22 |
| hypercenter | C2×C4×D4 | D7 |
| hypocenter | C7 | #2563 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6432 | C2 | semidirect |
| 4. #89984 | C2 | semidirect |
| 5. #89749 | C2 | semidirect |
| 6. #89719 | C2 | semidirect |
| 7. C4×C4⋊Dic7 | C2 | semidirect |
| 27. C2×C7×D4 | Q8 | semidirect |
| 112. C2×D4 | Dic14 | semidirect |
| 158. C7 | #2563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×D4 | 2 |
| C2×#6432 | 2 |
| #89993 | 2 |
| #89984 | 2 |
| #89749 | 2 |
| #89719 | 2 |
| C4×C4⋊Dic7 | 2 |
| #2563 | 7 |