Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×Q8 | C2 |
| prime radical | C22×D28 | C22 |
| hypercenter | C2×C4×Q8 | D7 |
| hypocenter | C7 | #2580 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×C7×Q8 | C2 | semidirect |
| 3. #90003 | C2 | semidirect |
| 4. #89764 | C2 | semidirect |
| 5. #89761 | C2 | semidirect |
| 7. #89708 | C2 | semidirect |
| 8. #89700 | C2 | semidirect |
| 40. C2×D28 | Q8 | semidirect |
| 53. C2×C4×Q8 | D7 | semidirect |
| 181. C7 | #2580 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×Q8 | 2 |
| C2×C4×D28 | 2 |
| #90003 | 2 |
| #89764 | 2 |
| #89761 | 2 |
| #89711 | 2 |
| #89708 | 2 |
| #89700 | 2 |
| #2580 | 7 |