Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
612 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 137 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×#386 | C2 |
| prime radical | C2×#6503 | C2 |
| hypercenter | #386 | D7 |
| hypocenter | C7 | #2709 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #386 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6498 | C2 | semidirect |
| 3. C7×#386 | C2 | semidirect |
| 4. #89985 | C2 | semidirect |
| 5. #89733 | C2 | semidirect |
| 6. #89733 | C2 | semidirect |
| 7. #89730 | C2 | semidirect |
| 8. #89444 | C2 | semidirect |
| 9. #89425 | C2 | semidirect |
| 79. #386 | D7 | semidirect |
| 260. C7 | #2709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6503 | 2 |
| C2×#6498 | 2 |
| C7×#386 | 2 |
| #89985 | 2 |
| #89733 | 2 |
| #89730 | 2 |
| #89444 | 2 |
| #89425 | 2 |
| #2709 | 7 |