Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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×#399 | C2 |
| prime radical | #89732 | C2 |
| hypercenter | #399 | D7 |
| hypocenter | C7 | #2826 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#399 | C2 | semidirect |
| 2. #89766 | C2 | semidirect |
| 3. #89763 | C2 | semidirect |
| 4. #89758 | C2 | semidirect |
| 5. #89733 | C2 | semidirect |
| 7. #89730 | C2 | semidirect |
| 8. #89718 | C2 | semidirect |
| 9. #89447 | C2 | semidirect |
| 61. #399 | D7 | semidirect |
| 219. C7 | #2826 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#399 | 2 |
| #89766 | 2 |
| #89763 | 2 |
| #89758 | 2 |
| #89733 | 2 |
| #89732 | 2 |
| #89730 | 2 |
| #89718 | 2 |
| #89447 | 2 |
| #2826 | 7 |