Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
604 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 604 conjugacy classes of subgroups — too many to draw. Only its 133 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×#397 | C2 |
| prime radical | #89732 | C2 |
| hypercenter | #397 | D7 |
| hypocenter | C7 | #2600 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #397 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×(C22⋊D7)⋊C4 | C2 | semidirect |
| 2. C7×#397 | C2 | semidirect |
| 3. #89761 | C2 | semidirect |
| 5. #89731 | C2 | semidirect |
| 6. #89715 | C2 | semidirect |
| 7. #89441 | C2 | semidirect |
| 8. #89437 | C2 | semidirect |
| 9. #89424 | C2 | semidirect |
| 44. C22×Dic7 | D4 | semidirect |
| 65. #397 | D7 | semidirect |
| 229. C7 | #2600 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×(C22⋊D7)⋊C4 | 2 |
| C7×#397 | 2 |
| #89761 | 2 |
| #89732 | 2 |
| #89731 | 2 |
| #89715 | 2 |
| #89441 | 2 |
| #89437 | 2 |
| #89424 | 2 |
| #2600 | 7 |