Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 582 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 | C22×C42×C7 | C2 |
| prime radical | C22×C22⋊D7 | C22 |
| hypercenter | C22×C42 | D7 |
| hypocenter | C7 | #2852 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42×C7 | C2 | semidirect |
| 2. C2×#6439 | C2 | semidirect |
| 4. #89982 | C2 | semidirect |
| 5. #89712 | C2 | semidirect |
| 6. #89709 | C2 | semidirect |
| 7. #89705 | C2 | semidirect |
| 36. C42×C7 | D4 | semidirect |
| 43. C22×C42 | D7 | semidirect |
| 88. C22×C7 | #126 | semidirect |
| 117. C42 | C22⋊D7 | semidirect |
| 145. C7 | #2852 | semidirect |
| 151. C22 | #6439 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C7 | 2 |
| C2×#6439 | 2 |
| #89985 | 2 |
| #89982 | 2 |
| #89712 | 2 |
| #89709 | 2 |
| #89705 | 2 |
| #2852 | 7 |