Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
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×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C22×C42×C53 | C2 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | #2852 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133211 | C2 | semidirect |
| 2. #16810514 | C2 | semidirect |
| 3. #16810507 | C2 | semidirect |
| 5. #16810496 | C2 | semidirect |
| 6. #16810491 | C2 | semidirect |
| 7. C22×C42×C53 | C2 | semidirect |
| 35. C42×C53 | D4 | semidirect |
| 65. C22×C53 | #126 | semidirect |
| 84. C22×C42 | D53 | semidirect |
| 90. C53 | #2852 | semidirect |
| 123. C42 | C22⋊D53 | semidirect |
| 151. C22 | #13133211 | semidirect |