Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
728 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 728 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 | C2×#117×C53 | C2 |
| hypercenter | C2×#117 | D53 |
| hypocenter | C53 | #2619 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133214 | C2 | semidirect |
| 2. C2×#13133191 | C2 | semidirect |
| 3. #16810540 | C2 | semidirect |
| 4. #16810532 | C2 | semidirect |
| 5. #16810525 | C2 | semidirect |
| 6. #16810489 | C2 | semidirect |
| 7. C2×#117×C53 | C2 | semidirect |
| 27. C22×C4×C53 | D4 | semidirect |
| 43. C42×C53 | D4 | semidirect |
| 100. C2×#117 | D53 | semidirect |
| 105. C53 | #2619 | semidirect |
| 133. C22×C4 | D212 | semidirect |
| 144. C42 | C22⋊D53 | semidirect |