Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 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×#124×C53 | C2 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | #2436 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D53 | C2 | semidirect |
| 2. C2×#13133204 | C2 | semidirect |
| 4. #16810535 | C2 | semidirect |
| 5. #16810530 | C2 | semidirect |
| 6. #16810529 | C2 | semidirect |
| 7. #16810499 | C2 | semidirect |
| 8. C4×#11719921 | C2 | semidirect |
| 9. C2×#124×C53 | C2 | semidirect |
| 37. C2×Q8×C53 | D4 | semidirect |
| 62. C2×Dic106 | D4 | semidirect |
| 122. C2×D53 | #124 | semidirect |
| 140. C2×#124 | D53 | semidirect |
| 147. C53 | #2436 | semidirect |
| 182. C2×Q8 | C22⋊D53 | semidirect |