Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
552 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 221 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 | C22×C53 | C23×C4 |
| Fitting | C2×#123×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2374 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810516 | C2 | semidirect |
| 3. #16810516 | C2 | semidirect |
| 4. #16810509 | C2 | semidirect |
| 6. C4⋊C4×Dic53 | C2 | semidirect |
| 7. C22⋊C4×Dic53 | C2 | semidirect |
| 8. C4×C22⋊Dic53 | C2 | semidirect |
| 28. #123×C53 | C4 | semidirect |
| 39. C22×C4×C53 | C2×C4 | semidirect |
| 49. C4⋊C4×C53 | C2×C4 | semidirect |
| 51. C22⋊C4×C53 | C2×C4 | semidirect |
| 136. C53 | #2374 | semidirect |
| 140. #123 | Dic53 | semidirect |
| 172. C22×C4 | C2×Dic53 | semidirect |
| 178. C4⋊C4 | C2×Dic53 | semidirect |
| 184. C22⋊C4 | C2×Dic53 | semidirect |