Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1102 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 197 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×#120×C53 | C2 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | #2849 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Dic53 | C2 | semidirect |
| 2. C2×#13133198 | C2 | semidirect |
| 3. #16810503 | C2 | semidirect |
| 4. #16810495 | C2 | semidirect |
| 5. #16810484 | C2 | semidirect |
| 6. #16810479 | C2 | semidirect |
| 13. C2×#11719919 | C22 | semidirect |
| 16. C2×C22⋊Dic53 | C22 | semidirect |
| 17. #13133124 | C22 | semidirect |
| 41. C22×Dic53 | D4 | semidirect |
| 87. C22×C53 | #123 | semidirect |
| 109. C53 | #2849 | semidirect |
| 189. C22 | #13133198 | semidirect |