Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 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×#120×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | C4×#120 |
| LCS | G ▷ C22×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×D4×Dic53 | C2 | semidirect |
| 3. #16810503 | C2 | semidirect |
| 4. #16810498 | C2 | semidirect |
| 5. #16810479 | C2 | semidirect |
| 6. C22⋊C4×Dic53 | C2 | semidirect |
| 11. C23×Dic53 | C22 | semidirect |
| 13. C2×C4⋊Dic53 | C22 | semidirect |
| 16. C2×C22⋊Dic53 | C22 | semidirect |
| 17. C2×C4×Dic53 | C22 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 21. #120×C53 | C4 | semidirect |
| 25. C24×C53 | C2×C4 | semidirect |
| 30. C2×D4×C53 | C2×C4 | semidirect |
| 37. C22×Dic53 | D4 | semidirect |
| 42. C22⋊C4×C53 | C2×C4 | semidirect |
| 56. C23×C53 | C22×C4 | semidirect |
| 63. C2×C4×C53 | C22×C4 | semidirect |
| 81. C22×C53 | C4×D4 | semidirect |
| 89. Dic53 | #120 | direct (N × H) |
| 103. C53 | C4×#120 | semidirect |
| 110. #120 | Dic53 | direct (N × H) |
| 121. C24 | C2×Dic53 | semidirect |
| 126. C2×D4 | C2×Dic53 | semidirect |
| 142. C22⋊C4 | C2×Dic53 | semidirect |
| 154. C23 | C22×Dic53 | semidirect |
| 168. C2×C4 | C22×Dic53 | semidirect |
| 177. C22 | D4×Dic53 | semidirect |