Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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×#121×C53 | C2 |
| hypercenter | C2×#121 | D53 |
| hypocenter | C53 | #2847 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C4×D53 | C2 | semidirect |
| 2. C2×#13133188 | C2 | semidirect |
| 3. C2×#13133185 | C2 | semidirect |
| 4. #16810503 | C2 | semidirect |
| 5. #16810499 | C2 | semidirect |
| 6. #16810483 | C2 | semidirect |
| 7. #16810480 | C2 | semidirect |
| 8. C2×#121×C53 | C2 | semidirect |
| 18. C2×C4⋊Dic53 | C22 | semidirect |
| 20. C2×C22⋊Dic53 | C22 | semidirect |
| 22. #13133124 | C22 | semidirect |
| 23. #13133124 | C22 | semidirect |
| 25. C22×D4×C53 | C22 | semidirect |
| 28. C2×C4⋊C4×C53 | C22 | semidirect |
| 29. C2×C22⋊C4×C53 | C22 | semidirect |
| 36. C23×D53 | D4 | semidirect |
| 44. C22×C4×C53 | D4 | semidirect |
| 61. C2×C4×D53 | D4 | semidirect |
| 120. C22×C53 | #121 | semidirect |
| 133. C2×D53 | #121 | semidirect |
| 136. C4×C53 | #120 | semidirect |
| 150. C2×#121 | D53 | semidirect |
| 156. C53 | #2847 | semidirect |
| 160. C22×D4 | C2×D53 | semidirect |
| 171. C2×C4⋊C4 | C2×D53 | semidirect |
| 175. C2×C22⋊C4 | C2×D53 | semidirect |
| 206. C22×C4 | C22⋊D53 | semidirect |
| 276. C22 | #13133188 | semidirect |
| 286. C4 | #13133185 | semidirect |