Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C53×#390 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #390 | D53 |
| hypocenter | C53 | C4×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×D4×Dic53 | C2 | semidirect |
| 3. C2×#13133179 | C2 | semidirect |
| 4. C2×#13133177 | C2 | semidirect |
| 5. #16810525 | C2 | semidirect |
| 6. #16810523 | C2 | semidirect |
| 7. #16810492 | C2 | semidirect |
| 8. C42×Dic53 | C2 | semidirect |
| 9. C53×#390 | C2 | semidirect |
| 14. #13133213 | C4 | semidirect |
| 35. C2×D4×C53 | C2×C4 | semidirect |
| 41. C2×C22⋊D53 | C2×C4 | semidirect |
| 50. C2×D212 | C2×C4 | semidirect |
| 62. C4×Dic53 | C2×C4 | semidirect |
| 63. C4×Dic53 | D4 | semidirect |
| 64. C4×Dic53 | D4 | semidirect |
| 115. C4×C53 | C4×D4 | semidirect |
| 118. Dic53 | C4×D4 | semidirect |
| 119. Dic53 | #127 | semidirect |
| 134. #390 | D53 | semidirect |
| 137. C53 | C4×#127 | semidirect |
| 169. C2×D4 | C4×D53 | semidirect |
| 247. C4 | #13133177 | semidirect |