Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1224 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 249 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×#379 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #379 | D53 |
| hypocenter | C53 | C4×#120 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Dic53 | C2 | semidirect |
| 3. C2×#13133177 | C2 | semidirect |
| 4. #16810500 | C2 | semidirect |
| 5. #16810495 | C2 | semidirect |
| 6. C22⋊C4×Dic53 | C2 | semidirect |
| 7. C53×#379 | C2 | semidirect |
| 12. C22×C4×D53 | C22 | semidirect |
| 13. #13133186 | C4 | semidirect |
| 15. C2×#11719919 | C22 | semidirect |
| 17. C2×#11719921 | C22 | semidirect |
| 18. C2×C4×Dic53 | C22 | semidirect |
| 19. #13133124 | C22 | semidirect |
| 25. C24×C53 | C2×C4 | semidirect |
| 32. C2×C22⋊D53 | C2×C4 | semidirect |
| 37. C22×Dic53 | D4 | semidirect |
| 44. C22⋊Dic53 | C2×C4 | semidirect |
| 59. C22×D53 | C22×C4 | semidirect |
| 69. C2×Dic53 | C22×C4 | semidirect |
| 78. C22×C53 | C4×D4 | semidirect |
| 89. Dic53 | #120 | semidirect |
| 101. #379 | D53 | semidirect |
| 103. C53 | C4×#120 | semidirect |
| 122. C24 | C4×D53 | semidirect |
| 177. C22 | #13133177 | semidirect |