Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
656 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#493 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #493 | D53 |
| hypocenter | C53 | #3151 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133241 | C2 | semidirect |
| 2. #16811171 | C2 | semidirect |
| 3. #127×D53 | C2 | semidirect |
| 4. #16811011 | C2 | semidirect |
| 5. #16810999 | C2 | semidirect |
| 6. #16810996 | C2 | semidirect |
| 7. #16810995 | C2 | semidirect |
| 8. C4×#11719912 | C2 | semidirect |
| 9. C53×#493 | C2 | semidirect |
| 14. C2×#11719913 | C22 | semidirect |
| 15. C2×#11719923 | C22 | semidirect |
| 16. C2×#11719922 | C22 | semidirect |
| 18. #13133217 | C22 | semidirect |
| 22. #13133204 | C22 | semidirect |
| 26. #13133127 | C22 | semidirect |
| 27. C4×#88322 | C22 | semidirect |
| 28. C2×D8×C53 | C22 | semidirect |
| 30. #127×C53 | C22 | semidirect |
| 31. C4×C8×C53 | C22 | semidirect |
| 60. #11719912 | D4 | semidirect |
| 85. C8×C53 | C2×D4 | semidirect |
| 86. #88322 | C2×D4 | semidirect |
| 97. C4×C53 | AGL(1,ℤ/8) | semidirect |
| 114. #493 | D53 | semidirect |
| 118. C53 | #3151 | semidirect |
| 126. C2×D8 | C2×D53 | semidirect |
| 133. #127 | C2×D53 | semidirect |
| 140. C4×C8 | C2×D53 | semidirect |
| 195. C8 | D4×D53 | semidirect |
| 209. C4 | #13133241 | semidirect |