Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#424 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #424 | D53 |
| hypocenter | C53 | #3041 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #424 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133245 | C2 | semidirect |
| 2. C22×D424 | C2 | semidirect |
| 3. #16811192 | C2 | semidirect |
| 4. #16811097 | C2 | semidirect |
| 5. #16810839 | C2 | semidirect |
| 6. #16810819 | C2 | semidirect |
| 7. #16810629 | C2 | semidirect |
| 8. #16810620 | C2 | semidirect |
| 9. C53×#424 | C2 | semidirect |
| 14. C2×#11719913 | C22 | semidirect |
| 17. C2×D424 | C22 | semidirect |
| 18. #13133214 | C22 | semidirect |
| 19. #13133205 | C22 | semidirect |
| 22. C4×D212 | C22 | semidirect |
| 26. #13133142 | C22 | semidirect |
| 27. #13133139 | C22 | semidirect |
| 28. #13133139 | C22 | semidirect |
| 32. C4⋊C8×C53 | C22 | semidirect |
| 45. C2×D212 | D4 | semidirect |
| 128. #424 | D53 | semidirect |
| 131. C53 | #3041 | semidirect |
| 151. C4⋊C8 | C2×D53 | semidirect |