Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C22×C42×C53 | C2 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | #3444 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811180 | C2 | semidirect |
| 2. #16811173 | C2 | semidirect |
| 3. #16811157 | C2 | semidirect |
| 4. #16811134 | C2 | semidirect |
| 5. C4×#11719945 | C2 | semidirect |
| 6. C22×C42×C53 | C2 | semidirect |
| 8. #13133217 | C22 | semidirect |
| 9. #13133214 | C22 | semidirect |
| 10. #13133211 | C22 | semidirect |
| 11. #13133209 | C22 | semidirect |
| 12. #13133205 | C22 | semidirect |
| 17. C4×Dic106 | C22 | semidirect |
| 18. C4×D212 | C22 | semidirect |
| 21. #13133174 | C22 | semidirect |
| 22. C42×D53 | C22 | semidirect |
| 55. C2×C4×C53 | #41 | semidirect |
| 82. C22×C42 | D53 | semidirect |
| 83. C53 | #3444 | semidirect |
| 137. C2×C4 | #11719945 | semidirect |