Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
958 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D53 |
| hypocenter | C53 | #3568 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811373 | C2 | semidirect |
| 2. #16811342 | C2 | semidirect |
| 3. #16811304 | C2 | semidirect |
| 4. #16811237 | C2 | semidirect |
| 5. #16811208 | C2 | semidirect |
| 6. #16811152 | C2 | semidirect |
| 7. C53×H(3,𝔽4) | C2 | semidirect |
| 8. #13133215 | C22 | semidirect |
| 9. #13133211 | C22 | semidirect |
| 10. #13133210 | C22 | semidirect |
| 11. #13133210 | C22 | semidirect |
| 12. #13133210 | C22 | semidirect |
| 13. #13133210 | C22 | semidirect |
| 14. #13133209 | C22 | semidirect |
| 16. #13133203 | C22 | semidirect |
| 17. #13133199 | C22 | semidirect |
| 20. #13133194 | C22 | semidirect |
| 21. #13133194 | C22 | semidirect |
| 22. #13133194 | C22 | semidirect |
| 23. #13133193 | C22 | semidirect |
| 26. #13133186 | C22 | semidirect |
| 27. Q8×Dic53 | C22 | semidirect |
| 28. C4×Dic106 | C22 | semidirect |
| 31. #13133177 | C22 | semidirect |
| 32. #13133173 | C22 | semidirect |
| 33. #13133173 | C22 | semidirect |
| 34. C2×C22⋊Dic53 | C22 | semidirect |
| 35. C42×D53 | C22 | semidirect |
| 102. H(3,𝔽4) | D53 | semidirect |
| 108. C53 | #3568 | semidirect |