Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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 | #3529 |
| 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. #16811367 | C2 | semidirect |
| 3. #16811362 | C2 | semidirect |
| 4. #16811172 | C2 | semidirect |
| 5. #16811152 | C2 | semidirect |
| 6. #16811129 | C2 | semidirect |
| 7. C53×H(3,𝔽4) | C2 | semidirect |
| 8. C2×#11719949 | C22 | semidirect |
| 10. C23×Dic53 | C22 | semidirect |
| 11. #13133215 | C22 | semidirect |
| 13. #13133210 | C22 | semidirect |
| 14. #13133209 | C22 | semidirect |
| 15. #13133205 | C22 | semidirect |
| 16. #13133201 | C22 | semidirect |
| 17. #13133199 | C22 | semidirect |
| 19. #13133194 | C22 | semidirect |
| 20. #13133193 | C22 | semidirect |
| 21. #13133190 | C22 | semidirect |
| 24. #13133186 | C22 | semidirect |
| 25. Q8×Dic53 | C22 | semidirect |
| 27. #13133177 | C22 | semidirect |
| 28. #13133173 | C22 | semidirect |
| 29. C42×D53 | C22 | semidirect |
| 100. H(3,𝔽4) | D53 | semidirect |
| 105. C53 | #3529 | semidirect |