Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#124×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | C2×H(3,𝔽4) |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811365 | C2 | direct (N × H) |
| 2. C2×#13133206 | C2 | semidirect |
| 3. C2×#13133205 | C2 | semidirect |
| 4. C2×#13133203 | C2 | semidirect |
| 5. C2×#13133187 | C2 | semidirect |
| 6. C2×#13133185 | C2 | semidirect |
| 7. C2×#124×C53 | C2 | semidirect |
| 8. C24×D53 | C22 | semidirect |
| 9. C22×Dic106 | C22 | semidirect |
| 13. #13133206 | C22 | semidirect |
| 14. #13133205 | C22 | semidirect |
| 15. #13133203 | C22 | semidirect |
| 16. #13133187 | C22 | semidirect |
| 17. #13133185 | C22 | semidirect |
| 18. C2×C22⋊Dic53 | C22 | semidirect |
| 19. C2×#11719921 | C22 | semidirect |
| 20. C2×#11719921 | C22 | semidirect |
| 21. C2×#11719921 | C22 | semidirect |
| 22. C2×C4×Dic53 | C22 | semidirect |
| 23. C22×Q8×C53 | C22 | semidirect |
| 25. #124×C53 | C22 | semidirect |
| 26. C2×C22⋊C4×C53 | C22 | semidirect |
| 27. C2×C42×C53 | C22 | semidirect |
| 32. C23×D53 | C23 | semidirect |
| 34. C2×Q8×C53 | C23 | semidirect |
| 48. C2×Dic106 | C23 | semidirect |
| 49. C22⋊C4×C53 | C23 | semidirect |
| 50. C42×C53 | C23 | semidirect |
| 51. C22⋊Dic53 | C23 | semidirect |
| 52. #11719921 | C23 | semidirect |
| 53. #11719921 | C23 | semidirect |
| 54. #11719921 | C23 | semidirect |
| 55. C4×Dic53 | C23 | semidirect |
| 98. C2×C53 | H(3,𝔽4) | semidirect |
| 106. C2×#124 | D53 | semidirect |
| 109. C53 | C2×H(3,𝔽4) | semidirect |
| 112. C22×Q8 | C2×D53 | semidirect |
| 120. #124 | C2×D53 | semidirect |
| 127. C2×C22⋊C4 | C2×D53 | semidirect |
| 129. C2×C42 | C2×D53 | semidirect |
| 138. C2×Q8 | C22×D53 | semidirect |
| 155. C22⋊C4 | C22×D53 | semidirect |
| 157. C42 | C22×D53 | semidirect |
| 200. C2 | #16811365 | direct (N × H) |