Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1408 conjugacy classes of subgroups; only the 459 normal subgroups are shown.
This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 459 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 | C2×H(3,𝔽4) |
| 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. #16811366 | C2 | semidirect |
| 2. #16811365 | C2 | semidirect |
| 3. #124×D53 | C2 | semidirect |
| 4. #120×D53 | C2 | semidirect |
| 5. C53×H(3,𝔽4) | C2 | semidirect |
| 6. C24×D53 | C22 | semidirect |
| 7. C2×Q8×D53 | C22 | semidirect |
| 9. #13133206 | C22 | semidirect |
| 10. #13133205 | C22 | semidirect |
| 11. #13133204 | C22 | semidirect |
| 12. #13133203 | C22 | semidirect |
| 13. #13133187 | C22 | semidirect |
| 14. #13133186 | C22 | semidirect |
| 15. #13133185 | C22 | semidirect |
| 16. C22⋊C4×D53 | C22 | semidirect |
| 17. C42×D53 | C22 | semidirect |
| 18. #124×C53 | C22 | semidirect |
| 19. #120×C53 | C22 | semidirect |
| 20. C24×C53 | C23 | semidirect |
| 23. C23×D53 | C23 | semidirect |
| 24. C2×Q8×C53 | C23 | semidirect |
| 31. C2×Dic106 | C23 | semidirect |
| 32. C22⋊C4×C53 | C23 | semidirect |
| 33. C42×C53 | C23 | semidirect |
| 34. C22⋊Dic53 | C23 | semidirect |
| 35. #11719921 | C23 | semidirect |
| 36. C4×Dic53 | C23 | semidirect |
| 65. D53 | H(3,𝔽4) | direct (N × H) |
| 68. H(3,𝔽4) | D53 | direct (N × H) |
| 71. C53 | C2×H(3,𝔽4) | semidirect |
| 79. #124 | C2×D53 | semidirect |
| 83. #120 | C2×D53 | semidirect |
| 89. C24 | C22×D53 | semidirect |
| 92. C2×Q8 | C22×D53 | semidirect |
| 100. C22⋊C4 | C22×D53 | semidirect |
| 102. C42 | C22×D53 | semidirect |