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×#120×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | C2×H(3,𝔽4) |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811366 | C2 | direct (N × H) |
| 2. C2×#13133204 | C2 | semidirect |
| 3. C2×#13133203 | C2 | semidirect |
| 4. C2×#13133186 | C2 | semidirect |
| 5. C2×#13133185 | C2 | semidirect |
| 6. C2×#120×C53 | C2 | semidirect |
| 7. C24×D53 | C22 | semidirect |
| 8. C22×Dic106 | C22 | semidirect |
| 10. #13133204 | C22 | semidirect |
| 11. #13133203 | C22 | semidirect |
| 12. #13133186 | C22 | semidirect |
| 13. #13133185 | C22 | semidirect |
| 14. C2×C22⋊Dic53 | C22 | semidirect |
| 15. C2×C22⋊Dic53 | C22 | semidirect |
| 16. C2×#11719921 | C22 | semidirect |
| 17. C2×C4×Dic53 | C22 | semidirect |
| 18. C25×C53 | C22 | semidirect |
| 20. #120×C53 | C22 | semidirect |
| 21. C2×C22⋊C4×C53 | C22 | semidirect |
| 25. C24×C53 | C23 | semidirect |
| 28. C23×D53 | C23 | semidirect |
| 36. C2×Dic106 | C23 | semidirect |
| 37. C22⋊C4×C53 | C23 | semidirect |
| 38. C22⋊Dic53 | C23 | semidirect |
| 39. C22⋊Dic53 | C23 | semidirect |
| 40. #11719921 | C23 | semidirect |
| 41. C4×Dic53 | C23 | semidirect |
| 77. C2×C53 | H(3,𝔽4) | semidirect |
| 85. C2×#120 | D53 | semidirect |
| 86. C53 | C2×H(3,𝔽4) | semidirect |
| 88. C25 | C2×D53 | semidirect |
| 96. #120 | C2×D53 | semidirect |
| 100. C2×C22⋊C4 | C2×D53 | semidirect |
| 108. C24 | C22×D53 | semidirect |
| 120. C22⋊C4 | C22×D53 | semidirect |
| 157. C2 | #16811366 | direct (N × H) |