Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#493 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #493 | D53 |
| hypocenter | C53 | D4×D8 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D8×D53 | C2 | semidirect |
| 2. #16811258 | C2 | semidirect |
| 3. #16810856 | C2 | semidirect |
| 4. #16810815 | C2 | semidirect |
| 5. #16810755 | C2 | semidirect |
| 6. C4×D424 | C2 | semidirect |
| 7. C8×D212 | C2 | semidirect |
| 8. C53×#493 | C2 | semidirect |
| 10. C2×D4×D53 | C22 | semidirect |
| 12. C2×#11719922 | C22 | semidirect |
| 14. C2×C8×D53 | C22 | semidirect |
| 16. #13133188 | C22 | semidirect |
| 19. C4×D212 | C22 | semidirect |
| 20. #13133151 | C22 | semidirect |
| 21. C4⋊#88322 | C22 | semidirect |
| 22. #13133139 | C22 | semidirect |
| 23. #13133138 | C22 | semidirect |
| 24. #13133129 | C22 | semidirect |
| 25. C2×D8×C53 | C22 | semidirect |
| 26. C2×D8×C53 | C22 | semidirect |
| 27. #127×C53 | C22 | semidirect |
| 28. C4×C8×C53 | C22 | semidirect |
| 54. D424 | D4 | semidirect |
| 74. D212 | D8 | semidirect |
| 75. D212 | C2×D4 | semidirect |
| 80. C8×C53 | C2×D4 | semidirect |
| 92. C2×D53 | C2×D8 | semidirect |
| 94. C4×C53 | C2×D8 | semidirect |
| 108. #493 | D53 | semidirect |
| 114. C53 | D4×D8 | semidirect |
| 120. C2×D8 | C2×D53 | semidirect |
| 121. C2×D8 | C2×D53 | semidirect |
| 123. #127 | C2×D53 | semidirect |
| 134. C4×C8 | C2×D53 | semidirect |
| 187. C8 | D4×D53 | semidirect |
| 202. C4 | D8×D53 | semidirect |