Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
592 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 145 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×#495 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #495 | D53 |
| hypocenter | C53 | C2×#495 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #495 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q16×D53 | C2 | semidirect |
| 2. C2×SD16×D53 | C2 | semidirect |
| 4. #124×D53 | C2 | semidirect |
| 5. #16811008 | C2 | semidirect |
| 6. #16811007 | C2 | semidirect |
| 7. #16811006 | C2 | semidirect |
| 8. #16811005 | C2 | semidirect |
| 9. #16811004 | C2 | semidirect |
| 10. C4×C8×D53 | C2 | semidirect |
| 11. C53×#495 | C2 | semidirect |
| 14. C2×Dic212 | C22 | semidirect |
| 15. C2×#11719925 | C22 | semidirect |
| 17. C2×#11719913 | C22 | semidirect |
| 18. C2×#11719923 | C22 | semidirect |
| 19. C2×#11719924 | C22 | semidirect |
| 26. #13133206 | C22 | semidirect |
| 27. #13133205 | C22 | semidirect |
| 28. #13133204 | C22 | semidirect |
| 32. C4×#88322 | C22 | semidirect |
| 33. C8×Dic53 | C22 | semidirect |
| 34. C8×Dic53 | C22 | semidirect |
| 35. C2×Q16×C53 | C22 | semidirect |
| 36. C2×SD16×C53 | C22 | semidirect |
| 38. #124×C53 | C22 | semidirect |
| 39. C4×C8×C53 | C22 | semidirect |
| 126. D53 | #495 | direct (N × H) |
| 136. #495 | D53 | direct (N × H) |
| 138. C53 | C2×#495 | semidirect |
| 144. C2×Q16 | C2×D53 | semidirect |
| 149. C2×SD16 | C2×D53 | semidirect |
| 160. #124 | C2×D53 | semidirect |
| 166. C4×C8 | C2×D53 | semidirect |