Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#496 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #496 | D53 |
| hypocenter | C53 | #3153 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #496 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133246 | C2 | semidirect |
| 2. C2×#13133241 | C2 | semidirect |
| 3. #16811171 | C2 | semidirect |
| 4. #124×D53 | C2 | semidirect |
| 5. #16811018 | C2 | semidirect |
| 6. #16811017 | C2 | semidirect |
| 7. #16811010 | C2 | semidirect |
| 8. #16811006 | C2 | semidirect |
| 9. #16811004 | C2 | semidirect |
| 10. #16810575 | C2 | semidirect |
| 11. C53×#496 | C2 | semidirect |
| 17. C2×Dic212 | C22 | semidirect |
| 18. C2×#11719925 | C22 | semidirect |
| 21. C2×#11719923 | C22 | semidirect |
| 24. #13133217 | C22 | semidirect |
| 28. #13133204 | C22 | semidirect |
| 34. #13133128 | C22 | semidirect |
| 35. C8×Dic53 | C22 | semidirect |
| 36. C2×SD16×C53 | C22 | semidirect |
| 38. #127×C53 | C22 | semidirect |
| 40. #97×C53 | C22 | semidirect |
| 79. #11719912 | D4 | semidirect |
| 142. #496 | D53 | semidirect |
| 146. C53 | #3153 | semidirect |
| 157. C2×SD16 | C2×D53 | semidirect |
| 162. #127 | C2×D53 | semidirect |
| 174. #97 | C2×D53 | semidirect |