Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
484 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 175 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 | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#441 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #441 | D53 |
| hypocenter | C53 | #2952 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #441 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811105 | C2 | semidirect |
| 3. #16810736 | C2 | semidirect |
| 4. #16810734 | C2 | semidirect |
| 5. #16810730 | C2 | semidirect |
| 6. C4×#11719924 | C2 | semidirect |
| 7. #16810708 | C2 | semidirect |
| 8. #106×D53 | C2 | semidirect |
| 9. #16810623 | C2 | semidirect |
| 10. #16810580 | C2 | semidirect |
| 11. C53×#441 | C2 | semidirect |
| 13. #13133240 | C4 | semidirect |
| 42. #11719949 | C2×C4 | semidirect |
| 50. Q16×C53 | C2×C4 | semidirect |
| 57. #11719924 | C2×C4 | semidirect |
| 69. D424 | C2×C4 | semidirect |
| 71. C8×D53 | C2×C4 | semidirect |
| 120. #441 | D53 | semidirect |
| 126. C53 | #2952 | semidirect |
| 165. Q16 | C4×D53 | semidirect |