Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
496 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 131 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×#460 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #460 | D53 |
| hypocenter | C53 | #3035 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #460 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133238 | C2 | semidirect |
| 2. #16811171 | C2 | semidirect |
| 3. C4×#11719949 | C2 | semidirect |
| 4. #16810843 | C2 | semidirect |
| 5. #16810837 | C2 | semidirect |
| 6. #16810833 | C2 | semidirect |
| 7. #16810829 | C2 | semidirect |
| 8. #16810815 | C2 | semidirect |
| 9. C4⋊C8×D53 | C2 | semidirect |
| 10. #16810616 | C2 | semidirect |
| 11. C53×#460 | C2 | semidirect |
| 51. #11719949 | D4 | semidirect |
| 136. C4×C53 | #135 | semidirect |
| 158. #460 | D53 | semidirect |
| 162. C53 | #3035 | semidirect |
| 292. C4 | #13133238 | semidirect |