Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
548 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 548 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 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#423 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #423 | D53 |
| hypocenter | C53 | #3036 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #423 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133245 | C2 | semidirect |
| 2. #16811173 | C2 | semidirect |
| 3. C2×C4×D212 | C2 | semidirect |
| 4. #16810839 | C2 | semidirect |
| 5. #16810829 | C2 | semidirect |
| 6. #16810819 | C2 | semidirect |
| 7. #16810625 | C2 | semidirect |
| 8. C53×#423 | C2 | semidirect |
| 15. #13133217 | C22 | semidirect |
| 16. #13133214 | C22 | semidirect |
| 17. #13133205 | C22 | semidirect |
| 28. #13133142 | C22 | semidirect |
| 29. #13133139 | C22 | semidirect |
| 32. C4⋊C8×C53 | C22 | semidirect |
| 33. C4⋊C8×C53 | C22 | semidirect |
| 46. C2×D212 | D4 | semidirect |
| 118. C4×C53 | AGL(1,ℤ/8) | semidirect |
| 136. #423 | D53 | semidirect |
| 138. C53 | #3036 | semidirect |
| 160. C4⋊C8 | C2×D53 | semidirect |
| 161. C4⋊C8 | C2×D53 | semidirect |
| 250. C4 | #13133245 | semidirect |