Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
532 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 141 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×#431 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #431 | D53 |
| hypocenter | C53 | #3205 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811173 | C2 | semidirect |
| 2. C2×#13133214 | C2 | semidirect |
| 3. #16811000 | C2 | semidirect |
| 4. #16810970 | C2 | semidirect |
| 5. #16810819 | C2 | semidirect |
| 6. #16810759 | C2 | semidirect |
| 7. C53×#431 | C2 | semidirect |
| 12. C2×D424 | C22 | semidirect |
| 13. #13133217 | C22 | semidirect |
| 18. #13133214 | C22 | semidirect |
| 23. C4×D212 | C22 | semidirect |
| 24. #13133139 | C22 | semidirect |
| 26. C4⋊C8×C53 | C22 | semidirect |
| 28. C4×C8×C53 | C22 | semidirect |
| 67. C2×C4×C53 | D8 | semidirect |
| 119. #431 | D53 | semidirect |
| 120. C53 | #3205 | semidirect |
| 138. C4⋊C8 | C2×D53 | semidirect |
| 141. C4×C8 | C2×D53 | semidirect |
| 193. C2×C4 | D424 | semidirect |