Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×#6157 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #104×C53 | C4 |
| socle | C2×C53 | C4×C22⋊C4 |
| hypercenter | #104 | #6157 |
| hypocenter | C53 | #1765 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #104 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16811589 | C2 | semidirect |
| 4. #16811582 | C2 | semidirect |
| 5. #16811498 | C2 | semidirect |
| 6. #16811494 | C2 | semidirect |
| 8. #13133305 | C4 | semidirect |
| 10. #13133304 | C4 | semidirect |
| 17. #13133280 | C4 | semidirect |
| 18. #13133279 | C4 | semidirect |
| 25. #13133145 | C4 | semidirect |
| 26. #13133144 | C4 | semidirect |
| 27. #13133143 | C4 | semidirect |
| 28. #104×C53 | C4 | semidirect |
| 128. C53 | #1765 | semidirect |
| 144. #104 | #6157 | semidirect |