Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×#519 | C2 |
| hypercenter | #519 | D53 |
| hypocenter | C53 | #3486 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133265 | C2 | semidirect |
| 2. C22×Q8×D53 | C2 | semidirect |
| 3. Q8×D212 | C2 | semidirect |
| 4. #16811285 | C2 | semidirect |
| 5. #16811199 | C2 | semidirect |
| 6. #16811135 | C2 | semidirect |
| 7. C53×#519 | C2 | semidirect |
| 9. C2×#11719949 | C22 | semidirect |
| 14. C2×Q8×D53 | C22 | semidirect |
| 16. #13133217 | C22 | semidirect |
| 17. #13133205 | C22 | semidirect |
| 20. #13133197 | C22 | semidirect |
| 25. C4×Q8×C53 | C22 | semidirect |
| 28. C2×Q8×C53 | D4 | semidirect |
| 78. C2×D53 | 2−1+4 | semidirect |
| 95. #519 | D53 | semidirect |
| 96. C53 | #3486 | semidirect |
| 113. C4×Q8 | C2×D53 | semidirect |
| 124. C2×Q8 | D212 | semidirect |