Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
516 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 168 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 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #126×C53 | C4 |
| hypercenter | #126 | #6157 |
| hypocenter | C53 | C4×#126 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811572 | C2 | semidirect |
| 3. #16811568 | C2 | semidirect |
| 4. C4⋊C4×#6157 | C2 | semidirect |
| 6. C42×#6157 | C2 | semidirect |
| 18. #13133212 | C4 | semidirect |
| 19. #13133211 | C4 | semidirect |
| 20. #13133210 | C4 | semidirect |
| 24. #126×C53 | C4 | semidirect |
| 38. C4⋊C4×C53 | C2×C4 | semidirect |
| 40. C42×C53 | C2×C4 | semidirect |
| 44. C4⋊Dic53 | C2×C4 | semidirect |
| 45. #11719919 | C2×C4 | semidirect |
| 46. #11719919 | C2×C4 | semidirect |
| 47. C4×Dic53 | C2×C4 | semidirect |
| 80. #6157 | #126 | direct (N × H) |
| 98. C53 | C4×#126 | semidirect |
| 103. #126 | #6157 | direct (N × H) |
| 128. C4⋊C4 | C2×#6157 | semidirect |
| 138. C42 | C2×#6157 | semidirect |