Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
924 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 210 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 | #127×C53 | C4 |
| hypercenter | #127 | #6157 |
| hypocenter | C53 | C4×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×#6157 | C2 | semidirect |
| 2. #16811575 | C2 | semidirect |
| 3. #16811569 | C2 | semidirect |
| 4. C42×#6157 | C2 | semidirect |
| 13. #13133214 | C4 | semidirect |
| 14. #13133213 | C4 | semidirect |
| 16. #127×C53 | C4 | semidirect |
| 22. C2×D4×C53 | C2×C4 | semidirect |
| 23. C2×C22⋊D53 | C2×C4 | semidirect |
| 26. C2×D212 | C2×C4 | semidirect |
| 31. C4×#6157 | D4 | semidirect |
| 32. C42×C53 | C2×C4 | semidirect |
| 33. C4×Dic53 | C2×C4 | semidirect |
| 60. #6157 | #127 | direct (N × H) |
| 61. C4×C53 | C4×D4 | semidirect |
| 62. Dic53 | C4×D4 | semidirect |
| 74. C53 | C4×#127 | semidirect |
| 79. #127 | #6157 | direct (N × H) |
| 91. C2×D4 | C2×#6157 | semidirect |
| 102. C42 | C2×#6157 | semidirect |
| 131. C4 | D4×#6157 | semidirect |