Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
652 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 182 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 | #124×C53 | C4 |
| hypercenter | #124 | #6157 |
| hypocenter | C53 | C4×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×#6157 | C2 | semidirect |
| 3. (C2×#6157)⋊Q8 | C2 | semidirect |
| 5. #16811573 | C2 | semidirect |
| 6. #16811567 | C2 | semidirect |
| 7. C22⋊C4×#6157 | C2 | semidirect |
| 8. C42×#6157 | C2 | semidirect |
| 27. #13133206 | C4 | semidirect |
| 28. #13133205 | C4 | semidirect |
| 29. #13133204 | C4 | semidirect |
| 30. #13133203 | C4 | semidirect |
| 33. #124×C53 | C4 | semidirect |
| 42. C2×Q8×C53 | C2×C4 | semidirect |
| 53. C2×Dic106 | C2×C4 | semidirect |
| 54. C2×Dic106 | C2×C4 | semidirect |
| 65. C22⋊C4×C53 | C2×C4 | semidirect |
| 66. C42×C53 | C2×C4 | semidirect |
| 67. C22⋊Dic53 | C2×C4 | semidirect |
| 68. #11719921 | C2×C4 | semidirect |
| 69. #11719921 | C2×C4 | semidirect |
| 70. C4×Dic53 | C2×C4 | semidirect |
| 71. C4×Dic53 | C2×C4 | semidirect |
| 125. #6157 | #124 | direct (N × H) |
| 149. C53 | C4×#124 | semidirect |
| 157. #124 | #6157 | direct (N × H) |
| 179. C2×Q8 | C2×#6157 | semidirect |
| 203. C22⋊C4 | C2×#6157 | semidirect |
| 211. C42 | C2×#6157 | semidirect |