Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
552 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | C2×#436 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#11719912 | C2 | semidirect |
| 3. #16810702 | C2 | direct (N × H) |
| 4. C2×C4⋊#88322 | C2 | semidirect |
| 5. C2×#13133129 | C2 | semidirect |
| 6. C2×C4×C8×C53 | C2 | semidirect |
| 10. C2×#11719912 | C22 | semidirect |
| 16. C4⋊#88322 | C22 | semidirect |
| 17. #13133129 | C22 | semidirect |
| 21. C4×C8×C53 | C22 | semidirect |
| 34. C2×C8×C53 | D4 | semidirect |
| 50. C2×C4×C53 | M16 | semidirect |
| 64. C8×C53 | C2×D4 | semidirect |
| 76. C4×C53 | C2×M16 | semidirect |
| 81. C2×C53 | #436 | semidirect |
| 91. C2×C4×C8 | D53 | semidirect |
| 92. C53 | C2×#436 | semidirect |
| 107. C4×C8 | C2×D53 | semidirect |
| 123. C2×C8 | D212 | semidirect |
| 142. C2×C4 | #11719912 | semidirect |
| 151. C8 | C2×D212 | semidirect |
| 163. C4 | C2×#11719912 | semidirect |
| 167. C2 | #16810702 | direct (N × H) |