Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
656 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 357 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D53 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | C2×C4×M16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C4×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#11719912 | C2 | direct (N × H) |
| 4. C2×#13133127 | C2 | semidirect |
| 5. C2×C4×#88322 | C2 | semidirect |
| 6. C2×C4×C8×C53 | C2 | semidirect |
| 9. C2×#11719912 | C4 | direct (N × H) |
| 16. #13133127 | C22 | semidirect |
| 17. C4×#88322 | C22 | semidirect |
| 20. C4×C8×C53 | C22 | semidirect |
| 31. C2×C8×C53 | C2×C4 | semidirect |
| 38. C2×#88322 | C2×C4 | semidirect |
| 40. #11719912 | C2×C4 | direct (N × H) |
| 45. C2×C4×C53 | M16 | semidirect |
| 57. C8×C53 | C22×C4 | semidirect |
| 58. #88322 | C22×C4 | semidirect |
| 66. C4×C53 | C2×M16 | semidirect |
| 71. C2×C53 | C4×M16 | semidirect |
| 80. C2×C4×C8 | D53 | semidirect |
| 81. C53 | C2×C4×M16 | semidirect |
| 95. C4×C8 | C2×D53 | semidirect |
| 109. C2×C8 | C4×D53 | semidirect |
| 127. C2×C4 | #11719912 | direct (N × H) |
| 133. C8 | C2×C4×D53 | semidirect |
| 143. C4 | C2×#11719912 | direct (N × H) |
| 146. C2 | C4×#11719912 | direct (N × H) |