Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
656 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#492 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #492 | D53 |
| hypocenter | C53 | C2×#492 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #492 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SD16×D53 | C2 | semidirect |
| 2. C4⋊Q8×D53 | C2 | semidirect |
| 4. #16810997 | C2 | semidirect |
| 5. #16810996 | C2 | semidirect |
| 6. #16810995 | C2 | semidirect |
| 7. #16810994 | C2 | semidirect |
| 8. C4×C8×D53 | C2 | semidirect |
| 9. C53×#492 | C2 | semidirect |
| 13. C2×#11719913 | C22 | semidirect |
| 14. C2×#11719923 | C22 | semidirect |
| 15. C2×#11719924 | C22 | semidirect |
| 18. #13133217 | C22 | semidirect |
| 20. #13133215 | C22 | semidirect |
| 26. C4×#88322 | C22 | semidirect |
| 27. C8×Dic53 | C22 | semidirect |
| 28. C2×SD16×C53 | C22 | semidirect |
| 29. C4⋊Q8×C53 | C22 | semidirect |
| 31. C4×C8×C53 | C22 | semidirect |
| 60. C8×D53 | D4 | semidirect |
| 80. C4×D53 | SD16 | semidirect |
| 84. C8×C53 | C2×D4 | semidirect |
| 85. #88322 | C2×D4 | semidirect |
| 95. C4×C53 | C2×SD16 | semidirect |
| 99. Dic53 | C2×SD16 | semidirect |
| 105. D53 | #492 | direct (N × H) |
| 113. #492 | D53 | direct (N × H) |
| 115. C53 | C2×#492 | semidirect |
| 123. C2×SD16 | C2×D53 | semidirect |
| 128. C4⋊Q8 | C2×D53 | semidirect |
| 137. C4×C8 | C2×D53 | semidirect |
| 191. C8 | D4×D53 | semidirect |
| 204. C4 | SD16×D53 | semidirect |