Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#380 | C2 |
| hypercenter | #380 | D53 |
| hypocenter | C53 | #2408 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D212 | C2 | semidirect |
| 2. C2×#13133192 | C2 | semidirect |
| 3. C2×#13133187 | C2 | semidirect |
| 4. C2×C4×C22⋊D53 | C2 | semidirect |
| 5. #16810541 | C2 | semidirect |
| 6. #16810533 | C2 | semidirect |
| 7. #16810524 | C2 | semidirect |
| 8. #16810478 | C2 | semidirect |
| 9. C53×#380 | C2 | semidirect |
| 15. C22×D212 | C22 | semidirect |
| 17. C22×C4×D53 | C22 | semidirect |
| 21. C2×#11719919 | C22 | semidirect |
| 23. C2×#11719921 | C22 | semidirect |
| 24. C2×#11719921 | C22 | semidirect |
| 25. C2×#11719921 | C22 | semidirect |
| 26. C2×#11719921 | C22 | semidirect |
| 27. C2×C4×Dic53 | C22 | semidirect |
| 29. C2×C4⋊C4×C53 | C22 | semidirect |
| 31. #95×C53 | C22 | semidirect |
| 41. C22×C4×C53 | D4 | semidirect |
| 47. C2×C22⋊D53 | D4 | semidirect |
| 59. C2×D212 | D4 | semidirect |
| 93. C22×D53 | C2×D4 | semidirect |
| 109. C2×Dic53 | C2×D4 | semidirect |
| 118. C22×C53 | #121 | semidirect |
| 139. C4×C53 | #120 | semidirect |
| 159. #380 | D53 | semidirect |
| 161. C53 | #2408 | semidirect |
| 175. C2×C4⋊C4 | C2×D53 | semidirect |
| 183. #95 | C2×D53 | semidirect |
| 210. C22×C4 | D212 | semidirect |
| 283. C22 | #13133192 | semidirect |
| 294. C4 | #13133187 | semidirect |