Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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 | C2×#120×C53 | C2 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | #2847 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Dic53 | C2 | semidirect |
| 2. C2×#13133189 | C2 | semidirect |
| 3. C2×#13133186 | C2 | semidirect |
| 4. #16810503 | C2 | semidirect |
| 5. #16810500 | C2 | semidirect |
| 6. #16810481 | C2 | semidirect |
| 7. C2×#120×C53 | C2 | semidirect |
| 9. C22×C22⋊D53 | C22 | semidirect |
| 15. C2×#11719919 | C22 | semidirect |
| 17. C2×C22⋊Dic53 | C22 | semidirect |
| 18. C2×#11719921 | C22 | semidirect |
| 19. #13133124 | C22 | semidirect |
| 26. C24×C53 | D4 | semidirect |
| 42. C22×Dic53 | D4 | semidirect |
| 88. C22×C53 | #121 | semidirect |
| 98. Dic53 | #120 | semidirect |
| 109. C2×#120 | D53 | semidirect |
| 113. C53 | #2847 | semidirect |
| 133. C24 | C22⋊D53 | semidirect |
| 197. C22 | #13133189 | semidirect |