Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
3422 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 3422 conjugacy classes of subgroups — too many to draw. Only its 245 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 | #2846 |
| 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. C25×D53 | C2 | semidirect |
| 2. C2×#13133185 | C2 | semidirect |
| 3. #16810479 | C2 | semidirect |
| 4. #16810478 | C2 | semidirect |
| 5. C2×#120×C53 | C2 | semidirect |
| 9. C22×C22⋊D53 | C22 | semidirect |
| 11. C2×C22⋊Dic53 | C22 | semidirect |
| 12. C2×#11719921 | C22 | semidirect |
| 14. C22×D4×C53 | C22 | semidirect |
| 16. C2×C22⋊C4×C53 | C22 | semidirect |
| 18. C24×C53 | D4 | semidirect |
| 20. C24×C53 | C23 | semidirect |
| 26. C23×D53 | D4 | semidirect |
| 30. C22×C4×C53 | C23 | semidirect |
| 32. C22×Dic53 | C23 | semidirect |
| 62. C22×C53 | #120 | semidirect |
| 68. C2×D53 | #120 | semidirect |
| 79. C2×#120 | D53 | semidirect |
| 82. C53 | #2846 | semidirect |
| 88. C22×D4 | C2×D53 | semidirect |
| 93. C2×C22⋊C4 | C2×D53 | semidirect |
| 102. C24 | C22⋊D53 | semidirect |
| 104. C24 | C22×D53 | semidirect |
| 109. C22×C4 | C22×D53 | semidirect |
| 143. C22 | #13133185 | semidirect |