Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C23×Q8×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C23×Q8 | D53 |
| hypocenter | C53 | C22×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133206 | C2 | direct (N × H) |
| 3. C22×#11719921 | C2 | semidirect |
| 4. C22×C4×Dic53 | C2 | semidirect |
| 5. C23×Q8×C53 | C2 | semidirect |
| 10. #13133206 | C22 | direct (N × H) |
| 11. C2×#11719921 | C22 | semidirect |
| 12. C2×C4×Dic53 | C22 | semidirect |
| 14. C22×Q8×C53 | C22 | semidirect |
| 22. C2×Q8×C53 | C23 | semidirect |
| 31. #11719921 | C23 | semidirect |
| 32. C4×Dic53 | C23 | semidirect |
| 48. C22×C53 | #124 | semidirect |
| 59. C2×C53 | C2×#124 | semidirect |
| 63. C23×Q8 | D53 | semidirect |
| 68. C53 | C22×#124 | semidirect |
| 71. C22×Q8 | C2×D53 | semidirect |
| 85. C2×Q8 | C22×D53 | semidirect |
| 114. C22 | #13133206 | direct (N × H) |
| 122. C2 | C2×#13133206 | direct (N × H) |