Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1840 conjugacy classes of subgroups; only the 685 normal subgroups are shown.
This group has 1840 conjugacy classes of subgroups — too many to draw. Only its 685 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×C22⋊Q8 |
| 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×#13133193 | C2 | direct (N × H) |
| 3. C22×C4⋊Dic53 | C2 | semidirect |
| 4. C22×#11719919 | C2 | semidirect |
| 6. C23×Q8×C53 | C2 | semidirect |
| 12. #13133193 | C22 | direct (N × H) |
| 13. C2×C4⋊Dic53 | C22 | semidirect |
| 14. C2×#11719919 | C22 | semidirect |
| 17. C22×Q8×C53 | C22 | semidirect |
| 23. C23×D53 | Q8 | semidirect |
| 26. C2×Q8×C53 | C23 | semidirect |
| 38. C4⋊Dic53 | C23 | semidirect |
| 39. #11719919 | C23 | semidirect |
| 47. C22×D53 | C2×Q8 | semidirect |
| 59. C22×C53 | C22⋊Q8 | semidirect |
| 67. C2×D53 | C22×Q8 | semidirect |
| 73. C2×C53 | C2×C22⋊Q8 | semidirect |
| 78. C23×Q8 | D53 | semidirect |
| 84. C53 | C22×C22⋊Q8 | semidirect |
| 87. C22×Q8 | C2×D53 | semidirect |
| 104. C2×Q8 | C22×D53 | semidirect |
| 141. C22 | #13133193 | direct (N × H) |
| 151. C2 | C2×#13133193 | direct (N × H) |