Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
588 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C4⋊Q8×C53 | C4 |
| hypercenter | C4⋊Q8 | #6157 |
| hypocenter | C53 | C4×C4⋊Q8 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 19. #13133217 | C4 | semidirect |
| 20. #13133216 | C4 | semidirect |
| 21. #13133215 | C4 | semidirect |
| 24. C4⋊Q8×C53 | C4 | semidirect |
| 44. C4×#6157 | Q8 | semidirect |
| 84. #6157 | C4⋊Q8 | direct (N × H) |
| 87. C4×C53 | C4×Q8 | semidirect |
| 90. Dic53 | C4×Q8 | semidirect |
| 104. C53 | C4×C4⋊Q8 | semidirect |
| 109. C4⋊Q8 | #6157 | direct (N × H) |
| 187. C4 | Q8×#6157 | semidirect |