Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
928 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 928 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C22×Q8×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C22×Q8 | #6157 |
| hypocenter | C53 | C2×#391 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. (C2×#6157)⋊Q8 | C2 | direct (N × H) |
| 13. C22×Dic106 | C4 | semidirect |
| 14. C22×Dic106 | C4 | semidirect |
| 17. C2×Q8×D53 | C4 | semidirect |
| 21. C22×Q8×C53 | C4 | semidirect |
| 27. C22×#6157 | Q8 | semidirect |
| 31. C2×Q8×C53 | C2×C4 | semidirect |
| 49. C2×Dic106 | C2×C4 | semidirect |
| 50. C2×Dic106 | C2×C4 | semidirect |
| 66. C2×#6157 | C2×Q8 | semidirect |
| 103. C2×C53 | #391 | semidirect |
| 106. D53 | #391 | semidirect |
| 115. C53 | C2×#391 | semidirect |
| 122. C22×Q8 | #6157 | semidirect |
| 141. C2×Q8 | C2×#6157 | semidirect |
| 216. C2 | (C2×#6157)⋊Q8 | direct (N × H) |