Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
388 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C8×Q8×C53 | C2 |
| hypercenter | C8×Q8 | D53 |
| hypocenter | C53 | Q8×M16 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810753 | C2 | semidirect |
| 3. #16810750 | C2 | semidirect |
| 4. Q8×#88322 | C2 | semidirect |
| 7. C8×Q8×C53 | C2 | semidirect |
| 42. #11719912 | Q8 | direct (N × H) |
| 44. Q8×C53 | M16 | semidirect |
| 57. C8×C53 | C2×Q8 | semidirect |
| 59. #88322 | C2×Q8 | semidirect |
| 80. C8×Q8 | D53 | semidirect |
| 83. C53 | Q8×M16 | semidirect |
| 121. Q8 | #11719912 | direct (N × H) |
| 135. C8 | Q8×D53 | semidirect |