Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810532 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#410 | C2 |
| socle | C2×C53 | #392 |
| hypercenter | #410 | D53 |
| hypocenter | C53 | #2008 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #410 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811421 | C2 | semidirect |
| 2. C2×#13133242 | C2 | semidirect |
| 3. #16811192 | C2 | semidirect |
| 4. #16810641 | C2 | semidirect |
| 5. #16810627 | C2 | semidirect |
| 6. #16810393 | C2 | semidirect |
| 7. C53×#410 | C2 | semidirect |
| 36. C2×Q8×C53 | D4 | semidirect |
| 38. C2×D4×C53 | D4 | semidirect |
| 72. #11719917 | D4 | semidirect |
| 74. C2×#88322 | D4 | semidirect |
| 148. #410 | D53 | semidirect |
| 150. C53 | #2008 | semidirect |
| 184. C2×Q8 | D212 | semidirect |
| 195. C2×D4 | D212 | semidirect |