Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
346 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810499 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#410 | C2 |
| socle | C2×C53 | #386 |
| hypercenter | #410 | D53 |
| hypocenter | C53 | #1904 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. #16811097 | C2 | semidirect |
| 3. #16810640 | C2 | semidirect |
| 4. #16810627 | C2 | semidirect |
| 5. #16810393 | C2 | semidirect |
| 6. #16810379 | C2 | semidirect |
| 7. C53×#410 | C2 | semidirect |
| 11. C2×#11719924 | C4 | semidirect |
| 12. C2×#11719922 | C4 | semidirect |
| 30. C2×Q8×C53 | C2×C4 | semidirect |
| 32. C2×D4×C53 | C2×C4 | semidirect |
| 59. C2×#88322 | C2×C4 | semidirect |
| 120. #410 | D53 | semidirect |
| 123. C53 | #1904 | semidirect |
| 146. C2×Q8 | C4×D53 | semidirect |
| 154. C2×D4 | C4×D53 | semidirect |