Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
680 conjugacy classes of subgroups; only the 378 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 378 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C2×#6157 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C22 | C23×#6157 |
| Fitting | C23×C4×C53 | C4 |
| socle | C24×C53 | C2×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | C22×#97 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133272 | C2 | direct (N × H) |
| 4. C22×#88324 | C4 | semidirect |
| 6. #13133272 | C22 | direct (N × H) |
| 20. C2×#88324 | C2×C4 | semidirect |
| 36. #88324 | C22×C4 | semidirect |
| 37. C22×C53 | #97 | semidirect |
| 47. C2×C53 | C2×#97 | semidirect |
| 54. C53 | C22×#97 | semidirect |
| 88. C22 | #13133272 | direct (N × H) |
| 97. C2 | C2×#13133272 | direct (N × H) |