Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
406 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C2×#41×C53 | C4 |
| hypercenter | C2×#41 | #6157 |
| hypocenter | C53 | #2970 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811674 | C2 | semidirect |
| 2. #16811654 | C2 | semidirect |
| 3. D4×#88324 | C2 | semidirect |
| 4. #16811644 | C2 | semidirect |
| 5. #16811643 | C2 | semidirect |
| 6. C4×#11719940 | C2 | semidirect |
| 7. C2×#13133272 | C2 | semidirect |
| 64. D4×C53 | M16 | semidirect |
| 82. #88324 | #41 | semidirect |
| 113. C53 | #2970 | semidirect |
| 169. D4 | #11719940 | semidirect |