Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
774 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 204 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 | #2880 |
| 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. C2×#13133305 | C2 | semidirect |
| 2. #16811597 | C2 | semidirect |
| 3. #16811594 | C2 | semidirect |
| 4. D53⋊(C22⋊C8) | C2 | semidirect |
| 7. C2×#11719936 | C22 | semidirect |
| 8. C2×D53⋊C8 | C22 | semidirect |
| 11. C22⋊#88324 | C22 | semidirect |
| 12. C22⋊#88324 | C22 | semidirect |
| 77. C2×D53 | #131 | semidirect |
| 94. C53 | #2880 | semidirect |