Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
412 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C4×Q8×C53 | C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | #2965 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811674 | C2 | semidirect |
| 2. #16811657 | C2 | semidirect |
| 3. #16811653 | C2 | semidirect |
| 4. #16811642 | C2 | semidirect |
| 5. #16811641 | C2 | semidirect |
| 6. C4×#11719936 | C2 | semidirect |
| 7. C4×D53⋊C8 | C2 | semidirect |
| 79. C4×D53 | M16 | semidirect |
| 82. #88324 | #41 | semidirect |
| 113. C53 | #2965 | semidirect |