Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
284 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C22×D4×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#97×C53 | C2 | semidirect |
| 5. C53×#389 | C2 | semidirect |
| 6. C53×#340 | C2 | semidirect |
| 7. C53×#340 | C2 | semidirect |
| 19. #102×C53 | C4 | semidirect |
| 110. #1936 | C53 | direct (N × H) |
| 121. C2×#97 | C2×C53 | semidirect |
| 123. #389 | C2×C53 | semidirect |
| 124. #340 | C2×C53 | semidirect |
| 125. #340 | C2×C53 | semidirect |
| 126. C53 | #1936 | direct (N × H) |
| 138. #102 | C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#102×C53 | 2 |
| C2×#97×C53 | 2 |
| C53×#390 | 2 |
| C53×#389 | 2 |
| C53×#340 | 2 |
| #1936 | 53 |