Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
600 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 360 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 | C24 |
| commutator | C22 | C23×C4×C53 |
| Frattini, cosocle | C23 | C24×C53 |
| prime radical | C24×C53 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#389 | C2 | semidirect |
| 8. C2×C4⋊C4×C53 | C4 | semidirect |
| 40. C2×C4×C53 | C4⋊C4 | semidirect |
| 58. #2316 | C53 | direct (N × H) |
| 67. #389 | C2×C53 | semidirect |
| 69. C53 | #2316 | direct (N × H) |
| 73. C2×C4⋊C4 | C4×C53 | semidirect |
| 105. C2×C4 | C4⋊C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#117×C53 | 2 |
| C22×C4⋊C4×C53 | 2 |
| C53×#389 | 2 |
| C53×#380 | 2 |
| #2316 | 53 |