Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
908 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 328 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C53 |
| prime radical | C25×C53 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊C4×C53 | C2 | semidirect |
| 3. C53×#399 | C2 | direct (N × H) |
| 4. C2×#95×C53 | C2 | semidirect |
| 9. C2×C4⋊C4×C53 | C22 | semidirect |
| 12. #95×C53 | C22 | semidirect |
| 54. C2×#399 | C53 | direct (N × H) |
| 56. C2×C53 | #399 | direct (N × H) |
| 60. C22×C4⋊C4 | C2×C53 | semidirect |
| 62. #399 | C2×C53 | direct (N × H) |
| 63. C2×#95 | C2×C53 | semidirect |
| 64. C53 | C2×#399 | direct (N × H) |
| 69. C2×C4⋊C4 | C22×C53 | semidirect |
| 72. #95 | C22×C53 | semidirect |
| 115. C2 | C53×#399 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4×C53 | 2 |
| C22×C22⋊C4×C53 | 2 |
| C53×#399 | 2 |
| C2×#95×C53 | 2 |
| C2×#399 | 53 |