Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
2186 conjugacy classes of subgroups; only the 990 normal subgroups are shown.
This group has 2186 conjugacy classes of subgroups — too many to draw. Only its 990 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 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C53 |
| prime radical | C25×C53 | C23 |
| 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. C53×#2866 | C2 | direct (N × H) |
| 3. C2×C53×#387 | C2 | semidirect |
| 4. C2×C43×C53 | C2 | semidirect |
| 9. C53×#387 | C22 | semidirect |
| 11. C43×C53 | C22 | semidirect |
| 44. C2×#2866 | C53 | direct (N × H) |
| 51. #2866 | C2×C53 | direct (N × H) |
| 53. C2×#387 | C2×C53 | semidirect |
| 54. C2×C43 | C2×C53 | semidirect |
| 56. C2×C53 | #2866 | direct (N × H) |
| 62. #387 | C22×C53 | semidirect |
| 64. C43 | C22×C53 | semidirect |
| 65. C53 | C2×#2866 | direct (N × H) |
| 105. C2 | C53×#2866 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2866 | 2 |
| C2×C53×#388 | 2 |
| C2×C53×#387 | 2 |
| C2×C43×C53 | 2 |
| C2×#2866 | 53 |