Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1326 conjugacy classes of subgroups; only the 622 normal subgroups are shown.
This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 622 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C24 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C53×#384 | C4 | semidirect |
| 8. C53×#383 | C4 | semidirect |
| 35. C42×C53 | C42 | semidirect |
| 50. #13830 | C53 | direct (N × H) |
| 70. #384 | C4×C53 | semidirect |
| 71. #383 | C4×C53 | semidirect |
| 74. C53 | #13830 | direct (N × H) |
| 99. C42 | C42×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#384 | 2 |
| C2×C53×#383 | 2 |
| C53×#1436 | 2 |
| #13830 | 53 |