Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
712 conjugacy classes of subgroups; only the 518 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 518 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C24 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C22×C42 |
| prime radical | C24×C53 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1455 | C2 | semidirect |
| 3. C53×#1454 | C2 | semidirect |
| 10. C53×#322 | C22 | semidirect |
| 11. C82×C53 | C22 | semidirect |
| 45. #17339 | C53 | direct (N × H) |
| 54. #1455 | C2×C53 | semidirect |
| 55. #1454 | C2×C53 | semidirect |
| 65. #322 | C22×C53 | semidirect |
| 66. C82 | C22×C53 | semidirect |
| 67. C53 | #17339 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2920 | 2 |
| C53×#1455 | 2 |
| C53×#1454 | 2 |
| #17339 | 53 |