Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
190 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #353 |
| commutator | C42 | C2×C8×C53 |
| Frattini, cosocle | #384 | C22×C53 |
| prime radical | C23×C53 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C53×#384 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 17. C4×C8×C53 | C8 | semidirect |
| 48. #7370 | C53 | direct (N × H) |
| 72. C53 | #7370 | direct (N × H) |
| 82. C4×C8 | C8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1845 | 2 |
| C53×#1325 | 2 |
| #7370 | 53 |