Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
166 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #357 |
| commutator | C16 | C42×C53 |
| Frattini, cosocle | C22×C16 | C22×C53 |
| socle | C2×C53 | #1380 |
| prime radical | C23×C53 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ C22×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C53×M64 | C4 | semidirect |
| 11. C53×#368 | C4 | semidirect |
| 58. #7480 | C53 | direct (N × H) |
| 79. M64 | C4×C53 | semidirect |
| 82. #368 | C4×C53 | semidirect |
| 85. C53 | #7480 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×M64 | 2 |
| C2×C53×#368 | 2 |
| C53×#2158 | 2 |
| #7480 | 53 |