Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
842 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #379 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 16. C2×M16×C53 | Q8 | semidirect |
| 20. C22×C8×C53 | Q8 | semidirect |
| 70. #11391 | C53 | direct (N × H) |
| 97. C53 | #11391 | direct (N × H) |
| 102. C2×M16 | Q8×C53 | semidirect |
| 106. C22×C8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3458 | 2 |
| C53×#1742 | 2 |
| C53×#1729 | 2 |
| #11391 | 53 |