Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1246 conjugacy classes of subgroups; only the 406 normal subgroups are shown.
This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 406 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | C23 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle, prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C22×Q8×C53 | C8 | semidirect |
| 17. C22×C8×C53 | Q8 | semidirect |
| 62. #11380 | C53 | direct (N × H) |
| 87. C53 | #11380 | direct (N × H) |
| 90. C22×Q8 | C8×C53 | semidirect |
| 96. C22×C8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×Q8×C53 | 2 |
| C2×C53×#336 | 2 |
| #11380 | 53 |