Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
852 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C2×C53×#447 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3193 | C2 | semidirect |
| 2. C2×C53×#450 | C2 | semidirect |
| 4. C53×#1783 | C2 | semidirect |
| 5. C53×#1472 | C2 | semidirect |
| 6. C53×#1471 | C2 | semidirect |
| 7. C53×#1298 | C2 | semidirect |
| 194. #9987 | C53 | direct (N × H) |
| 224. #3193 | C2×C53 | semidirect |
| 225. C2×#450 | C2×C53 | semidirect |
| 227. #1783 | C2×C53 | semidirect |
| 228. #1472 | C2×C53 | semidirect |
| 229. #1471 | C2×C53 | semidirect |
| 230. #1298 | C2×C53 | semidirect |
| 262. C53 | #9987 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3193 | 2 |
| C2×C53×#450 | 2 |
| C2×C53×#447 | 2 |
| C53×#1783 | 2 |
| C53×#1472 | 2 |
| C53×#1471 | 2 |
| C53×#1298 | 2 |
| #9987 | 53 |