Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
724 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 724 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×#121×C53 | C22 |
| 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 |
|---|---|---|
| 2. C2×C53×#451 | C2 | semidirect |
| 4. C53×#1784 | C2 | semidirect |
| 5. C53×#1472 | C2 | semidirect |
| 7. C53×#1298 | C2 | semidirect |
| 185. #9989 | C53 | direct (N × H) |
| 210. C2×#451 | C2×C53 | semidirect |
| 212. #1784 | C2×C53 | semidirect |
| 213. #1472 | C2×C53 | semidirect |
| 215. #1298 | C2×C53 | semidirect |
| 246. C53 | #9989 | 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×#451 | 2 |
| C2×C53×#448 | 2 |
| C53×#1784 | 2 |
| C53×#1472 | 2 |
| C53×#1471 | 2 |
| C53×#1298 | 2 |
| #9989 | 53 |