Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
820 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #379 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3549 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1963 | C2 | semidirect |
| 3. C53×#1557 | C2 | semidirect |
| 4. C53×#1536 | C2 | semidirect |
| 5. C53×#1505 | C2 | semidirect |
| 8. C53×#555 | C4 | semidirect |
| 10. D42×C53 | C4 | semidirect |
| 15. C53×#432 | C22 | semidirect |
| 16. C53×#423 | C22 | semidirect |
| 18. C53×#376 | C22 | semidirect |
| 132. #8491 | C53 | direct (N × H) |
| 151. #1963 | C2×C53 | semidirect |
| 152. #1557 | C2×C53 | semidirect |
| 153. #1536 | C2×C53 | semidirect |
| 154. #1505 | C2×C53 | semidirect |
| 162. #555 | C4×C53 | semidirect |
| 164. D42 | C4×C53 | semidirect |
| 169. #432 | C22×C53 | semidirect |
| 170. #423 | C22×C53 | semidirect |
| 172. #376 | C22×C53 | semidirect |
| 177. C53 | #8491 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3549 | 2 |
| C53×#1963 | 2 |
| C53×#1557 | 2 |
| C53×#1536 | 2 |
| C53×#1505 | 2 |
| #8491 | 53 |