Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
492 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 492 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 | C2×C42 | C23×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 3. C53×#3165 | C2 | semidirect |
| 4. C53×#1966 | C2 | semidirect |
| 6. C53×#1485 | C2 | semidirect |
| 7. C53×#1299 | C2 | semidirect |
| 156. #10116 | C53 | direct (N × H) |
| 177. #3165 | C2×C53 | semidirect |
| 178. #1966 | C2×C53 | semidirect |
| 180. #1485 | C2×C53 | semidirect |
| 181. #1299 | C2×C53 | semidirect |
| 210. C53 | #10116 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3254 | 2 |
| C53×#3227 | 2 |
| C53×#3165 | 2 |
| C53×#1966 | 2 |
| C53×#1525 | 2 |
| C53×#1485 | 2 |
| C53×#1299 | 2 |
| #10116 | 53 |