Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
492 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2634 | C2 | semidirect |
| 4. C53×#1534 | C2 | semidirect |
| 6. C53×#1491 | C2 | semidirect |
| 7. C53×#1288 | C2 | semidirect |
| 165. #10021 | C53 | direct (N × H) |
| 187. #2634 | C2×C53 | semidirect |
| 188. #1534 | C2×C53 | semidirect |
| 190. #1491 | C2×C53 | semidirect |
| 191. #1288 | C2×C53 | semidirect |
| 222. C53 | #10021 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3142 | 2 |
| C53×#3112 | 2 |
| C53×#2634 | 2 |
| C53×#1534 | 2 |
| C53×#1514 | 2 |
| C53×#1491 | 2 |
| C53×#1288 | 2 |
| #10021 | 53 |