Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 366 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 | #394 |
| 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 |
|---|---|---|
| 2. C53×#1997 | C2 | semidirect |
| 3. C53×#1574 | C2 | semidirect |
| 5. C53×#1549 | C2 | semidirect |
| 7. C53×#1299 | C2 | semidirect |
| 131. #10490 | C53 | direct (N × H) |
| 150. #1997 | C2×C53 | semidirect |
| 151. #1574 | C2×C53 | semidirect |
| 153. #1549 | C2×C53 | semidirect |
| 155. #1299 | C2×C53 | semidirect |
| 180. C53 | #10490 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3226 | 2 |
| C53×#1997 | 2 |
| C53×#1574 | 2 |
| C53×#1560 | 2 |
| C53×#1549 | 2 |
| C53×#1484 | 2 |
| C53×#1299 | 2 |
| #10490 | 53 |