Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
786 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 786 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 | C2×#99 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C53×#2408 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×#99×C53 | C2 | semidirect |
| 3. C53×#1466 | C2 | semidirect |
| 4. C53×#1343 | C2 | semidirect |
| 5. C23×D4×C53 | C4 | semidirect |
| 11. C53×#394 | C4 | semidirect |
| 14. C53×#380 | C4 | semidirect |
| 17. C53×#342 | C22 | semidirect |
| 19. C53×#323 | C22 | semidirect |
| 153. #11772 | C53 | direct (N × H) |
| 186. C4×#99 | C2×C53 | semidirect |
| 187. #1466 | C2×C53 | semidirect |
| 188. #1343 | C2×C53 | semidirect |
| 197. C23×D4 | C4×C53 | semidirect |
| 203. #394 | C4×C53 | semidirect |
| 206. #380 | C4×C53 | semidirect |
| 209. #342 | C22×C53 | semidirect |
| 211. #323 | C22×C53 | semidirect |
| 212. C53 | #11772 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2408 | 2 |
| C4×#99×C53 | 2 |
| C53×#1466 | 2 |
| C53×#1343 | 2 |
| #11772 | 53 |