Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
878 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #386 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C53×#2408 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2975 | C2 | semidirect |
| 3. C53×#1783 | C2 | semidirect |
| 4. C53×#1465 | C2 | semidirect |
| 5. C53×#1305 | C2 | semidirect |
| 35. C2×M16×C53 | D4 | semidirect |
| 152. C2×C4×C53 | #100 | semidirect |
| 158. #9504 | C53 | direct (N × H) |
| 190. #2975 | C2×C53 | semidirect |
| 192. #1783 | C2×C53 | semidirect |
| 193. #1465 | C2×C53 | semidirect |
| 194. #1305 | C2×C53 | semidirect |
| 220. C53 | #9504 | direct (N × H) |
| 233. C2×M16 | D4×C53 | semidirect |
| 350. C2×C4 | #100×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2975 | 2 |
| C53×#2408 | 2 |
| C53×#1783 | 2 |
| C53×#1465 | 2 |
| C53×#1305 | 2 |
| #9504 | 53 |