Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
2750 conjugacy classes of subgroups.
The subgroup lattice has 352 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | C23 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| socle, prime radical | C24×C53 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2295 | C2 | direct (N × H) |
| 11. C2×C4×Q8×C53 | C4 | semidirect |
| 46. C4×Q8×C53 | C2×C4 | semidirect |
| 61. C2×C42×C53 | Q8 | semidirect |
| 115. C42×C53 | C2×Q8 | semidirect |
| 151. C2×#2295 | C53 | direct (N × H) |
| 171. #2295 | C2×C53 | direct (N × H) |
| 179. C2×C53 | #2295 | direct (N × H) |
| 186. C2×C4×Q8 | C4×C53 | semidirect |
| 210. C53 | C2×#2295 | direct (N × H) |
| 222. C4×Q8 | C2×C4×C53 | semidirect |
| 237. C2×C42 | Q8×C53 | semidirect |
| 291. C42 | C2×Q8×C53 | semidirect |
| 345. C2 | C53×#2295 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×Q8×C53 | 2 |
| C53×#2295 | 2 |
| C2×C53×#391 | 2 |
| C2×C53×#389 | 2 |
| C2×C53×#387 | 2 |
| C2×C4×C4⋊C4×C53 | 2 |
| C2×C53×#376 | 2 |
| C2×C43×C53 | 2 |
| C2×#2295 | 53 |