Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3158 conjugacy classes of subgroups.
The subgroup lattice has 392 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C24 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#2365 | C2 | direct (N × H) |
| 11. C2×C4⋊Q8×C53 | C4 | semidirect |
| 48. C4⋊Q8×C53 | C2×C4 | semidirect |
| 56. C2×C4⋊C4×C53 | Q8 | semidirect |
| 119. C4⋊C4×C53 | C2×Q8 | semidirect |
| 154. C2×C4×C53 | C4×Q8 | semidirect |
| 165. C2×#2365 | C53 | direct (N × H) |
| 187. C4×C53 | C2×C4×Q8 | semidirect |
| 190. #2365 | C2×C53 | direct (N × H) |
| 197. C2×C53 | #2365 | direct (N × H) |
| 206. C2×C4⋊Q8 | C4×C53 | semidirect |
| 231. C53 | C2×#2365 | direct (N × H) |
| 244. C4⋊Q8 | C2×C4×C53 | semidirect |
| 252. C2×C4⋊C4 | Q8×C53 | semidirect |
| 315. C4⋊C4 | C2×Q8×C53 | semidirect |
| 350. C2×C4 | C4×Q8×C53 | semidirect |
| 382. C4 | C2×C4×Q8×C53 | semidirect |
| 383. C2 | C53×#2365 | 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 |
| C22×C4×Q8×C53 | 2 |
| C53×#2365 | 2 |
| C2×C53×#391 | 2 |
| C2×C53×#389 | 2 |
| C2×C53×#387 | 2 |
| C2×C4×C4⋊C4×C53 | 2 |
| C2×#2365 | 53 |