Order 13568.
1334 conjugacy classes of subgroups.
The subgroup lattice has 396 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D53 |
| commutator, socle, prime radical, cosocle | C24×C53 | C24 |
| Fitting | C53×#1434 | C2 |
| hypercenter | #1434 | D53 |
| hypocenter | C53 | #15104 |
| LCS | G ▷ C24×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ #1434 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 44. C2×C4⋊Dic53 | Q8 | semidirect |
| 61. #95×C53 | Q8 | semidirect |
| 224. C53 | #15104 | semidirect |
| 259. #95 | Dic106 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16810529 | 2 |
| C2×C4×C4⋊Dic53 | 2 |
| #20272985 | 2 |
| #20272984 | 2 |
| #20272980 | 2 |
| #20272949 | 2 |
| C53×#1434 | 2 |
| #15104 | 53 |