Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
308 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2092 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2161 | C2 | semidirect |
| 2. C53×#2155 | C2 | semidirect |
| 6. C53×#1833 | C2 | semidirect |
| 7. C53×#1378 | C2 | semidirect |
| 111. #13523 | C53 | direct (N × H) |
| 126. #2161 | C2×C53 | semidirect |
| 127. #2155 | C2×C53 | semidirect |
| 131. #1833 | C2×C53 | semidirect |
| 132. #1378 | C2×C53 | semidirect |
| 153. C53 | #13523 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2161 | 2 |
| C53×#2155 | 2 |
| C2×C53×#361 | 2 |
| C53×#2140 | 2 |
| C53×#1978 | 2 |
| C53×#1833 | 2 |
| C53×#1378 | 2 |
| #13523 | 53 |