Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
438 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C2×#121×C53 | C22 |
| LCS | G ▷ C23×C4 ▷ 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×#2667 | C2 | semidirect |
| 5. C53×#1305 | C2 | semidirect |
| 6. C53×#1297 | C2 | semidirect |
| 7. C53×#1284 | C2 | semidirect |
| 170. #10384 | C53 | direct (N × H) |
| 201. #2667 | C2×C53 | semidirect |
| 205. #1305 | C2×C53 | semidirect |
| 206. #1297 | C2×C53 | semidirect |
| 207. #1284 | C2×C53 | semidirect |
| 235. C53 | #10384 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2667 | 2 |
| C53×#1964 | 2 |
| C53×#1510 | 2 |
| C53×#1507 | 2 |
| C53×#1305 | 2 |
| C53×#1297 | 2 |
| C53×#1284 | 2 |
| #10384 | 53 |