Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
432 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 74 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C53×#534 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2763 | C2 | semidirect |
| 5. C53×#1351 | C2 | semidirect |
| 6. C53×#1351 | C2 | semidirect |
| 7. C53×#1326 | C2 | semidirect |
| 108. C22⋊C4×C53 | SD16 | semidirect |
| 164. #12177 | C53 | direct (N × H) |
| 192. #2763 | C2×C53 | semidirect |
| 196. #1351 | C2×C53 | semidirect |
| 197. #1351 | C2×C53 | semidirect |
| 198. #1326 | C2×C53 | semidirect |
| 226. C53 | #12177 | direct (N × H) |
| 308. C22⋊C4 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2763 | 2 |
| C53×#2044 | 2 |
| C53×#1601 | 2 |
| C53×#1351 | 2 |
| C53×#1326 | 2 |
| #12177 | 53 |