Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
726 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #99 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C25×C53 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#1317 | C2 | direct (N × H) |
| 22. C23×C4×C53 | C8 | semidirect |
| 69. C22×C4×C53 | C2×C8 | semidirect |
| 120. C2×#1317 | C53 | direct (N × H) |
| 151. #1317 | C2×C53 | direct (N × H) |
| 158. C2×C53 | #1317 | direct (N × H) |
| 171. C53 | C2×#1317 | direct (N × H) |
| 180. C23×C4 | C8×C53 | semidirect |
| 227. C22×C4 | C2×C8×C53 | semidirect |
| 312. C2 | C53×#1317 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#381 | 2 |
| C2×C53×#323 | 2 |
| C53×#1317 | 2 |
| C2×#1317 | 53 |