Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
372 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #342 |
| commutator | C22×C4 | C42×C53 |
| Frattini, cosocle | C2×C4×D4 | C22×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C23×C8×C53 | C4 | semidirect |
| 10. C2×C22⋊C8×C53 | C4 | semidirect |
| 12. C53×#343 | C4 | semidirect |
| 125. #7236 | C53 | direct (N × H) |
| 166. C23×C8 | C4×C53 | semidirect |
| 172. C2×C22⋊C8 | C4×C53 | semidirect |
| 174. #343 | C4×C53 | semidirect |
| 177. C53 | #7236 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1869 | 2 |
| C53×#1800 | 2 |
| C53×#1799 | 2 |
| #7236 | 53 |