Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
468 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 126 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 | C2×#99 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#117×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C2×C4⋊Q8×C53 | C4 | semidirect |
| 11. C53×#398 | C4 | semidirect |
| 15. C53×#382 | C4 | semidirect |
| 127. #11694 | C53 | direct (N × H) |
| 161. C2×C4⋊Q8 | C4×C53 | semidirect |
| 167. #398 | C4×C53 | semidirect |
| 171. #382 | C4×C53 | semidirect |
| 176. C53 | #11694 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2621 | 2 |
| C53×#1740 | 2 |
| C53×#1458 | 2 |
| C53×#1344 | 2 |
| #11694 | 53 |