Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C22×C4 | C42×C53 |
| Frattini, cosocle | C2×#117 | C22×C53 |
| socle | C22×C53 | #342 |
| prime radical | C24×C53 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C2×#117×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×C4⋊C8×C53 | C4 | semidirect |
| 8. C2×C4⋊C8×C53 | C4 | semidirect |
| 13. C53×#324 | C4 | semidirect |
| 116. #7252 | C53 | direct (N × H) |
| 153. C2×C4⋊C8 | C4×C53 | semidirect |
| 154. C2×C4⋊C8 | C4×C53 | semidirect |
| 159. #324 | C4×C53 | semidirect |
| 161. C53 | #7252 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1798 | 2 |
| C53×#1740 | 2 |
| C53×#1458 | 2 |
| #7252 | 53 |