Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1538 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 366 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 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| prime radical | C2×C53×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2312 | C2 | semidirect |
| 3. C53×#1964 | C2 | semidirect |
| 4. C53×#1781 | C2 | semidirect |
| 7. C2×#123×C53 | C4 | semidirect |
| 9. C2×#120×C53 | C4 | semidirect |
| 11. C2×#99×C53 | C22 | semidirect |
| 13. C53×#380 | C22 | semidirect |
| 15. C4×C22⋊C4×C53 | C22 | semidirect |
| 103. #21795 | C53 | direct (N × H) |
| 123. #2312 | C2×C53 | semidirect |
| 124. #1964 | C2×C53 | semidirect |
| 125. #1781 | C2×C53 | semidirect |
| 134. C2×#123 | C4×C53 | semidirect |
| 136. C2×#120 | C4×C53 | semidirect |
| 138. C2×#99 | C22×C53 | semidirect |
| 140. #380 | C22×C53 | semidirect |
| 142. C4×C22⋊C4 | C22×C53 | semidirect |
| 143. C53 | #21795 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#534 | 2 |
| C53×#2312 | 2 |
| C53×#1964 | 2 |
| C53×#1781 | 2 |
| #21795 | 53 |