Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4390 conjugacy classes of subgroups; only the 702 normal subgroups are shown.
This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 702 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| prime radical | C23×#41×C53 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C53×#380 | C2 | semidirect |
| 3. C53×#1446 | C2 | semidirect |
| 4. C53×#1438 | C2 | semidirect |
| 6. C22×#41×C53 | C4 | semidirect |
| 50. C22×C4×C53 | C22⋊C4 | semidirect |
| 82. #14337 | C53 | direct (N × H) |
| 98. C2×#380 | C2×C53 | semidirect |
| 99. #1446 | C2×C53 | semidirect |
| 100. #1438 | C2×C53 | semidirect |
| 107. C22×#41 | C4×C53 | semidirect |
| 115. C53 | #14337 | direct (N × H) |
| 152. C22×C4 | C22⋊C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41×C53 | 2 |
| C2×C53×#380 | 2 |
| C53×#1446 | 2 |
| C53×#1438 | 2 |
| #14337 | 53 |