Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
366 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #418 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1824 |
| prime radical | C2×#41×C53 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#1389 | C2 | semidirect |
| 5. C53×#1382 | C2 | semidirect |
| 11. C53×#430 | C4 | semidirect |
| 83. #12877 | C53 | direct (N × H) |
| 97. #1389 | C2×C53 | semidirect |
| 98. #1382 | C2×C53 | semidirect |
| 108. #430 | C4×C53 | semidirect |
| 115. C53 | #12877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2916 | 2 |
| C53×#2113 | 2 |
| C53×#1812 | 2 |
| C53×#1389 | 2 |
| C53×#1382 | 2 |
| #12877 | 53 |