Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #427 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C2×C53 | #1568 |
| prime radical | C23×C53 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ #97×C53 ◁ C53×#405 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#1373 | C2 | semidirect |
| 4. C53×#1369 | C2 | semidirect |
| 58. #12027 | C53 | direct (N × H) |
| 69. #1373 | C2×C53 | semidirect |
| 70. #1369 | C2×C53 | semidirect |
| 84. C53 | #12027 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2166 | 2 |
| C53×#2108 | 2 |
| C53×#1373 | 2 |
| C53×#1369 | 2 |
| #12027 | 53 |