Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
370 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 134 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 | #390 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×C53 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C53×#500 | C4 | semidirect |
| 8. C53×#498 | C4 | semidirect |
| 64. #9759 | C53 | direct (N × H) |
| 82. #500 | C4×C53 | semidirect |
| 84. #498 | C4×C53 | semidirect |
| 92. C53 | #9759 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3160 | 2 |
| C53×#1835 | 2 |
| C53×#1518 | 2 |
| C53×#1452 | 2 |
| #9759 | 53 |