Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1846 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1846 conjugacy classes of subgroups — too many to draw. Only its 174 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 | #379 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C25 | C23×C53 |
| prime radical | C22×#127×C53 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1781 | C2 | semidirect |
| 3. C23×D4×C53 | C4 | semidirect |
| 6. C22×C42×C53 | C4 | semidirect |
| 62. C2×C4×C53 | #99 | semidirect |
| 63. #8546 | C53 | direct (N × H) |
| 77. #1781 | C2×C53 | semidirect |
| 82. C23×D4 | C4×C53 | semidirect |
| 85. C22×C42 | C4×C53 | semidirect |
| 88. C53 | #8546 | direct (N × H) |
| 142. C2×C4 | #99×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127×C53 | 2 |
| C53×#1781 | 2 |
| #8546 | 53 |