Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
170 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 170 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 | C8×C53 | #117 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| socle | C2×C53 | #2163 |
| prime radical | C23×C53 | M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C53×#346 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1422 | C2 | semidirect |
| 4. C53×#1398 | C2 | semidirect |
| 56. #13642 | C53 | direct (N × H) |
| 66. #1422 | C2×C53 | semidirect |
| 68. #1398 | C2×C53 | semidirect |
| 80. C53 | #13642 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×M32×C53 | 2 |
| C53×#1422 | 2 |
| C53×#1400 | 2 |
| C53×#1398 | 2 |
| #13642 | 53 |