Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C53 | C23 |
| commutator | C22 | C22×C16×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| socle | C22×C53 | C22×C16 |
| prime radical | C2×D4×C53 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×M64 | C2 | semidirect |
| 3. C53×C4⋊C32 | C2 | semidirect |
| 4. C53×C22⋊C32 | C2 | semidirect |
| 5. C4×C32×C53 | C2 | semidirect |
| 27. C32×C53 | D4 | semidirect |
| 54. #13646 | C53 | direct (N × H) |
| 61. C4×C53 | M64 | semidirect |
| 64. C2×M64 | C2×C53 | semidirect |
| 66. C4⋊C32 | C2×C53 | semidirect |
| 67. C22⋊C32 | C2×C53 | semidirect |
| 68. C4×C32 | C2×C53 | semidirect |
| 82. C53 | #13646 | direct (N × H) |
| 95. C32 | D4×C53 | semidirect |
| 128. C4 | C53×M64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×M64 | 2 |
| D4×C16×C53 | 2 |
| C53×C4⋊C32 | 2 |
| C53×C22⋊C32 | 2 |
| C4×C32×C53 | 2 |
| #13646 | 53 |