Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
302 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 302 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, socle | C22×C53 | C2×D16 |
| commutator | C16 | C22×C4×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| prime radical | C2×C53×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ C22×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C53×M64 | C2 | semidirect |
| 3. C2×C53×#366 | C2 | semidirect |
| 4. C53×#1416 | C2 | semidirect |
| 5. C53×#1415 | C2 | semidirect |
| 8. C53×#508 | C4 | semidirect |
| 27. Q32×C53 | C2×C4 | semidirect |
| 30. D16×C53 | C2×C4 | semidirect |
| 78. #13616 | C53 | direct (N × H) |
| 92. C2×M64 | C2×C53 | semidirect |
| 93. C2×#366 | C2×C53 | semidirect |
| 94. #1416 | C2×C53 | semidirect |
| 95. #1415 | C2×C53 | semidirect |
| 103. #508 | C4×C53 | semidirect |
| 113. C53 | #13616 | direct (N × H) |
| 123. Q32 | C2×C4×C53 | semidirect |
| 126. D16 | C2×C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#508 | 2 |
| C2×C53×M64 | 2 |
| C2×C53×#366 | 2 |
| C53×#1416 | 2 |
| C53×#1415 | 2 |
| #13616 | 53 |