Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
232 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C53×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C8×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×C16×C53 | C2 | semidirect |
| 3. C53×#1670 | C2 | semidirect |
| 5. C53×#1372 | C2 | semidirect |
| 6. C53×#1362 | C2 | semidirect |
| 70. C16×C53 | SD16 | semidirect |
| 75. Q8×C53 | SD32 | semidirect |
| 93. #12441 | C53 | direct (N × H) |
| 107. Q8×C16 | C2×C53 | semidirect |
| 108. #1670 | C2×C53 | semidirect |
| 110. #1372 | C2×C53 | semidirect |
| 111. #1362 | C2×C53 | semidirect |
| 132. C53 | #12441 | direct (N × H) |
| 180. C16 | SD16×C53 | semidirect |
| 185. Q8 | SD32×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2248 | 2 |
| Q8×C16×C53 | 2 |
| C53×#1670 | 2 |
| C53×#1668 | 2 |
| C53×#1372 | 2 |
| C53×#1362 | 2 |
| C53×#1361 | 2 |
| #12441 | 53 |