Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
392 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #385 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#1569 | C2 | semidirect |
| 5. C53×#1300 | C2 | semidirect |
| 19. C53×#329 | C4 | semidirect |
| 20. C53×#329 | C4 | semidirect |
| 21. C53×#329 | C4 | semidirect |
| 117. #8918 | C53 | direct (N × H) |
| 135. #1569 | C2×C53 | semidirect |
| 137. #1300 | C2×C53 | semidirect |
| 156. #329 | C4×C53 | semidirect |
| 157. #329 | C4×C53 | semidirect |
| 158. #329 | C4×C53 | semidirect |
| 159. C53 | #8918 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2980 | 2 |
| C53×#1963 | 2 |
| C53×#1569 | 2 |
| C53×#1525 | 2 |
| C53×#1300 | 2 |
| #8918 | 53 |