Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D53 |
| commutator, cosocle | C23×C53 | C25 |
| Frattini | C23 | C24×D53 |
| Fitting | C2×C53×#383 | C2 |
| socle | C24×C53 | C24 |
| hypercenter | C2×#383 | D53 |
| hypocenter | C53 | C2×#2818 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C2×#383 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13133214 | C2 | semidirect |
| 2. C22×#13133211 | C2 | semidirect |
| 3. #20282758 | C2 | direct (N × H) |
| 4. C2×#16810541 | C2 | semidirect |
| 5. C2×#16810533 | C2 | semidirect |
| 6. C2×C53×#383 | C2 | semidirect |
| 10. C2×#13133214 | C22 | semidirect |
| 12. C2×#13133211 | C22 | semidirect |
| 16. #16810541 | C22 | semidirect |
| 17. #16810533 | C22 | semidirect |
| 19. C53×#383 | C22 | semidirect |
| 39. C2×C42×C53 | D4 | semidirect |
| 63. C42×C53 | C2×D4 | semidirect |
| 110. C2×#383 | D53 | semidirect |
| 111. C2×C53 | #2818 | semidirect |
| 128. #383 | C2×D53 | semidirect |
| 130. C53 | C2×#2818 | semidirect |
| 149. C2×C42 | D212 | semidirect |
| 176. C42 | C2×D212 | semidirect |
| 216. C2 | #20282758 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13133214 | 2 |
| C22×#13133211 | 2 |
| #20282758 | 2 |
| C2×#16810541 | 2 |
| C2×#16810533 | 2 |
| C2×C53×#383 | 2 |
| C2×#2818 | 53 |