Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
5150 conjugacy classes of subgroups; only the 939 normal subgroups are shown.
This group has 5150 conjugacy classes of subgroups — too many to draw. Only its 939 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22×D53 |
| commutator, cosocle | C23×C53 | C25 |
| Frattini | C23 | C24×D53 |
| Fitting | C2×C53×#376 | C2 |
| socle | C24×C53 | C24 |
| hypercenter | C2×#376 | D53 |
| hypocenter | C53 | C2×#2397 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23×C4 ◁ C2×#376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4×D212 | C2 | semidirect |
| 2. #20277996 | C2 | direct (N × H) |
| 3. C2×#16810560 | C2 | semidirect |
| 4. C2×#16810533 | C2 | semidirect |
| 5. C2×C53×#376 | C2 | semidirect |
| 9. C2×C4×D212 | C22 | semidirect |
| 12. #16810560 | C22 | semidirect |
| 13. #16810533 | C22 | semidirect |
| 15. C53×#376 | C22 | semidirect |
| 31. C2×C42×C53 | D4 | semidirect |
| 51. C42×C53 | C2×D4 | semidirect |
| 88. C2×#376 | D53 | semidirect |
| 89. C2×C53 | #2397 | semidirect |
| 102. #376 | C2×D53 | semidirect |
| 104. C53 | C2×#2397 | semidirect |
| 119. C2×C42 | D212 | semidirect |
| 141. C42 | C2×D212 | semidirect |
| 172. C2 | #20277996 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D212 | 2 |
| #20277996 | 2 |
| C2×#16810560 | 2 |
| C2×#16810533 | 2 |
| C2×C53×#376 | 2 |
| C2×#2397 | 53 |