Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
8094 conjugacy classes of subgroups; only the 1123 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Fitting | C53×#2867 | C2 |
| hypercenter | #2867 | D53 |
| hypocenter | C53 | C2×#2867 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2867 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#127×D53 | C2 | semidirect |
| 2. #20283110 | C2 | semidirect |
| 3. #20283109 | C2 | semidirect |
| 4. C43×D53 | C2 | semidirect |
| 5. C53×#2867 | C2 | semidirect |
| 7. C2×#13133214 | C22 | semidirect |
| 8. C2×#13133213 | C22 | semidirect |
| 11. C42×Dic53 | C22 | semidirect |
| 12. C2×#127×C53 | C22 | semidirect |
| 13. C43×C53 | C22 | semidirect |
| 22. C42×D53 | D4 | semidirect |
| 37. C42×C53 | C2×D4 | semidirect |
| 38. C4×Dic53 | C2×D4 | semidirect |
| 49. C4×D53 | #127 | semidirect |
| 56. C4×C53 | C2×#127 | semidirect |
| 57. Dic53 | C2×#127 | semidirect |
| 61. #2867 | D53 | direct (N × H) |
| 66. D53 | #2867 | direct (N × H) |
| 71. C2×#127 | C2×D53 | semidirect |
| 74. C43 | C2×D53 | semidirect |
| 75. C53 | C2×#2867 | semidirect |
| 100. C42 | D4×D53 | semidirect |
| 118. C4 | #127×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×D53 | 2 |
| #20283110 | 2 |
| #20283109 | 2 |
| C43×D53 | 2 |
| C53×#2867 | 2 |
| C2×#2867 | 53 |