Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3294 conjugacy classes of subgroups; only the 974 normal subgroups are shown.
This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 974 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C53 |
| prime radical | C26×C53 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2842 | C2 | direct (N × H) |
| 2. C2×C53×#399 | C2 | semidirect |
| 3. C2×C53×#396 | C2 | semidirect |
| 6. C22×C4⋊C4×C53 | C22 | semidirect |
| 9. C53×#399 | C22 | semidirect |
| 10. C53×#396 | C22 | semidirect |
| 12. C2×#95×C53 | C22 | semidirect |
| 20. C2×C4⋊C4×C53 | C23 | semidirect |
| 25. #95×C53 | C23 | semidirect |
| 60. C2×#2842 | C53 | direct (N × H) |
| 73. #2842 | C2×C53 | direct (N × H) |
| 74. C2×#399 | C2×C53 | semidirect |
| 75. C2×#396 | C2×C53 | semidirect |
| 79. C2×C53 | #2842 | direct (N × H) |
| 83. C22×C4⋊C4 | C22×C53 | semidirect |
| 86. #399 | C22×C53 | semidirect |
| 87. #396 | C22×C53 | semidirect |
| 89. C2×#95 | C22×C53 | semidirect |
| 90. C53 | C2×#2842 | direct (N × H) |
| 98. C2×C4⋊C4 | C23×C53 | semidirect |
| 103. #95 | C23×C53 | semidirect |
| 152. C2 | C53×#2842 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2842 | 2 |
| C2×C53×#399 | 2 |
| C2×C53×#396 | 2 |
| C2×C53×#379 | 2 |
| C2×#2842 | 53 |