Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
13310 conjugacy classes of subgroups; only the 6878 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 6878 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C53 | D4 |
| commutator, Frattini, cosocle | C4 | C26×C53 |
| prime radical | C24×D4×C53 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C53 ◁ C24×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8×C53 | C2 | semidirect |
| 3. C23×SD16×C53 | C2 | direct (N × H) |
| 4. C24×C8×C53 | C2 | semidirect |
| 7. C23×Q8×C53 | C22 | semidirect |
| 12. C22×SD16×C53 | C22 | direct (N × H) |
| 13. C23×C8×C53 | C22 | semidirect |
| 18. C22×Q8×C53 | C23 | semidirect |
| 24. C2×SD16×C53 | C23 | direct (N × H) |
| 25. C22×C8×C53 | C23 | semidirect |
| 28. C24×C53 | SD16 | direct (N × H) |
| 31. C2×Q8×C53 | C24 | semidirect |
| 37. SD16×C53 | C24 | direct (N × H) |
| 38. C2×C8×C53 | C24 | semidirect |
| 41. C23×C53 | C2×SD16 | direct (N × H) |
| 44. Q8×C53 | C25 | semidirect |
| 50. C8×C53 | C25 | semidirect |
| 51. C24×SD16 | C53 | direct (N × H) |
| 54. C22×C53 | C22×SD16 | direct (N × H) |
| 59. C24×Q8 | C2×C53 | semidirect |
| 61. C23×SD16 | C2×C53 | direct (N × H) |
| 62. C24×C8 | C2×C53 | semidirect |
| 64. C2×C53 | C23×SD16 | direct (N × H) |
| 68. C23×Q8 | C22×C53 | semidirect |
| 73. C22×SD16 | C22×C53 | direct (N × H) |
| 74. C23×C8 | C22×C53 | semidirect |
| 75. C53 | C24×SD16 | direct (N × H) |
| 80. C22×Q8 | C23×C53 | semidirect |
| 86. C2×SD16 | C23×C53 | direct (N × H) |
| 87. C22×C8 | C23×C53 | semidirect |
| 90. C24 | SD16×C53 | direct (N × H) |
| 93. C2×Q8 | C24×C53 | semidirect |
| 99. SD16 | C24×C53 | direct (N × H) |
| 100. C2×C8 | C24×C53 | semidirect |
| 103. C23 | C2×SD16×C53 | direct (N × H) |
| 106. Q8 | C25×C53 | semidirect |
| 112. C8 | C25×C53 | semidirect |
| 115. C22 | C22×SD16×C53 | direct (N × H) |
| 121. C2 | C23×SD16×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8×C53 | 2 |
| C24×D4×C53 | 2 |
| C23×SD16×C53 | 2 |
| C24×C8×C53 | 2 |
| C24×SD16 | 53 |