Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
10246 conjugacy classes of subgroups; only the 5774 normal subgroups are shown.
This group has 10246 conjugacy classes of subgroups — too many to draw. Only its 5774 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 | C22 | C26×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×H(3,𝔽4) | C2 | direct (N × H) |
| 2. C22×#124×C53 | C2 | semidirect |
| 3. C22×#120×C53 | C2 | semidirect |
| 4. C26×C53 | C22 | semidirect |
| 5. C23×Q8×C53 | C22 | semidirect |
| 7. C53×H(3,𝔽4) | C22 | direct (N × H) |
| 8. C2×#124×C53 | C22 | semidirect |
| 9. C2×#120×C53 | C22 | semidirect |
| 10. C22×C22⋊C4×C53 | C22 | semidirect |
| 11. C22×C42×C53 | C22 | semidirect |
| 14. C25×C53 | C23 | semidirect |
| 16. C22×Q8×C53 | C23 | semidirect |
| 20. #124×C53 | C23 | semidirect |
| 21. #120×C53 | C23 | semidirect |
| 22. C2×C22⋊C4×C53 | C23 | semidirect |
| 23. C2×C42×C53 | C23 | semidirect |
| 29. C24×C53 | C24 | semidirect |
| 31. C2×Q8×C53 | C24 | semidirect |
| 36. C22⋊C4×C53 | C24 | semidirect |
| 37. C42×C53 | C24 | semidirect |
| 49. C22×H(3,𝔽4) | C53 | direct (N × H) |
| 53. C22×C53 | H(3,𝔽4) | direct (N × H) |
| 57. C2×H(3,𝔽4) | C2×C53 | direct (N × H) |
| 58. C22×#124 | C2×C53 | semidirect |
| 59. C22×#120 | C2×C53 | semidirect |
| 61. C2×C53 | C2×H(3,𝔽4) | direct (N × H) |
| 63. C26 | C22×C53 | semidirect |
| 64. C23×Q8 | C22×C53 | semidirect |
| 66. H(3,𝔽4) | C22×C53 | direct (N × H) |
| 67. C2×#124 | C22×C53 | semidirect |
| 68. C2×#120 | C22×C53 | semidirect |
| 69. C22×C22⋊C4 | C22×C53 | semidirect |
| 70. C22×C42 | C22×C53 | semidirect |
| 71. C53 | C22×H(3,𝔽4) | direct (N × H) |
| 74. C25 | C23×C53 | semidirect |
| 76. C22×Q8 | C23×C53 | semidirect |
| 80. #124 | C23×C53 | semidirect |
| 81. #120 | C23×C53 | semidirect |
| 82. C2×C22⋊C4 | C23×C53 | semidirect |
| 83. C2×C42 | C23×C53 | semidirect |
| 89. C24 | C24×C53 | semidirect |
| 91. C2×Q8 | C24×C53 | semidirect |
| 96. C22⋊C4 | C24×C53 | semidirect |
| 97. C42 | C24×C53 | semidirect |
| 112. C22 | C53×H(3,𝔽4) | direct (N × H) |
| 117. C2 | C2×C53×H(3,𝔽4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×H(3,𝔽4) | 2 |
| C22×#124×C53 | 2 |
| C22×#120×C53 | 2 |
| C22×H(3,𝔽4) | 53 |