← C53×#63091 #27270603
Subgroups — C53×#63091 #27270603
Order 13568.
10894 conjugacy classes of subgroups.
The subgroup lattice has 196 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×H(7,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C53×#3585 | C2 | 1 | 30 | yes | maximal, complemented |
| 3 | C53×#3584 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | C2×C53×#576 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C53×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C53×#585 | — | 2 | 12 | no | — |
| 7 | C53×#585 | C22 | 1 | 1 | yes | characteristic |
| 8 | C53×#585 | C22 | 1 | 30 | yes | complemented |
| 9 | C2×H(5,𝔽2)×C53 | — | 2 | 20 | no | — |
| 10 | C2×H(5,𝔽2)×C53 | C22 | 1 | 30 | yes | complemented |
| 11 | C53×#578 | C22 | 1 | 20 | yes | complemented |
| 12 | C53×#577 | C22 | 1 | 120 | yes | complemented |
| 13 | C53×#576 | C22 | 1 | 60 | yes | complemented |
| 14 | C53×#576 | C22 | 1 | 120 | yes | complemented |
| 15 | C53×#575 | C22 | 1 | 90 | yes | complemented |
| 16 | C2×AGL(1,ℤ/8)×C53 | C22 | 1 | 90 | yes | complemented |
| 17 | C2×#135×C53 | C22 | 1 | 45 | yes | complemented |
| 18 | C22×D8×C53 | C22 | 1 | 15 | yes | complemented |
| 19 | C53×#568 | C22 | 1 | 15 | yes | complemented |
| 20 | C2×#131×C53 | C22 | 1 | 15 | yes | complemented |
| 21 | 2−1+4×C53 | — | 2 | 30 | no | — |
| 22 | 2−1+4×C53 | D4 | 1 | 6 | yes | — |
| 23 | 2−1+4×C53 | C23 | 1 | 40 | yes | complemented |
| 24 | H(5,𝔽2)×C53 | — | 2 | 120 | no | — |
| 25 | H(5,𝔽2)×C53 | — | 2 | 90 | no | — |
| 26 | H(5,𝔽2)×C53 | D4 | 1 | 10 | yes | — |
| 27 | H(5,𝔽2)×C53 | C23 | 1 | 120 | yes | complemented |
| 28 | C2×#41×C53 | — | 2 | 120 | no | — |
| 29 | C2×#41×C53 | — | 2 | 30 | no | — |
| 30 | C2×#41×C53 | C23 | 1 | 15 | yes | — |
| 31 | C2×#41×C53 | C23 | 1 | 90 | yes | complemented |
| 32 | C22×D4×C53 | — | 2 | 90 | no | — |
| 33 | C22×D4×C53 | C23 | 1 | 30 | yes | complemented |
| 34 | #137×C53 | C23 | 1 | 120 | yes | complemented |
| 35 | AGL(1,ℤ/8)×C53 | C23 | 1 | 360 | yes | complemented |
| 36 | #135×C53 | C23 | 1 | 60 | yes | complemented |
| 37 | #135×C53 | C23 | 1 | 180 | yes | complemented |
| 38 | C2×Q16×C53 | C23 | 1 | 15 | yes | complemented |
| 39 | C2×SD16×C53 | C23 | 1 | 90 | yes | complemented |
| 40 | C2×D8×C53 | C23 | 1 | 45 | yes | complemented |
| 41 | C2×D8×C53 | C23 | 1 | 90 | yes | complemented |
| 42 | #131×C53 | C23 | 1 | 20 | yes | complemented |
| 43 | #131×C53 | C23 | 1 | 60 | yes | complemented |
| 44 | C2×M16×C53 | C23 | 1 | 45 | yes | complemented |
| 45 | C22×C8×C53 | C23 | 1 | 15 | yes | complemented |
| 46 | C24×C53 | — | 2 | 30 | no | — |
| 47 | #41×C53 | — | 2 | 120 | no | — |
| 48 | #41×C53 | — | 2 | 120 | no | — |
| 49 | #41×C53 | — | 2 | 180 | no | — |
| 50 | #41×C53 | C2×D4 | 1 | 60 | yes | — |
| 51 | #41×C53 | C24 | 1 | 120 | yes | — |
| 52 | #41×C53 | C24 | 1 | 20 | yes | — |
| 53 | C2×Q8×C53 | — | 2 | 40 | no | — |
| 54 | C2×Q8×C53 | C2×D4 | 1 | 15 | yes | — |
| 55 | C2×Q8×C53 | C24 | 1 | 30 | yes | — |
| 56 | C2×D4×C53 | — | 2 | 90 | no | — |
| 57 | C2×D4×C53 | — | 2 | 360 | no | — |
| 58 | C2×D4×C53 | — | 2 | 90 | no | — |
| 59 | C2×D4×C53 | C2×D4 | 1 | 45 | yes | — |
| 60 | C2×D4×C53 | C24 | 1 | 90 | yes | — |
| 61 | C22×C4×C53 | — | 2 | 90 | no | — |
| 62 | C22×C4×C53 | C24 | 1 | 15 | yes | — |
| 63 | Q16×C53 | C24 | 1 | 6 | yes | — |
| 64 | Q16×C53 | C24 | 1 | 30 | yes | — |
| 65 | SD16×C53 | C24 | 1 | 120 | yes | — |
| 66 | D8×C53 | C24 | 1 | 10 | yes | — |
| 67 | D8×C53 | C24 | 1 | 90 | yes | — |
| 68 | M16×C53 | C24 | 1 | 60 | yes | — |
| 69 | C2×C8×C53 | C24 | 1 | 15 | yes | — |
| 70 | C2×C8×C53 | C24 | 1 | 45 | yes | — |
| 71 | C23×C53 | — | 2 | 90 | no | — |
| 72 | C23×C53 | — | 16 | 30 | no | — |
| 73 | C23×C53 | C22×D4 | 1 | 15 | yes | — |
| 74 | Q8×C53 | — | 2 | 30 | no | — |
| 75 | Q8×C53 | C22×D4 | 1 | 20 | yes | — |
| 76 | Q8×C53 | C25 | 1 | 12 | yes | — |
| 77 | D4×C53 | — | 2 | 120 | no | — |
| 78 | D4×C53 | — | 2 | 90 | no | — |
| 79 | D4×C53 | C22×D4 | 1 | 60 | yes | — |
| 80 | D4×C53 | C25 | 1 | 20 | yes | — |
| 81 | C2×C4×C53 | — | 2 | 30 | no | — |
| 82 | C2×C4×C53 | — | 2 | 120 | no | — |
| 83 | C2×C4×C53 | C22×D4 | 1 | 45 | yes | — |
| 84 | C2×C4×C53 | C25 | 1 | 15 | yes | — |
| 85 | C8×C53 | C25 | 1 | 1 | yes | characteristic |
| 86 | C8×C53 | C25 | 1 | 15 | yes | — |
| 87 | #63091 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 88 | C22×C53 | — | 2 | 20 | no | — |
| 89 | C22×C53 | — | 4 | 15 | no | — |
| 90 | C22×C53 | — | 8 | 90 | no | — |
| 91 | C22×C53 | C23×D4 | 1 | 15 | yes | — |
| 92 | C4×C53 | — | 2 | 12 | no | — |
| 93 | C4×C53 | C23×D4 | 1 | 15 | yes | — |
| 94 | C4×C53 | C26 | 1 | 1 | yes | characteristic |
| 95 | H(7,𝔽2) | C2×C53 | 1 | 2 | yes | complemented |
| 96 | #3585 | C2×C53 | 1 | 30 | yes | complemented |
| 97 | #3584 | C2×C53 | 1 | 15 | yes | complemented |
| 98 | C2×#576 | C2×C53 | 1 | 15 | yes | complemented |
| 99 | #3573 | C2×C53 | 1 | 1 | yes | characteristic, complemented |
| 100 | C2×C53 | — | 2 | 15 | no | — |
| 101 | C2×C53 | — | 4 | 20 | no | — |
| 102 | C2×C53 | C24×D4 | 1 | 1 | yes | characteristic |
| 103 | #585 | — | 2 | 12 | no | — |
| 104 | #585 | C22×C53 | 1 | 1 | yes | characteristic |
| 105 | #585 | C22×C53 | 1 | 30 | yes | complemented |
| 106 | C2×H(5,𝔽2) | — | 2 | 20 | no | — |
| 107 | C2×H(5,𝔽2) | C22×C53 | 1 | 30 | yes | complemented |
| 108 | #578 | C22×C53 | 1 | 20 | yes | complemented |
| 109 | #577 | C22×C53 | 1 | 120 | yes | complemented |
| 110 | #576 | C22×C53 | 1 | 60 | yes | complemented |
| 111 | #576 | C22×C53 | 1 | 120 | yes | complemented |
| 112 | #575 | C22×C53 | 1 | 90 | yes | complemented |
| 113 | C2×AGL(1,ℤ/8) | C22×C53 | 1 | 90 | yes | complemented |
| 114 | C2×#135 | C22×C53 | 1 | 45 | yes | complemented |
| 115 | C22×D8 | C22×C53 | 1 | 15 | yes | complemented |
| 116 | #568 | C22×C53 | 1 | 15 | yes | complemented |
| 117 | C2×#131 | C22×C53 | 1 | 15 | yes | complemented |
| 118 | C53 | #63091 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 119 | 2−1+4 | — | 2 | 30 | no | — |
| 120 | 2−1+4 | D4×C53 | 1 | 6 | yes | — |
| 121 | 2−1+4 | C23×C53 | 1 | 40 | yes | complemented |
| 122 | H(5,𝔽2) | — | 2 | 120 | no | — |
| 123 | H(5,𝔽2) | — | 2 | 90 | no | — |
| 124 | H(5,𝔽2) | D4×C53 | 1 | 10 | yes | — |
| 125 | H(5,𝔽2) | C23×C53 | 1 | 120 | yes | complemented |
| 126 | C2×#41 | — | 2 | 120 | no | — |
| 127 | C2×#41 | — | 2 | 30 | no | — |
| 128 | C2×#41 | C23×C53 | 1 | 15 | yes | — |
| 129 | C2×#41 | C23×C53 | 1 | 90 | yes | complemented |
| 130 | C22×D4 | — | 2 | 90 | no | — |
| 131 | C22×D4 | C23×C53 | 1 | 30 | yes | complemented |
| 132 | #137 | C23×C53 | 1 | 120 | yes | complemented |
| 133 | AGL(1,ℤ/8) | C23×C53 | 1 | 360 | yes | complemented |
| 134 | #135 | C23×C53 | 1 | 60 | yes | complemented |
| 135 | #135 | C23×C53 | 1 | 180 | yes | complemented |
| 136 | C2×Q16 | C23×C53 | 1 | 15 | yes | complemented |
| 137 | C2×SD16 | C23×C53 | 1 | 90 | yes | complemented |
| 138 | C2×D8 | C23×C53 | 1 | 45 | yes | complemented |
| 139 | C2×D8 | C23×C53 | 1 | 90 | yes | complemented |
| 140 | #131 | C23×C53 | 1 | 20 | yes | complemented |
| 141 | #131 | C23×C53 | 1 | 60 | yes | complemented |
| 142 | C2×M16 | C23×C53 | 1 | 45 | yes | complemented |
| 143 | C22×C8 | C23×C53 | 1 | 15 | yes | complemented |
| 144 | C24 | — | 2 | 30 | no | — |
| 145 | #41 | — | 2 | 120 | no | — |
| 146 | #41 | — | 2 | 120 | no | — |
| 147 | #41 | — | 2 | 180 | no | — |
| 148 | #41 | C2×D4×C53 | 1 | 60 | yes | — |
| 149 | #41 | C24×C53 | 1 | 120 | yes | — |
| 150 | #41 | C24×C53 | 1 | 20 | yes | — |
| 151 | C2×Q8 | — | 2 | 40 | no | — |
| 152 | C2×Q8 | C2×D4×C53 | 1 | 15 | yes | — |
| 153 | C2×Q8 | C24×C53 | 1 | 30 | yes | — |
| 154 | C2×D4 | — | 2 | 90 | no | — |
| 155 | C2×D4 | — | 2 | 360 | no | — |
| 156 | C2×D4 | — | 2 | 90 | no | — |
| 157 | C2×D4 | C2×D4×C53 | 1 | 45 | yes | — |
| 158 | C2×D4 | C24×C53 | 1 | 90 | yes | — |
| 159 | C22×C4 | — | 2 | 90 | no | — |
| 160 | C22×C4 | C24×C53 | 1 | 15 | yes | — |
| 161 | Q16 | C24×C53 | 1 | 6 | yes | — |
| 162 | Q16 | C24×C53 | 1 | 30 | yes | — |
| 163 | SD16 | C24×C53 | 1 | 120 | yes | — |
| 164 | D8 | C24×C53 | 1 | 10 | yes | — |
| 165 | D8 | C24×C53 | 1 | 90 | yes | — |
| 166 | M16 | C24×C53 | 1 | 60 | yes | — |
| 167 | C2×C8 | C24×C53 | 1 | 15 | yes | — |
| 168 | C2×C8 | C24×C53 | 1 | 45 | yes | — |
| 169 | C23 | — | 2 | 90 | no | — |
| 170 | C23 | — | 16 | 30 | no | — |
| 171 | C23 | C22×D4×C53 | 1 | 15 | yes | — |
| 172 | Q8 | — | 2 | 30 | no | — |
| 173 | Q8 | C22×D4×C53 | 1 | 20 | yes | — |
| 174 | Q8 | C25×C53 | 1 | 12 | yes | — |
| 175 | D4 | — | 2 | 120 | no | — |
| 176 | D4 | — | 2 | 90 | no | — |
| 177 | D4 | C22×D4×C53 | 1 | 60 | yes | — |
| 178 | D4 | C25×C53 | 1 | 20 | yes | — |
| 179 | C2×C4 | — | 2 | 30 | no | — |
| 180 | C2×C4 | — | 2 | 120 | no | — |
| 181 | C2×C4 | C22×D4×C53 | 1 | 45 | yes | — |
| 182 | C2×C4 | C25×C53 | 1 | 15 | yes | — |
| 183 | C8 | C25×C53 | 1 | 1 | yes | characteristic |
| 184 | C8 | C25×C53 | 1 | 15 | yes | — |
| 185 | C22 | — | 2 | 20 | no | — |
| 186 | C22 | — | 4 | 15 | no | — |
| 187 | C22 | — | 8 | 90 | no | — |
| 188 | C22 | C23×D4×C53 | 1 | 15 | yes | — |
| 189 | C4 | — | 2 | 12 | no | — |
| 190 | C4 | C23×D4×C53 | 1 | 15 | yes | — |
| 191 | C4 | C26×C53 | 1 | 1 | yes | characteristic |
| 192 | C2 | — | 2 | 15 | no | — |
| 193 | C2 | — | 4 | 20 | no | — |
| 194 | C2 | C24×D4×C53 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C53×H(7,𝔽2) | C2 | semidirect |
| 2. C53×#3585 | C2 | semidirect |
| 3. C53×#3584 | C2 | semidirect |
| 4. C2×C53×#576 | C2 | semidirect |
| 5. C53×#3573 | C2 | semidirect |
| 8. C53×#585 | C22 | semidirect |
| 10. C2×H(5,𝔽2)×C53 | C22 | semidirect |
| 11. C53×#578 | C22 | semidirect |
| 12. C53×#577 | C22 | semidirect |
| 13. C53×#576 | C22 | semidirect |
| 14. C53×#576 | C22 | semidirect |
| 15. C53×#575 | C22 | semidirect |
| 16. C2×AGL(1,ℤ/8)×C53 | C22 | semidirect |
| 17. C2×#135×C53 | C22 | semidirect |
| 18. C22×D8×C53 | C22 | semidirect |
| 19. C53×#568 | C22 | semidirect |
| 20. C2×#131×C53 | C22 | semidirect |
| 23. 2−1+4×C53 | C23 | semidirect |
| 27. H(5,𝔽2)×C53 | C23 | semidirect |
| 31. C2×#41×C53 | C23 | semidirect |
| 33. C22×D4×C53 | C23 | semidirect |
| 34. #137×C53 | C23 | semidirect |
| 35. AGL(1,ℤ/8)×C53 | C23 | semidirect |
| 36. #135×C53 | C23 | semidirect |
| 37. #135×C53 | C23 | semidirect |
| 38. C2×Q16×C53 | C23 | semidirect |
| 39. C2×SD16×C53 | C23 | semidirect |
| 40. C2×D8×C53 | C23 | semidirect |
| 41. C2×D8×C53 | C23 | semidirect |
| 42. #131×C53 | C23 | semidirect |
| 43. #131×C53 | C23 | semidirect |
| 44. C2×M16×C53 | C23 | semidirect |
| 45. C22×C8×C53 | C23 | semidirect |
| 87. #63091 | C53 | direct (N × H) |
| 95. H(7,𝔽2) | C2×C53 | semidirect |
| 96. #3585 | C2×C53 | semidirect |
| 97. #3584 | C2×C53 | semidirect |
| 98. C2×#576 | C2×C53 | semidirect |
| 99. #3573 | C2×C53 | semidirect |
| 105. #585 | C22×C53 | semidirect |
| 107. C2×H(5,𝔽2) | C22×C53 | semidirect |
| 108. #578 | C22×C53 | semidirect |
| 109. #577 | C22×C53 | semidirect |
| 110. #576 | C22×C53 | semidirect |
| 111. #576 | C22×C53 | semidirect |
| 112. #575 | C22×C53 | semidirect |
| 113. C2×AGL(1,ℤ/8) | C22×C53 | semidirect |
| 114. C2×#135 | C22×C53 | semidirect |
| 115. C22×D8 | C22×C53 | semidirect |
| 116. #568 | C22×C53 | semidirect |
| 117. C2×#131 | C22×C53 | semidirect |
| 118. C53 | #63091 | direct (N × H) |
| 121. 2−1+4 | C23×C53 | semidirect |
| 125. H(5,𝔽2) | C23×C53 | semidirect |
| 129. C2×#41 | C23×C53 | semidirect |
| 131. C22×D4 | C23×C53 | semidirect |
| 132. #137 | C23×C53 | semidirect |
| 133. AGL(1,ℤ/8) | C23×C53 | semidirect |
| 134. #135 | C23×C53 | semidirect |
| 135. #135 | C23×C53 | semidirect |
| 136. C2×Q16 | C23×C53 | semidirect |
| 137. C2×SD16 | C23×C53 | semidirect |
| 138. C2×D8 | C23×C53 | semidirect |
| 139. C2×D8 | C23×C53 | semidirect |
| 140. #131 | C23×C53 | semidirect |
| 141. #131 | C23×C53 | semidirect |
| 142. C2×M16 | C23×C53 | semidirect |
| 143. C22×C8 | C23×C53 | semidirect |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta