← C2×#585×D53 #28297983
Subgroups — C2×#585×D53 #28297983
Order 13568.
60366 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16811460 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×#16811459 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | C2×#16811458 | C2 | 1 | 10 | yes | maximal, complemented |
| 4 | #585×D53 | C2 | 1 | 64 | yes | maximal, direct product factor |
| 5 | C2×2−1+4×D53 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×H(5,𝔽2)×D53 | C2 | 1 | 10 | yes | maximal, complemented |
| 7 | C22×#41×D53 | C2 | 1 | 15 | yes | maximal, complemented |
| 8 | C2×C53×#585 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #16811460 | C22 | 1 | 192 | yes | complemented |
| 10 | #16811459 | C22 | 1 | 480 | yes | complemented |
| 11 | #16811458 | C22 | 1 | 320 | yes | complemented |
| 12 | C2×#13133265 | C22 | 1 | 30 | yes | complemented |
| 13 | C2×#13133264 | C22 | 1 | 60 | yes | complemented |
| 14 | 2−1+4×D53 | C22 | 1 | 192 | yes | complemented |
| 15 | C2×#13133263 | C22 | 1 | 60 | yes | complemented |
| 16 | C2×#13133262 | C22 | 1 | 90 | yes | complemented |
| 17 | H(5,𝔽2)×D53 | C22 | 1 | 320 | yes | complemented |
| 18 | C22×#11719949 | C22 | 1 | 15 | yes | complemented |
| 19 | C22×#11719945 | C22 | 1 | 45 | yes | complemented |
| 20 | C22×#11719947 | C22 | 1 | 45 | yes | complemented |
| 21 | C2×#41×D53 | C22 | 1 | 20 | yes | complemented |
| 22 | C2×#41×D53 | C22 | 1 | 60 | yes | complemented |
| 23 | C2×#41×D53 | C22 | 1 | 480 | yes | complemented |
| 24 | C2×#41×D53 | C22 | 1 | 60 | yes | complemented |
| 25 | C2×#41×D53 | C22 | 1 | 60 | yes | complemented |
| 26 | C22×Q8×D53 | C22 | 1 | 15 | yes | complemented |
| 27 | C22×D4×D53 | C22 | 1 | 45 | yes | complemented |
| 28 | C23×C4×D53 | C22 | 1 | 15 | yes | complemented |
| 29 | C53×#585 | C22 | 1 | 32 | yes | complemented |
| 30 | C2×2−1+4×C53 | C22 | 1 | 6 | yes | complemented |
| 31 | C2×H(5,𝔽2)×C53 | C22 | 1 | 10 | yes | complemented |
| 32 | C22×#41×C53 | C22 | 1 | 15 | yes | complemented |
| 33 | C24×D53 | C23 | 1 | 15 | yes | — |
| 34 | #13133265 | C23 | 1 | 480 | yes | complemented |
| 35 | #13133264 | C23 | 1 | 960 | yes | complemented |
| 36 | #13133263 | C23 | 1 | 960 | yes | complemented |
| 37 | #13133262 | C23 | 1 | 1440 | yes | complemented |
| 38 | C2×#11719949 | C23 | 1 | 20 | yes | complemented |
| 39 | C2×#11719949 | C23 | 1 | 60 | yes | complemented |
| 40 | C2×#11719949 | C23 | 1 | 240 | yes | complemented |
| 41 | C2×#11719945 | C23 | 1 | 60 | yes | complemented |
| 42 | C2×#11719945 | C23 | 1 | 180 | yes | complemented |
| 43 | C2×#11719945 | C23 | 1 | 720 | yes | complemented |
| 44 | C2×#11719947 | C23 | 1 | 60 | yes | complemented |
| 45 | C2×#11719947 | C23 | 1 | 180 | yes | complemented |
| 46 | C2×#11719947 | C23 | 1 | 720 | yes | complemented |
| 47 | #41×D53 | C23 | 1 | 320 | yes | complemented |
| 48 | #41×D53 | C23 | 1 | 960 | yes | complemented |
| 49 | #41×D53 | C23 | 1 | 960 | yes | complemented |
| 50 | #41×D53 | C23 | 1 | 960 | yes | complemented |
| 51 | C22×Dic106 | C23 | 1 | 45 | yes | complemented |
| 52 | C2×Q8×D53 | C23 | 1 | 20 | yes | — |
| 53 | C2×Q8×D53 | C23 | 1 | 60 | yes | complemented |
| 54 | C2×Q8×D53 | C23 | 1 | 240 | yes | complemented |
| 55 | C22×D212 | C23 | 1 | 45 | yes | complemented |
| 56 | C22×C22⋊D53 | C23 | 1 | 90 | yes | complemented |
| 57 | C2×D4×D53 | C23 | 1 | 60 | yes | — |
| 58 | C2×D4×D53 | C23 | 1 | 180 | yes | complemented |
| 59 | C2×D4×D53 | C23 | 1 | 720 | yes | complemented |
| 60 | C23×Dic53 | C23 | 1 | 15 | yes | complemented |
| 61 | C22×C4×D53 | C23 | 1 | 15 | yes | — |
| 62 | C22×C4×D53 | C23 | 1 | 45 | yes | — |
| 63 | C22×C4×D53 | C23 | 1 | 240 | yes | complemented |
| 64 | C22×C4×D53 | C23 | 1 | 45 | yes | complemented |
| 65 | C22×C4×D53 | C23 | 1 | 45 | yes | complemented |
| 66 | 2−1+4×C53 | C23 | 1 | 96 | yes | complemented |
| 67 | H(5,𝔽2)×C53 | C23 | 1 | 160 | yes | complemented |
| 68 | C2×#41×C53 | C23 | 1 | 20 | yes | complemented |
| 69 | C2×#41×C53 | C23 | 1 | 60 | yes | complemented |
| 70 | C2×#41×C53 | C23 | 1 | 240 | yes | complemented |
| 71 | C22×Q8×C53 | C23 | 1 | 15 | yes | complemented |
| 72 | C22×D4×C53 | C23 | 1 | 45 | yes | complemented |
| 73 | C23×C4×C53 | C23 | 1 | 15 | yes | complemented |
| 74 | C24×C53 | C24 | 1 | 15 | yes | — |
| 75 | C23×D53 | — | 4 | 60 | no | — |
| 76 | C23×D53 | C24 | 1 | 15 | yes | — |
| 77 | C23×D53 | C24 | 1 | 120 | yes | — |
| 78 | C23×D53 | C24 | 1 | 45 | yes | — |
| 79 | #41×C53 | C24 | 1 | 160 | yes | complemented |
| 80 | #41×C53 | C24 | 1 | 480 | yes | complemented |
| 81 | C2×Q8×C53 | C24 | 1 | 20 | yes | — |
| 82 | C2×Q8×C53 | C24 | 1 | 120 | yes | complemented |
| 83 | C2×D4×C53 | C24 | 1 | 60 | yes | — |
| 84 | C2×D4×C53 | C24 | 1 | 360 | yes | complemented |
| 85 | C22×C4×C53 | C24 | 1 | 15 | yes | — |
| 86 | C22×C4×C53 | C24 | 1 | 45 | yes | — |
| 87 | C22×C4×C53 | C24 | 1 | 120 | yes | complemented |
| 88 | C2×C22⋊D53 | C24 | 1 | 120 | yes | — |
| 89 | C2×C22⋊D53 | C24 | 1 | 720 | yes | complemented |
| 90 | C22×Dic53 | C24 | 1 | 15 | yes | — |
| 91 | C22×Dic53 | C24 | 1 | 45 | yes | — |
| 92 | C22×Dic53 | C24 | 1 | 120 | yes | complemented |
| 93 | #11719949 | C24 | 1 | 160 | yes | complemented |
| 94 | #11719949 | C24 | 1 | 480 | yes | complemented |
| 95 | Q8×D53 | C24 | 1 | 160 | yes | — |
| 96 | Q8×D53 | C24 | 1 | 480 | yes | complemented |
| 97 | #11719947 | C24 | 1 | 480 | yes | complemented |
| 98 | #11719947 | C24 | 1 | 1440 | yes | complemented |
| 99 | D4×D53 | C24 | 1 | 480 | yes | — |
| 100 | D4×D53 | C24 | 1 | 1440 | yes | complemented |
| 101 | #11719945 | C24 | 1 | 480 | yes | complemented |
| 102 | #11719945 | C24 | 1 | 1440 | yes | complemented |
| 103 | C2×D212 | C24 | 1 | 60 | yes | — |
| 104 | C2×D212 | C24 | 1 | 360 | yes | complemented |
| 105 | C2×C4×D53 | C24 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×D53 | C24 | 1 | 15 | yes | — |
| 107 | C2×C4×D53 | C24 | 1 | 120 | yes | — |
| 108 | C2×C4×D53 | C24 | 1 | 15 | yes | — |
| 109 | C2×C4×D53 | C24 | 1 | 15 | yes | — |
| 110 | C2×C4×D53 | C24 | 1 | 360 | yes | — |
| 111 | C2×C4×D53 | C24 | 1 | 90 | yes | — |
| 112 | C2×C4×D53 | C24 | 1 | 360 | yes | complemented |
| 113 | C2×C4×D53 | C24 | 1 | 360 | yes | complemented |
| 114 | C2×Dic106 | C24 | 1 | 60 | yes | — |
| 115 | C2×Dic106 | C24 | 1 | 360 | yes | complemented |
| 116 | C23×C53 | — | 4 | 30 | no | — |
| 117 | C23×C53 | C25 | 1 | 15 | yes | — |
| 118 | C23×C53 | C25 | 1 | 60 | yes | — |
| 119 | C22×D53 | — | 2 | 60 | no | — |
| 120 | C22×D53 | — | 4 | 240 | no | — |
| 121 | C22×D53 | — | 4 | 90 | no | — |
| 122 | C22×D53 | C25 | 1 | 15 | yes | — |
| 123 | C22×D53 | C25 | 1 | 60 | yes | — |
| 124 | C22×D53 | C25 | 1 | 180 | yes | — |
| 125 | C22×D53 | C25 | 1 | 1 | yes | characteristic |
| 126 | Q8×C53 | C25 | 1 | 80 | yes | — |
| 127 | D4×C53 | C25 | 1 | 240 | yes | — |
| 128 | C2×C4×C53 | C25 | 1 | 15 | yes | — |
| 129 | C2×C4×C53 | C25 | 1 | 180 | yes | — |
| 130 | C2×C4×C53 | C25 | 1 | 60 | yes | — |
| 131 | C2×C4×C53 | C25 | 1 | 1 | yes | characteristic |
| 132 | C22⋊D53 | C25 | 1 | 480 | yes | — |
| 133 | C2×Dic53 | C25 | 1 | 15 | yes | — |
| 134 | C2×Dic53 | C25 | 1 | 180 | yes | — |
| 135 | C2×Dic53 | C25 | 1 | 1 | yes | characteristic |
| 136 | C2×Dic53 | C25 | 1 | 60 | yes | — |
| 137 | D212 | C25 | 1 | 240 | yes | — |
| 138 | C4×D53 | C25 | 1 | 4 | yes | — |
| 139 | C4×D53 | C25 | 1 | 60 | yes | — |
| 140 | C4×D53 | C25 | 1 | 60 | yes | — |
| 141 | C4×D53 | C25 | 1 | 60 | yes | — |
| 142 | C4×D53 | C25 | 1 | 360 | yes | — |
| 143 | Dic106 | C25 | 1 | 240 | yes | — |
| 144 | C22×#585 | — | 53 | 1 | no | maximal, Sylow |
| 145 | C22×C53 | — | 2 | 30 | no | — |
| 146 | C22×C53 | — | 4 | 60 | no | — |
| 147 | C22×C53 | C26 | 1 | 30 | yes | — |
| 148 | C22×C53 | C26 | 1 | 1 | yes | characteristic |
| 149 | C2×D53 | — | 2 | 30 | no | — |
| 150 | C2×D53 | — | 2 | 120 | no | — |
| 151 | C2×D53 | — | 4 | 180 | no | — |
| 152 | C2×D53 | #585 | 1 | 4 | yes | direct product factor |
| 153 | C2×D53 | C26 | 1 | 30 | yes | — |
| 154 | C2×D53 | C26 | 1 | 2 | yes | — |
| 155 | C4×C53 | C26 | 1 | 30 | yes | — |
| 156 | C4×C53 | C26 | 1 | 2 | yes | — |
| 157 | Dic53 | C26 | 1 | 30 | yes | — |
| 158 | Dic53 | C26 | 1 | 2 | yes | — |
| 159 | C2×#585 | — | 53 | 64 | no | — |
| 160 | C2×#585 | — | 53 | 6 | no | — |
| 161 | C2×#585 | — | 53 | 10 | no | — |
| 162 | C2×#585 | — | 53 | 15 | no | — |
| 163 | C2×#585 | D53 | 1 | 1 | yes | characteristic, direct product factor |
| 164 | C22×2−1+4 | — | 53 | 6 | no | — |
| 165 | C22×H(5,𝔽2) | — | 53 | 10 | no | — |
| 166 | C23×#41 | — | 53 | 15 | no | — |
| 167 | C2×C53 | — | 2 | 30 | no | — |
| 168 | C2×C53 | C2×#585 | 1 | 2 | yes | complemented |
| 169 | C2×C53 | C27 | 1 | 1 | yes | characteristic |
| 170 | D53 | — | 2 | 30 | no | — |
| 171 | D53 | C2×#585 | 1 | 4 | yes | direct product factor |
| 172 | #585 | — | 53 | 192 | no | — |
| 173 | #585 | — | 53 | 320 | no | — |
| 174 | #585 | — | 53 | 480 | no | — |
| 175 | #585 | C2×D53 | 1 | 32 | yes | direct product factor |
| 176 | C2×2−1+4 | — | 53 | 192 | no | — |
| 177 | C2×2−1+4 | — | 53 | 60 | no | — |
| 178 | C2×2−1+4 | — | 53 | 30 | no | — |
| 179 | C2×2−1+4 | C2×D53 | 1 | 6 | yes | complemented |
| 180 | C2×H(5,𝔽2) | — | 53 | 320 | no | — |
| 181 | C2×H(5,𝔽2) | — | 53 | 60 | no | — |
| 182 | C2×H(5,𝔽2) | — | 53 | 90 | no | — |
| 183 | C2×H(5,𝔽2) | C2×D53 | 1 | 10 | yes | complemented |
| 184 | C22×#41 | — | 53 | 20 | no | — |
| 185 | C22×#41 | — | 53 | 60 | no | — |
| 186 | C22×#41 | — | 53 | 60 | no | — |
| 187 | C22×#41 | — | 53 | 60 | no | — |
| 188 | C22×#41 | — | 53 | 480 | no | — |
| 189 | C22×#41 | — | 53 | 15 | no | — |
| 190 | C22×#41 | — | 53 | 45 | no | — |
| 191 | C22×#41 | — | 53 | 45 | no | — |
| 192 | C22×#41 | C2×D53 | 1 | 15 | yes | complemented |
| 193 | C23×Q8 | — | 53 | 15 | no | — |
| 194 | C23×D4 | — | 53 | 45 | no | — |
| 195 | C24×C4 | — | 53 | 15 | no | — |
| 196 | C53 | C22×#585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 197 | C25 | — | 53 | 15 | no | — |
| 198 | 2−1+4 | — | 53 | 960 | no | — |
| 199 | 2−1+4 | — | 53 | 480 | no | — |
| 200 | 2−1+4 | C22×D53 | 1 | 96 | yes | complemented |
| 201 | H(5,𝔽2) | — | 53 | 960 | no | — |
| 202 | H(5,𝔽2) | — | 53 | 1440 | no | — |
| 203 | H(5,𝔽2) | C22×D53 | 1 | 160 | yes | complemented |
| 204 | C2×#41 | — | 53 | 320 | no | — |
| 205 | C2×#41 | — | 53 | 960 | no | — |
| 206 | C2×#41 | — | 53 | 960 | no | — |
| 207 | C2×#41 | — | 53 | 960 | no | — |
| 208 | C2×#41 | — | 53 | 20 | no | — |
| 209 | C2×#41 | — | 53 | 60 | no | — |
| 210 | C2×#41 | — | 53 | 240 | no | — |
| 211 | C2×#41 | — | 53 | 60 | no | — |
| 212 | C2×#41 | — | 53 | 180 | no | — |
| 213 | C2×#41 | — | 53 | 720 | no | — |
| 214 | C2×#41 | — | 53 | 60 | no | — |
| 215 | C2×#41 | — | 53 | 180 | no | — |
| 216 | C2×#41 | — | 53 | 720 | no | — |
| 217 | C2×#41 | C22×D53 | 1 | 20 | yes | complemented |
| 218 | C2×#41 | C22×D53 | 1 | 60 | yes | complemented |
| 219 | C2×#41 | C22×D53 | 1 | 240 | yes | complemented |
| 220 | C22×Q8 | — | 53 | 20 | no | — |
| 221 | C22×Q8 | — | 53 | 60 | no | — |
| 222 | C22×Q8 | — | 53 | 240 | no | — |
| 223 | C22×Q8 | — | 53 | 45 | no | — |
| 224 | C22×Q8 | C22×D53 | 1 | 15 | yes | complemented |
| 225 | C22×D4 | — | 53 | 90 | no | — |
| 226 | C22×D4 | — | 53 | 60 | no | — |
| 227 | C22×D4 | — | 53 | 180 | no | — |
| 228 | C22×D4 | — | 53 | 720 | no | — |
| 229 | C22×D4 | — | 53 | 45 | no | — |
| 230 | C22×D4 | C22×D53 | 1 | 45 | yes | complemented |
| 231 | C23×C4 | — | 53 | 15 | no | — |
| 232 | C23×C4 | — | 53 | 15 | no | — |
| 233 | C23×C4 | — | 53 | 45 | no | — |
| 234 | C23×C4 | — | 53 | 45 | no | — |
| 235 | C23×C4 | — | 53 | 45 | no | — |
| 236 | C23×C4 | — | 53 | 240 | no | — |
| 237 | C23×C4 | C22×D53 | 1 | 15 | yes | complemented |
| 238 | C24 | — | 53 | 15 | no | — |
| 239 | C24 | — | 53 | 120 | no | — |
| 240 | C24 | — | 53 | 45 | no | — |
| 241 | C24 | — | 212 | 60 | no | — |
| 242 | C24 | C23×D53 | 1 | 15 | yes | — |
| 243 | #41 | — | 53 | 160 | no | — |
| 244 | #41 | — | 53 | 480 | no | — |
| 245 | #41 | — | 53 | 480 | no | — |
| 246 | #41 | — | 53 | 1440 | no | — |
| 247 | #41 | — | 53 | 480 | no | — |
| 248 | #41 | — | 53 | 1440 | no | — |
| 249 | #41 | C23×D53 | 1 | 160 | yes | complemented |
| 250 | #41 | C23×D53 | 1 | 480 | yes | complemented |
| 251 | C2×Q8 | — | 53 | 160 | no | — |
| 252 | C2×Q8 | — | 53 | 480 | no | — |
| 253 | C2×Q8 | — | 53 | 60 | no | — |
| 254 | C2×Q8 | — | 53 | 360 | no | — |
| 255 | C2×Q8 | C23×D53 | 1 | 20 | yes | — |
| 256 | C2×Q8 | C23×D53 | 1 | 120 | yes | complemented |
| 257 | C2×D4 | — | 53 | 120 | no | — |
| 258 | C2×D4 | — | 53 | 720 | no | — |
| 259 | C2×D4 | — | 53 | 480 | no | — |
| 260 | C2×D4 | — | 53 | 1440 | no | — |
| 261 | C2×D4 | — | 53 | 60 | no | — |
| 262 | C2×D4 | — | 53 | 360 | no | — |
| 263 | C2×D4 | C23×D53 | 1 | 60 | yes | — |
| 264 | C2×D4 | C23×D53 | 1 | 360 | yes | complemented |
| 265 | C22×C4 | — | 53 | 15 | no | — |
| 266 | C22×C4 | — | 53 | 45 | no | — |
| 267 | C22×C4 | — | 53 | 120 | no | — |
| 268 | C22×C4 | — | 53 | 15 | no | — |
| 269 | C22×C4 | — | 53 | 15 | no | — |
| 270 | C22×C4 | — | 53 | 120 | no | — |
| 271 | C22×C4 | — | 53 | 90 | no | — |
| 272 | C22×C4 | — | 53 | 360 | no | — |
| 273 | C22×C4 | — | 53 | 360 | no | — |
| 274 | C22×C4 | — | 53 | 360 | no | — |
| 275 | C22×C4 | — | 53 | 1 | no | — |
| 276 | C22×C4 | — | 53 | 15 | no | — |
| 277 | C22×C4 | C23×D53 | 1 | 45 | yes | — |
| 278 | C22×C4 | C23×D53 | 1 | 15 | yes | — |
| 279 | C22×C4 | C23×D53 | 1 | 120 | yes | complemented |
| 280 | C23 | — | 4 | 30 | no | — |
| 281 | C23 | — | 53 | 1 | no | — |
| 282 | C23 | — | 53 | 60 | no | — |
| 283 | C23 | — | 53 | 15 | no | — |
| 284 | C23 | — | 53 | 180 | no | — |
| 285 | C23 | — | 106 | 60 | no | — |
| 286 | C23 | — | 212 | 90 | no | — |
| 287 | C23 | — | 212 | 240 | no | — |
| 288 | C23 | C24×D53 | 1 | 15 | yes | — |
| 289 | C23 | C24×D53 | 1 | 60 | yes | — |
| 290 | Q8 | — | 53 | 240 | no | — |
| 291 | Q8 | C24×D53 | 1 | 80 | yes | — |
| 292 | D4 | — | 53 | 480 | no | — |
| 293 | D4 | — | 53 | 240 | no | — |
| 294 | D4 | C24×D53 | 1 | 240 | yes | — |
| 295 | C2×C4 | — | 53 | 1 | no | — |
| 296 | C2×C4 | — | 53 | 15 | no | — |
| 297 | C2×C4 | — | 53 | 60 | no | — |
| 298 | C2×C4 | — | 53 | 180 | no | — |
| 299 | C2×C4 | — | 53 | 4 | no | — |
| 300 | C2×C4 | — | 53 | 60 | no | — |
| 301 | C2×C4 | — | 53 | 60 | no | — |
| 302 | C2×C4 | — | 53 | 60 | no | — |
| 303 | C2×C4 | — | 53 | 360 | no | — |
| 304 | C2×C4 | C24×D53 | 1 | 15 | yes | — |
| 305 | C2×C4 | C24×D53 | 1 | 60 | yes | — |
| 306 | C2×C4 | C24×D53 | 1 | 180 | yes | — |
| 307 | C2×C4 | C24×D53 | 1 | 1 | yes | characteristic |
| 308 | C22 | — | 2 | 30 | no | — |
| 309 | C22 | — | 4 | 60 | no | — |
| 310 | C22 | — | 53 | 4 | no | — |
| 311 | C22 | — | 53 | 2 | no | — |
| 312 | C22 | — | 53 | 30 | no | — |
| 313 | C22 | — | 106 | 120 | no | — |
| 314 | C22 | — | 106 | 30 | no | — |
| 315 | C22 | — | 212 | 180 | no | — |
| 316 | C22 | C25×D53 | 1 | 30 | yes | — |
| 317 | C22 | C25×D53 | 1 | 1 | yes | characteristic |
| 318 | C4 | — | 53 | 2 | no | — |
| 319 | C4 | — | 53 | 30 | no | — |
| 320 | C4 | C25×D53 | 1 | 2 | yes | — |
| 321 | C4 | C25×D53 | 1 | 30 | yes | — |
| 322 | C2 | — | 2 | 30 | no | — |
| 323 | C2 | — | 53 | 4 | no | — |
| 324 | C2 | — | 106 | 30 | no | — |
| 325 | C2 | #585×D53 | 1 | 2 | yes | direct product factor |
| 326 | C2 | C26×D53 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta