← C53×#8685 #27216197
Subgroups — C53×#8685 #27216197
Order 13568.
362 conjugacy classes of subgroups.
The subgroup lattice has 268 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#2960 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#1848 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C53×#335 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53×#1550 | C2 | 1 | 2 | yes | maximal |
| 5 | C53×#1292 | C2 | 1 | 2 | yes | maximal |
| 6 | C53×#533 | C22 | 1 | 1 | yes | characteristic |
| 7 | C53×#446 | — | 2 | 1 | no | — |
| 8 | C53×#446 | C4 | 1 | 2 | yes | — |
| 9 | C8×Q8×C53 | — | 2 | 1 | no | — |
| 10 | C8×Q8×C53 | C4 | 1 | 2 | yes | — |
| 11 | C2×#108×C53 | — | 2 | 1 | no | — |
| 12 | C2×#108×C53 | — | 2 | 1 | no | — |
| 13 | C53×#424 | — | 2 | 1 | no | — |
| 14 | C53×#423 | C22 | 1 | 2 | yes | — |
| 15 | C53×#423 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C4⋊C8×C53 | C22 | 1 | 2 | yes | — |
| 17 | C53×#405 | — | 2 | 1 | no | — |
| 18 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 19 | C53×#335 | — | 2 | 1 | no | — |
| 20 | C53×#335 | — | 2 | 1 | no | — |
| 21 | C53×#333 | — | 2 | 2 | no | — |
| 22 | C53×#332 | — | 2 | 2 | no | — |
| 23 | C2×M16×C53 | — | 2 | 1 | no | — |
| 24 | C2×M16×C53 | — | 2 | 2 | no | — |
| 25 | C22×C8×C53 | — | 2 | 2 | no | — |
| 26 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 28 | C4⋊Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 29 | C4⋊Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 30 | #125×C53 | — | 2 | 1 | no | — |
| 31 | C22⋊Q8×C53 | — | 2 | 1 | no | — |
| 32 | C22⋊Q8×C53 | Q8 | 1 | 2 | yes | — |
| 33 | C4×Q8×C53 | — | 2 | 1 | no | — |
| 34 | C4×Q8×C53 | C2×C4 | 1 | 2 | yes | — |
| 35 | #117×C53 | — | 2 | 1 | no | — |
| 36 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 37 | #108×C53 | — | 4 | 2 | no | — |
| 38 | #108×C53 | — | 4 | 2 | no | — |
| 39 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 40 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 41 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 42 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 43 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 44 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 45 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 46 | C4⋊C8×C53 | C2×C4 | 1 | 2 | yes | — |
| 47 | #97×C53 | — | 2 | 2 | no | — |
| 48 | C4×C8×C53 | — | 2 | 2 | no | — |
| 49 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 52 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 54 | C2×Q8×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 55 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | C22×C4×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 57 | C22×C4×C53 | #41 | 1 | 1 | yes | characteristic |
| 58 | M16×C53 | — | 4 | 1 | no | — |
| 59 | M16×C53 | — | 4 | 1 | no | — |
| 60 | M16×C53 | — | 4 | 2 | no | — |
| 61 | M16×C53 | — | 4 | 2 | no | — |
| 62 | C2×C8×C53 | — | 2 | 1 | no | — |
| 63 | C2×C8×C53 | — | 2 | 1 | no | — |
| 64 | C2×C8×C53 | — | 2 | 2 | no | — |
| 65 | C2×C8×C53 | — | 4 | 2 | no | — |
| 66 | C2×C8×C53 | — | 4 | 2 | no | — |
| 67 | C2×C8×C53 | — | 8 | 2 | no | — |
| 68 | C2×C8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C2×C8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C53 | SD16 | 1 | 2 | yes | — |
| 73 | C2×C8×C53 | Q16 | 1 | 2 | yes | — |
| 74 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 75 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 76 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 77 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 78 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 79 | C4⋊C4×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 80 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 81 | C42×C53 | — | 2 | 2 | no | — |
| 82 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 86 | C23×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 87 | Q8×C53 | — | 4 | 2 | no | — |
| 88 | Q8×C53 | — | 4 | 1 | no | — |
| 89 | Q8×C53 | — | 4 | 1 | no | — |
| 90 | C2×C4×C53 | — | 2 | 1 | no | — |
| 91 | C2×C4×C53 | — | 2 | 1 | no | — |
| 92 | C2×C4×C53 | — | 2 | 2 | no | — |
| 93 | C2×C4×C53 | — | 2 | 2 | no | — |
| 94 | C2×C4×C53 | — | 4 | 1 | no | — |
| 95 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C53 | #117 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 99 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 100 | C2×C4×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C53 | C2×SD16 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C53 | C2×Q16 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C53 | #137 | 1 | 1 | yes | characteristic |
| 105 | C8×C53 | — | 2 | 1 | no | — |
| 106 | C8×C53 | — | 2 | 1 | no | — |
| 107 | C8×C53 | — | 4 | 2 | no | — |
| 108 | C8×C53 | — | 8 | 2 | no | — |
| 109 | C8×C53 | — | 8 | 2 | no | — |
| 110 | C8×C53 | #103 | 1 | 2 | yes | — |
| 111 | C8×C53 | #103 | 1 | 2 | yes | — |
| 112 | #8685 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 113 | C22×C53 | — | 2 | 1 | no | — |
| 114 | C22×C53 | — | 2 | 1 | no | — |
| 115 | C22×C53 | #380 | 1 | 1 | yes | characteristic |
| 116 | C22×C53 | #475 | 1 | 1 | yes | characteristic |
| 117 | C22×C53 | #477 | 1 | 1 | yes | characteristic |
| 118 | C4×C53 | — | 2 | 1 | no | — |
| 119 | C4×C53 | — | 2 | 1 | no | — |
| 120 | C4×C53 | — | 2 | 1 | no | — |
| 121 | C4×C53 | — | 4 | 2 | no | — |
| 122 | C4×C53 | — | 4 | 2 | no | — |
| 123 | C4×C53 | C2×#103 | 1 | 1 | yes | characteristic |
| 124 | C4×C53 | #417 | 1 | 1 | yes | characteristic |
| 125 | C4×C53 | #460 | 1 | 1 | yes | characteristic |
| 126 | C4×C53 | #462 | 1 | 1 | yes | characteristic |
| 127 | C4×C53 | #472 | 1 | 1 | yes | characteristic |
| 128 | C4×C53 | #473 | 1 | 1 | yes | characteristic |
| 129 | #2960 | C2×C53 | 1 | 1 | yes | characteristic |
| 130 | #1848 | C2×C53 | 1 | 1 | yes | characteristic |
| 131 | C2×#335 | C2×C53 | 1 | 1 | yes | characteristic |
| 132 | #1550 | C2×C53 | 1 | 2 | yes | — |
| 133 | #1292 | C2×C53 | 1 | 2 | yes | — |
| 134 | C2×C53 | — | 2 | 1 | no | — |
| 135 | C2×C53 | — | 2 | 1 | no | — |
| 136 | C2×C53 | #1803 | 1 | 1 | yes | characteristic |
| 137 | C2×C53 | #1813 | 1 | 1 | yes | characteristic |
| 138 | C2×C53 | #1816 | 1 | 1 | yes | characteristic |
| 139 | #533 | C22×C53 | 1 | 1 | yes | characteristic |
| 140 | #446 | — | 2 | 1 | no | — |
| 141 | #446 | C4×C53 | 1 | 2 | yes | — |
| 142 | C8×Q8 | — | 2 | 1 | no | — |
| 143 | C8×Q8 | C4×C53 | 1 | 2 | yes | — |
| 144 | C2×#108 | — | 2 | 1 | no | — |
| 145 | C2×#108 | — | 2 | 1 | no | — |
| 146 | #424 | — | 2 | 1 | no | — |
| 147 | #423 | C22×C53 | 1 | 2 | yes | — |
| 148 | #423 | C22×C53 | 1 | 1 | yes | characteristic |
| 149 | C2×C4⋊C8 | C22×C53 | 1 | 2 | yes | — |
| 150 | #405 | — | 2 | 1 | no | — |
| 151 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 152 | #335 | — | 2 | 1 | no | — |
| 153 | #335 | — | 2 | 1 | no | — |
| 154 | #333 | — | 2 | 2 | no | — |
| 155 | #332 | — | 2 | 2 | no | — |
| 156 | C53 | #8685 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 157 | C2×M16 | — | 2 | 1 | no | — |
| 158 | C2×M16 | — | 2 | 2 | no | — |
| 159 | C22×C8 | — | 2 | 2 | no | — |
| 160 | C22×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 161 | C22×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 162 | C4⋊Q8 | D4×C53 | 1 | 1 | yes | characteristic |
| 163 | C4⋊Q8 | D4×C53 | 1 | 1 | yes | characteristic |
| 164 | #125 | — | 2 | 1 | no | — |
| 165 | C22⋊Q8 | — | 2 | 1 | no | — |
| 166 | C22⋊Q8 | Q8×C53 | 1 | 2 | yes | — |
| 167 | C4×Q8 | — | 2 | 1 | no | — |
| 168 | C4×Q8 | C2×C4×C53 | 1 | 2 | yes | — |
| 169 | #117 | — | 2 | 1 | no | — |
| 170 | C2×C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 171 | #108 | — | 4 | 2 | no | — |
| 172 | #108 | — | 4 | 2 | no | — |
| 173 | C4⋊C8 | — | 2 | 2 | no | — |
| 174 | C4⋊C8 | — | 2 | 2 | no | — |
| 175 | C4⋊C8 | — | 2 | 1 | no | — |
| 176 | C4⋊C8 | — | 2 | 2 | no | — |
| 177 | C4⋊C8 | — | 2 | 2 | no | — |
| 178 | C4⋊C8 | — | 2 | 2 | no | — |
| 179 | C4⋊C8 | — | 2 | 2 | no | — |
| 180 | C4⋊C8 | C2×C4×C53 | 1 | 2 | yes | — |
| 181 | #97 | — | 2 | 2 | no | — |
| 182 | C4×C8 | — | 2 | 2 | no | — |
| 183 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 184 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 185 | C4×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 187 | C2×Q8 | — | 2 | 1 | no | — |
| 188 | C2×Q8 | C4⋊C4×C53 | 1 | 2 | yes | — |
| 189 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 190 | C22×C4 | C2×Q8×C53 | 1 | 1 | yes | characteristic |
| 191 | C22×C4 | #41×C53 | 1 | 1 | yes | characteristic |
| 192 | M16 | — | 4 | 1 | no | — |
| 193 | M16 | — | 4 | 1 | no | — |
| 194 | M16 | — | 4 | 2 | no | — |
| 195 | M16 | — | 4 | 2 | no | — |
| 196 | C2×C8 | — | 2 | 1 | no | — |
| 197 | C2×C8 | — | 2 | 1 | no | — |
| 198 | C2×C8 | — | 2 | 2 | no | — |
| 199 | C2×C8 | — | 4 | 2 | no | — |
| 200 | C2×C8 | — | 4 | 2 | no | — |
| 201 | C2×C8 | — | 8 | 2 | no | — |
| 202 | C2×C8 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 203 | C2×C8 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 204 | C2×C8 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | SD16×C53 | 1 | 2 | yes | — |
| 207 | C2×C8 | Q16×C53 | 1 | 2 | yes | — |
| 208 | C4⋊C4 | — | 2 | 2 | no | — |
| 209 | C4⋊C4 | — | 2 | 1 | no | — |
| 210 | C4⋊C4 | — | 2 | 1 | no | — |
| 211 | C4⋊C4 | — | 2 | 1 | no | — |
| 212 | C4⋊C4 | — | 2 | 2 | no | — |
| 213 | C4⋊C4 | C4⋊C4×C53 | 1 | 2 | yes | — |
| 214 | C22⋊C4 | — | 2 | 2 | no | — |
| 215 | C42 | — | 2 | 2 | no | — |
| 216 | C42 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 217 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 218 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 219 | C42 | #41×C53 | 1 | 1 | yes | characteristic |
| 220 | C23 | C22⋊Q8×C53 | 1 | 1 | yes | characteristic |
| 221 | Q8 | — | 4 | 2 | no | — |
| 222 | Q8 | — | 4 | 1 | no | — |
| 223 | Q8 | — | 4 | 1 | no | — |
| 224 | C2×C4 | — | 2 | 1 | no | — |
| 225 | C2×C4 | — | 2 | 1 | no | — |
| 226 | C2×C4 | — | 2 | 2 | no | — |
| 227 | C2×C4 | — | 2 | 2 | no | — |
| 228 | C2×C4 | — | 4 | 1 | no | — |
| 229 | C2×C4 | C2×C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | C2×C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | #117×C53 | 1 | 1 | yes | characteristic |
| 232 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 233 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | C22⋊Q8×C53 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | C2×SD16×C53 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | C2×Q16×C53 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | AGL(1,ℤ/8)×C53 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #137×C53 | 1 | 1 | yes | characteristic |
| 239 | C8 | — | 2 | 1 | no | — |
| 240 | C8 | — | 2 | 1 | no | — |
| 241 | C8 | — | 4 | 2 | no | — |
| 242 | C8 | — | 8 | 2 | no | — |
| 243 | C8 | — | 8 | 2 | no | — |
| 244 | C8 | #103×C53 | 1 | 2 | yes | — |
| 245 | C8 | #103×C53 | 1 | 2 | yes | — |
| 246 | C22 | — | 2 | 1 | no | — |
| 247 | C22 | — | 2 | 1 | no | — |
| 248 | C22 | C53×#380 | 1 | 1 | yes | characteristic |
| 249 | C22 | C53×#475 | 1 | 1 | yes | characteristic |
| 250 | C22 | C53×#477 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 1 | no | — |
| 252 | C4 | — | 2 | 1 | no | — |
| 253 | C4 | — | 2 | 1 | no | — |
| 254 | C4 | — | 4 | 2 | no | — |
| 255 | C4 | — | 4 | 2 | no | — |
| 256 | C4 | C2×#103×C53 | 1 | 1 | yes | characteristic |
| 257 | C4 | C53×#417 | 1 | 1 | yes | characteristic |
| 258 | C4 | C53×#460 | 1 | 1 | yes | characteristic |
| 259 | C4 | C53×#462 | 1 | 1 | yes | characteristic |
| 260 | C4 | C53×#472 | 1 | 1 | yes | characteristic |
| 261 | C4 | C53×#473 | 1 | 1 | yes | characteristic |
| 262 | C2 | — | 2 | 1 | no | — |
| 263 | C2 | — | 2 | 1 | no | — |
| 264 | C2 | C53×#1803 | 1 | 1 | yes | characteristic |
| 265 | C2 | C53×#1813 | 1 | 1 | yes | characteristic |
| 266 | C2 | C53×#1816 | 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