← C53×#10298 #27217810
Subgroups — C53×#10298 #27217810
Order 13568.
516 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 | C53×#3229 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#3209 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C53×#2862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53×#1994 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C53×#1548 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C53×#1546 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#1276 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C53×#533 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4⋊Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4⋊Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 11 | C53×#498 | — | 2 | 1 | no | — |
| 12 | C53×#494 | — | 2 | 1 | no | — |
| 13 | C53×#487 | — | 2 | 1 | no | — |
| 14 | C53×#487 | — | 2 | 1 | no | — |
| 15 | C53×#483 | — | 2 | 2 | no | — |
| 16 | C53×#475 | — | 2 | 2 | no | — |
| 17 | C53×#462 | — | 2 | 2 | no | — |
| 18 | C53×#451 | — | 2 | 2 | no | — |
| 19 | C53×#431 | C22 | 1 | 1 | yes | characteristic |
| 20 | C53×#431 | C22 | 1 | 1 | yes | characteristic |
| 21 | C53×#423 | C22 | 1 | 1 | yes | characteristic |
| 22 | C53×#391 | — | 2 | 2 | no | — |
| 23 | C53×#391 | — | 2 | 2 | no | — |
| 24 | C53×#382 | — | 2 | 2 | no | — |
| 25 | C43×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×#339 | — | 2 | 2 | no | — |
| 27 | C53×#333 | — | 2 | 2 | no | — |
| 28 | C53×#332 | — | 2 | 2 | no | — |
| 29 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 30 | C22×Q8×C53 | D4 | 1 | 2 | yes | — |
| 31 | C2×Q16×C53 | — | 2 | 4 | no | — |
| 32 | C2×M16×C53 | — | 2 | 1 | no | — |
| 33 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 34 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 35 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 36 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 37 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 38 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 39 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 40 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 41 | C4⋊Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 42 | C4⋊Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 43 | #125×C53 | — | 2 | 1 | no | — |
| 44 | C22⋊Q8×C53 | — | 2 | 1 | no | — |
| 45 | C22⋊Q8×C53 | — | 2 | 1 | no | — |
| 46 | C4×Q8×C53 | — | 2 | 1 | no | — |
| 47 | C4×Q8×C53 | — | 2 | 1 | no | — |
| 48 | #117×C53 | — | 2 | 1 | no | — |
| 49 | C2×C4⋊C4×C53 | — | 2 | 1 | no | — |
| 50 | C2×C4⋊C4×C53 | — | 2 | 1 | no | — |
| 51 | C2×C4⋊C4×C53 | — | 2 | 1 | no | — |
| 52 | C2×C4⋊C4×C53 | D4 | 1 | 2 | yes | — |
| 53 | C2×C42×C53 | — | 2 | 1 | no | — |
| 54 | C2×C42×C53 | — | 2 | 1 | no | — |
| 55 | C2×C42×C53 | D4 | 1 | 2 | yes | — |
| 56 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 57 | #106×C53 | — | 2 | 4 | no | — |
| 58 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 59 | C4⋊C8×C53 | — | 2 | 1 | no | — |
| 60 | C4⋊C8×C53 | D4 | 1 | 2 | yes | — |
| 61 | C4⋊C8×C53 | D4 | 1 | 2 | yes | — |
| 62 | #103×C53 | — | 2 | 4 | no | — |
| 63 | #103×C53 | — | 2 | 4 | no | — |
| 64 | C22⋊C8×C53 | — | 2 | 1 | no | — |
| 65 | C22⋊C8×C53 | — | 2 | 1 | no | — |
| 66 | C4×C8×C53 | — | 2 | 1 | no | — |
| 67 | C4×C8×C53 | — | 2 | 1 | no | — |
| 68 | #95×C53 | — | 2 | 8 | no | — |
| 69 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 70 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 71 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 72 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 73 | C2×Q8×C53 | — | 4 | 4 | no | — |
| 74 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 75 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 76 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 77 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 78 | C22×C4×C53 | — | 2 | 2 | no | — |
| 79 | C22×C4×C53 | — | 2 | 2 | no | — |
| 80 | C22×C4×C53 | — | 2 | 2 | no | — |
| 81 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | Q16×C53 | — | 8 | 4 | no | — |
| 85 | M16×C53 | — | 4 | 2 | no | — |
| 86 | C2×C8×C53 | — | 2 | 2 | no | — |
| 87 | C2×C8×C53 | — | 2 | 2 | no | — |
| 88 | C2×C8×C53 | — | 4 | 1 | no | — |
| 89 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 90 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 91 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 92 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 93 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 94 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 95 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 96 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 97 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 98 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 99 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 100 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 101 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 102 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 103 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 104 | C42×C53 | — | 2 | 4 | no | — |
| 105 | C42×C53 | — | 2 | 2 | no | — |
| 106 | C42×C53 | — | 2 | 2 | no | — |
| 107 | C42×C53 | — | 4 | 2 | no | — |
| 108 | C42×C53 | — | 4 | 1 | no | — |
| 109 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 113 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 114 | Q8×C53 | — | 4 | 4 | no | — |
| 115 | Q8×C53 | — | 4 | 2 | no | — |
| 116 | Q8×C53 | — | 4 | 2 | no | — |
| 117 | Q8×C53 | — | 8 | 2 | no | — |
| 118 | Q8×C53 | — | 8 | 2 | no | — |
| 119 | Q8×C53 | — | 8 | 1 | no | — |
| 120 | Q8×C53 | — | 8 | 1 | no | — |
| 121 | C2×C4×C53 | — | 2 | 2 | no | — |
| 122 | C2×C4×C53 | — | 2 | 2 | no | — |
| 123 | C2×C4×C53 | — | 2 | 4 | no | — |
| 124 | C2×C4×C53 | — | 4 | 2 | no | — |
| 125 | C2×C4×C53 | — | 4 | 2 | no | — |
| 126 | C2×C4×C53 | — | 4 | 4 | no | — |
| 127 | C2×C4×C53 | — | 4 | 2 | no | — |
| 128 | C2×C4×C53 | — | 4 | 2 | no | — |
| 129 | C2×C4×C53 | — | 4 | 1 | no | — |
| 130 | C2×C4×C53 | — | 4 | 1 | no | — |
| 131 | C2×C4×C53 | — | 4 | 4 | no | — |
| 132 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C53 | #127 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C53 | #135 | 1 | 2 | yes | — |
| 139 | C2×C4×C53 | #137 | 1 | 2 | yes | — |
| 140 | C8×C53 | — | 4 | 2 | no | — |
| 141 | C8×C53 | — | 4 | 2 | no | — |
| 142 | C8×C53 | — | 8 | 1 | no | — |
| 143 | #10298 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 144 | C22×C53 | — | 2 | 1 | no | — |
| 145 | C22×C53 | — | 2 | 1 | no | — |
| 146 | C22×C53 | #392 | 1 | 1 | yes | characteristic |
| 147 | C22×C53 | #452 | 1 | 2 | yes | — |
| 148 | C4×C53 | — | 2 | 1 | no | — |
| 149 | C4×C53 | — | 2 | 1 | no | — |
| 150 | C4×C53 | — | 4 | 4 | no | — |
| 151 | C4×C53 | — | 8 | 2 | no | — |
| 152 | C4×C53 | — | 8 | 1 | no | — |
| 153 | C4×C53 | — | 8 | 1 | no | — |
| 154 | C4×C53 | — | 8 | 2 | no | — |
| 155 | C4×C53 | #455 | 1 | 2 | yes | — |
| 156 | C4×C53 | #456 | 1 | 2 | yes | — |
| 157 | C4×C53 | #464 | 1 | 2 | yes | — |
| 158 | C4×C53 | #495 | 1 | 1 | yes | characteristic |
| 159 | C4×C53 | #497 | 1 | 1 | yes | characteristic |
| 160 | #3229 | C2×C53 | 1 | 1 | yes | characteristic |
| 161 | #3209 | C2×C53 | 1 | 1 | yes | characteristic |
| 162 | #2862 | C2×C53 | 1 | 1 | yes | characteristic |
| 163 | #1994 | C2×C53 | 1 | 1 | yes | characteristic |
| 164 | #1548 | C2×C53 | 1 | 1 | yes | characteristic |
| 165 | #1546 | C2×C53 | 1 | 1 | yes | characteristic |
| 166 | #1276 | C2×C53 | 1 | 1 | yes | characteristic |
| 167 | C2×C53 | — | 2 | 2 | no | — |
| 168 | C2×C53 | #2004 | 1 | 1 | yes | characteristic |
| 169 | C2×C53 | #2005 | 1 | 1 | yes | characteristic |
| 170 | C2×C53 | #2015 | 1 | 1 | yes | characteristic |
| 171 | #533 | C22×C53 | 1 | 1 | yes | characteristic |
| 172 | C2×C4⋊Q8 | C22×C53 | 1 | 1 | yes | characteristic |
| 173 | C2×C4⋊Q8 | C22×C53 | 1 | 1 | yes | characteristic |
| 174 | #498 | — | 2 | 1 | no | — |
| 175 | #494 | — | 2 | 1 | no | — |
| 176 | #487 | — | 2 | 1 | no | — |
| 177 | #487 | — | 2 | 1 | no | — |
| 178 | #483 | — | 2 | 2 | no | — |
| 179 | #475 | — | 2 | 2 | no | — |
| 180 | #462 | — | 2 | 2 | no | — |
| 181 | #451 | — | 2 | 2 | no | — |
| 182 | #431 | C22×C53 | 1 | 1 | yes | characteristic |
| 183 | #431 | C22×C53 | 1 | 1 | yes | characteristic |
| 184 | #423 | C22×C53 | 1 | 1 | yes | characteristic |
| 185 | #391 | — | 2 | 2 | no | — |
| 186 | #391 | — | 2 | 2 | no | — |
| 187 | #382 | — | 2 | 2 | no | — |
| 188 | C43 | C22×C53 | 1 | 1 | yes | characteristic |
| 189 | #339 | — | 2 | 2 | no | — |
| 190 | #333 | — | 2 | 2 | no | — |
| 191 | #332 | — | 2 | 2 | no | — |
| 192 | C53 | #10298 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 193 | C22×Q8 | — | 2 | 1 | no | — |
| 194 | C22×Q8 | D4×C53 | 1 | 2 | yes | — |
| 195 | C2×Q16 | — | 2 | 4 | no | — |
| 196 | C2×M16 | — | 2 | 1 | no | — |
| 197 | C4⋊Q8 | — | 2 | 1 | no | — |
| 198 | C4⋊Q8 | — | 2 | 1 | no | — |
| 199 | C4⋊Q8 | — | 2 | 1 | no | — |
| 200 | C4⋊Q8 | — | 2 | 1 | no | — |
| 201 | C4⋊Q8 | — | 2 | 1 | no | — |
| 202 | C4⋊Q8 | — | 2 | 1 | no | — |
| 203 | C4⋊Q8 | — | 2 | 1 | no | — |
| 204 | C4⋊Q8 | — | 2 | 1 | no | — |
| 205 | C4⋊Q8 | D4×C53 | 1 | 1 | yes | characteristic |
| 206 | C4⋊Q8 | D4×C53 | 1 | 1 | yes | characteristic |
| 207 | #125 | — | 2 | 1 | no | — |
| 208 | C22⋊Q8 | — | 2 | 1 | no | — |
| 209 | C22⋊Q8 | — | 2 | 1 | no | — |
| 210 | C4×Q8 | — | 2 | 1 | no | — |
| 211 | C4×Q8 | — | 2 | 1 | no | — |
| 212 | #117 | — | 2 | 1 | no | — |
| 213 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 214 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 215 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 216 | C2×C4⋊C4 | D4×C53 | 1 | 2 | yes | — |
| 217 | C2×C42 | — | 2 | 1 | no | — |
| 218 | C2×C42 | — | 2 | 1 | no | — |
| 219 | C2×C42 | D4×C53 | 1 | 2 | yes | — |
| 220 | C2×C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 221 | #106 | — | 2 | 4 | no | — |
| 222 | C4⋊C8 | — | 2 | 1 | no | — |
| 223 | C4⋊C8 | — | 2 | 1 | no | — |
| 224 | C4⋊C8 | D4×C53 | 1 | 2 | yes | — |
| 225 | C4⋊C8 | D4×C53 | 1 | 2 | yes | — |
| 226 | #103 | — | 2 | 4 | no | — |
| 227 | #103 | — | 2 | 4 | no | — |
| 228 | C22⋊C8 | — | 2 | 1 | no | — |
| 229 | C22⋊C8 | — | 2 | 1 | no | — |
| 230 | C4×C8 | — | 2 | 1 | no | — |
| 231 | C4×C8 | — | 2 | 1 | no | — |
| 232 | #95 | — | 2 | 8 | no | — |
| 233 | C2×Q8 | — | 2 | 1 | no | — |
| 234 | C2×Q8 | — | 2 | 2 | no | — |
| 235 | C2×Q8 | — | 2 | 1 | no | — |
| 236 | C2×Q8 | — | 2 | 2 | no | — |
| 237 | C2×Q8 | — | 4 | 4 | no | — |
| 238 | C2×Q8 | — | 4 | 1 | no | — |
| 239 | C2×Q8 | — | 4 | 1 | no | — |
| 240 | C2×Q8 | — | 4 | 2 | no | — |
| 241 | C2×Q8 | — | 4 | 2 | no | — |
| 242 | C22×C4 | — | 2 | 2 | no | — |
| 243 | C22×C4 | — | 2 | 2 | no | — |
| 244 | C22×C4 | — | 2 | 2 | no | — |
| 245 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 246 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 247 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 248 | Q16 | — | 8 | 4 | no | — |
| 249 | M16 | — | 4 | 2 | no | — |
| 250 | C2×C8 | — | 2 | 2 | no | — |
| 251 | C2×C8 | — | 2 | 2 | no | — |
| 252 | C2×C8 | — | 4 | 1 | no | — |
| 253 | C4⋊C4 | — | 2 | 1 | no | — |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 2 | 1 | no | — |
| 256 | C4⋊C4 | — | 2 | 2 | no | — |
| 257 | C4⋊C4 | — | 2 | 1 | no | — |
| 258 | C4⋊C4 | — | 2 | 1 | no | — |
| 259 | C4⋊C4 | — | 4 | 1 | no | — |
| 260 | C4⋊C4 | — | 4 | 1 | no | — |
| 261 | C4⋊C4 | — | 4 | 1 | no | — |
| 262 | C4⋊C4 | — | 4 | 1 | no | — |
| 263 | C4⋊C4 | — | 4 | 1 | no | — |
| 264 | C4⋊C4 | — | 4 | 1 | no | — |
| 265 | C4⋊C4 | — | 4 | 1 | no | — |
| 266 | C4⋊C4 | — | 4 | 1 | no | — |
| 267 | C22⋊C4 | — | 4 | 1 | no | — |
| 268 | C42 | — | 2 | 4 | no | — |
| 269 | C42 | — | 2 | 2 | no | — |
| 270 | C42 | — | 2 | 2 | no | — |
| 271 | C42 | — | 4 | 2 | no | — |
| 272 | C42 | — | 4 | 1 | no | — |
| 273 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 274 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 275 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 276 | C42 | #41×C53 | 1 | 1 | yes | characteristic |
| 277 | C23 | #120×C53 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 4 | 2 | no | — |
| 279 | Q8 | — | 4 | 4 | no | — |
| 280 | Q8 | — | 4 | 2 | no | — |
| 281 | Q8 | — | 8 | 2 | no | — |
| 282 | Q8 | — | 8 | 1 | no | — |
| 283 | Q8 | — | 8 | 1 | no | — |
| 284 | Q8 | — | 8 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 4 | no | — |
| 288 | C2×C4 | — | 4 | 2 | no | — |
| 289 | C2×C4 | — | 4 | 2 | no | — |
| 290 | C2×C4 | — | 4 | 4 | no | — |
| 291 | C2×C4 | — | 4 | 1 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 4 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 4 | 4 | no | — |
| 296 | C2×C4 | #120×C53 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #120×C53 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #127×C53 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #135×C53 | 1 | 2 | yes | — |
| 303 | C2×C4 | #137×C53 | 1 | 2 | yes | — |
| 304 | C8 | — | 4 | 2 | no | — |
| 305 | C8 | — | 4 | 2 | no | — |
| 306 | C8 | — | 8 | 1 | no | — |
| 307 | C22 | — | 2 | 1 | no | — |
| 308 | C22 | — | 2 | 1 | no | — |
| 309 | C22 | C53×#392 | 1 | 1 | yes | characteristic |
| 310 | C22 | C53×#452 | 1 | 2 | yes | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 4 | 4 | no | — |
| 314 | C4 | — | 8 | 2 | no | — |
| 315 | C4 | — | 8 | 1 | no | — |
| 316 | C4 | — | 8 | 1 | no | — |
| 317 | C4 | — | 8 | 2 | no | — |
| 318 | C4 | C53×#455 | 1 | 2 | yes | — |
| 319 | C4 | C53×#456 | 1 | 2 | yes | — |
| 320 | C4 | C53×#464 | 1 | 2 | yes | — |
| 321 | C4 | C53×#495 | 1 | 1 | yes | characteristic |
| 322 | C4 | C53×#497 | 1 | 1 | yes | characteristic |
| 323 | C2 | — | 2 | 2 | no | — |
| 324 | C2 | C53×#2004 | 1 | 1 | yes | characteristic |
| 325 | C2 | C53×#2005 | 1 | 1 | yes | characteristic |
| 326 | C2 | C53×#2015 | 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