← C53×#21360 #27228872
Subgroups — C53×#21360 #27228872
Order 13568.
870 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C53×#431 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#2283 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4×#117×C53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53×#1949 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C53×#1854 | C2 | 1 | 4 | yes | maximal |
| 6 | C4×#107×C53 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C4×#106×C53 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×#117×C53 | C22 | 1 | 2 | yes | — |
| 9 | C22×C42×C53 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#431 | C4 | 1 | 8 | yes | complemented |
| 11 | C2×#107×C53 | C22 | 1 | 4 | yes | — |
| 12 | C2×#106×C53 | C22 | 1 | 4 | yes | — |
| 13 | C2×C4⋊C8×C53 | C22 | 1 | 2 | yes | — |
| 14 | C2×C22⋊C8×C53 | C22 | 1 | 2 | yes | — |
| 15 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 17 | C53×#383 | — | 2 | 1 | no | — |
| 18 | C53×#382 | C22 | 1 | 2 | yes | — |
| 19 | C53×#381 | — | 2 | 1 | no | — |
| 20 | C53×#380 | C22 | 1 | 2 | yes | — |
| 21 | C4×C4⋊C4×C53 | C22 | 1 | 2 | yes | — |
| 22 | C4×C4⋊C4×C53 | C22 | 1 | 2 | yes | — |
| 23 | C4×C4⋊C4×C53 | C22 | 1 | 2 | yes | — |
| 24 | C4×C22⋊C4×C53 | — | 2 | 1 | no | — |
| 25 | C4×C22⋊C4×C53 | — | 2 | 1 | no | — |
| 26 | C4×C22⋊C4×C53 | — | 2 | 1 | no | — |
| 27 | C53×#376 | — | 2 | 1 | no | — |
| 28 | C53×#376 | — | 2 | 1 | no | — |
| 29 | C43×C53 | — | 2 | 1 | no | — |
| 30 | C53×#340 | C22 | 1 | 4 | yes | — |
| 31 | C53×#340 | C22 | 1 | 4 | yes | — |
| 32 | C23×C4×C53 | D4 | 1 | 1 | yes | characteristic |
| 33 | C23×C4×C53 | D4 | 1 | 1 | yes | characteristic |
| 34 | C23×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 35 | C22×C8×C53 | C23 | 1 | 2 | yes | — |
| 36 | C22×C8×C53 | C23 | 1 | 2 | yes | — |
| 37 | #117×C53 | — | 2 | 8 | no | — |
| 38 | C2×C4⋊C4×C53 | C23 | 1 | 4 | yes | — |
| 39 | C2×C4⋊C4×C53 | C23 | 1 | 4 | yes | — |
| 40 | C2×C22⋊C4×C53 | — | 2 | 2 | no | — |
| 41 | C2×C22⋊C4×C53 | — | 2 | 2 | no | — |
| 42 | C2×C42×C53 | — | 2 | 2 | no | — |
| 43 | C2×C42×C53 | — | 2 | 2 | no | — |
| 44 | C2×C42×C53 | — | 2 | 2 | no | — |
| 45 | C2×C42×C53 | — | 2 | 2 | no | — |
| 46 | C2×C42×C53 | C2×C4 | 1 | 4 | yes | — |
| 47 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C42×C53 | Q8 | 1 | 4 | yes | — |
| 50 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 52 | #107×C53 | — | 2 | 8 | no | — |
| 53 | #106×C53 | — | 2 | 8 | no | — |
| 54 | C4⋊C8×C53 | C2×C4 | 1 | 8 | yes | — |
| 55 | C22⋊C8×C53 | C2×C4 | 1 | 8 | yes | — |
| 56 | C4×C8×C53 | C2×C4 | 1 | 4 | yes | — |
| 57 | C4×C8×C53 | C2×C4 | 1 | 4 | yes | — |
| 58 | #95×C53 | — | 2 | 2 | no | — |
| 59 | #95×C53 | — | 2 | 2 | no | — |
| 60 | #95×C53 | — | 2 | 2 | no | — |
| 61 | #95×C53 | — | 2 | 2 | no | — |
| 62 | #95×C53 | — | 2 | 2 | no | — |
| 63 | #95×C53 | — | 2 | 2 | no | — |
| 64 | C24×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C22×C4×C53 | — | 2 | 4 | no | — |
| 66 | C22×C4×C53 | — | 2 | 4 | no | — |
| 67 | C22×C4×C53 | — | 2 | 4 | no | — |
| 68 | C22×C4×C53 | — | 2 | 4 | no | — |
| 69 | C22×C4×C53 | — | 4 | 2 | no | — |
| 70 | C22×C4×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 71 | C22×C4×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 72 | C22×C4×C53 | C22×C4 | 1 | 2 | yes | — |
| 73 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C53 | C2×Q8 | 1 | 2 | yes | — |
| 79 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C53 | — | 2 | 4 | no | — |
| 81 | C2×C8×C53 | — | 2 | 4 | no | — |
| 82 | C2×C8×C53 | C22×C4 | 1 | 4 | yes | — |
| 83 | C2×C8×C53 | C22×C4 | 1 | 4 | yes | — |
| 84 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 85 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 86 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 87 | C4⋊C4×C53 | #41 | 1 | 8 | yes | — |
| 88 | C22⋊C4×C53 | — | 2 | 8 | no | — |
| 89 | C22⋊C4×C53 | — | 4 | 4 | no | — |
| 90 | C42×C53 | — | 2 | 8 | no | — |
| 91 | C42×C53 | — | 4 | 4 | no | — |
| 92 | C42×C53 | — | 4 | 4 | no | — |
| 93 | C42×C53 | — | 4 | 4 | no | — |
| 94 | C42×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 95 | C42×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 96 | C42×C53 | C22×C4 | 1 | 2 | yes | — |
| 97 | C42×C53 | C22×C4 | 1 | 2 | yes | — |
| 98 | C42×C53 | C2×Q8 | 1 | 2 | yes | — |
| 99 | C42×C53 | C2×Q8 | 1 | 2 | yes | — |
| 100 | C23×C53 | — | 2 | 2 | no | — |
| 101 | C23×C53 | — | 2 | 2 | no | — |
| 102 | C23×C53 | — | 2 | 2 | no | — |
| 103 | C23×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 104 | C23×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C53 | — | 2 | 2 | no | — |
| 106 | C2×C4×C53 | — | 2 | 2 | no | — |
| 107 | C2×C4×C53 | — | 2 | 2 | no | — |
| 108 | C2×C4×C53 | — | 2 | 2 | no | — |
| 109 | C2×C4×C53 | — | 2 | 4 | no | — |
| 110 | C2×C4×C53 | — | 2 | 4 | no | — |
| 111 | C2×C4×C53 | — | 2 | 8 | no | — |
| 112 | C2×C4×C53 | — | 2 | 8 | no | — |
| 113 | C2×C4×C53 | — | 4 | 4 | no | — |
| 114 | C2×C4×C53 | — | 4 | 4 | no | — |
| 115 | C2×C4×C53 | — | 4 | 4 | no | — |
| 116 | C2×C4×C53 | — | 4 | 4 | no | — |
| 117 | C2×C4×C53 | — | 4 | 4 | no | — |
| 118 | C2×C4×C53 | — | 4 | 4 | no | — |
| 119 | C2×C4×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 120 | C2×C4×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 121 | C2×C4×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 122 | C2×C4×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 123 | C2×C4×C53 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 124 | C2×C4×C53 | #135 | 1 | 4 | yes | — |
| 125 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C53 | C2×#41 | 1 | 4 | yes | — |
| 128 | C8×C53 | — | 2 | 4 | no | — |
| 129 | C8×C53 | — | 2 | 4 | no | — |
| 130 | #21360 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 131 | C22×C53 | — | 2 | 2 | no | — |
| 132 | C22×C53 | — | 2 | 2 | no | — |
| 133 | C22×C53 | — | 2 | 2 | no | — |
| 134 | C22×C53 | — | 4 | 4 | no | — |
| 135 | C22×C53 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 136 | C22×C53 | #525 | 1 | 2 | yes | — |
| 137 | C22×C53 | C2×#135 | 1 | 2 | yes | — |
| 138 | C22×C53 | #575 | 1 | 2 | yes | — |
| 139 | C4×C53 | — | 2 | 2 | no | — |
| 140 | C4×C53 | — | 2 | 2 | no | — |
| 141 | C4×C53 | — | 2 | 2 | no | — |
| 142 | C4×C53 | — | 2 | 2 | no | — |
| 143 | C4×C53 | — | 4 | 8 | no | — |
| 144 | C4×C53 | — | 8 | 4 | no | — |
| 145 | C4×C53 | #427 | 1 | 4 | yes | — |
| 146 | C4×C53 | C4×#41 | 1 | 2 | yes | — |
| 147 | C4×C53 | #533 | 1 | 2 | yes | — |
| 148 | C2×#431 | C2×C53 | 1 | 1 | yes | characteristic |
| 149 | #2283 | C2×C53 | 1 | 1 | yes | characteristic |
| 150 | C4×#117 | C2×C53 | 1 | 1 | yes | characteristic |
| 151 | #1949 | C2×C53 | 1 | 4 | yes | complemented |
| 152 | #1854 | C2×C53 | 1 | 4 | yes | — |
| 153 | C4×#107 | C2×C53 | 1 | 2 | yes | complemented |
| 154 | C4×#106 | C2×C53 | 1 | 2 | yes | complemented |
| 155 | C2×C53 | — | 4 | 2 | no | — |
| 156 | C2×C53 | #2296 | 1 | 1 | yes | characteristic |
| 157 | C2×C53 | C2×#427 | 1 | 1 | yes | characteristic |
| 158 | C2×C53 | #2911 | 1 | 1 | yes | characteristic |
| 159 | C2×C53 | #3102 | 1 | 4 | yes | — |
| 160 | C2×#117 | C22×C53 | 1 | 2 | yes | — |
| 161 | C22×C42 | C22×C53 | 1 | 1 | yes | characteristic |
| 162 | #431 | C4×C53 | 1 | 8 | yes | complemented |
| 163 | C2×#107 | C22×C53 | 1 | 4 | yes | — |
| 164 | C2×#106 | C22×C53 | 1 | 4 | yes | — |
| 165 | C2×C4⋊C8 | C22×C53 | 1 | 2 | yes | — |
| 166 | C2×C22⋊C8 | C22×C53 | 1 | 2 | yes | — |
| 167 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 168 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 169 | #383 | — | 2 | 1 | no | — |
| 170 | #382 | C22×C53 | 1 | 2 | yes | — |
| 171 | #381 | — | 2 | 1 | no | — |
| 172 | #380 | C22×C53 | 1 | 2 | yes | — |
| 173 | C4×C4⋊C4 | C22×C53 | 1 | 2 | yes | — |
| 174 | C4×C4⋊C4 | C22×C53 | 1 | 2 | yes | — |
| 175 | C4×C4⋊C4 | C22×C53 | 1 | 2 | yes | — |
| 176 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 177 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 178 | C4×C22⋊C4 | — | 2 | 1 | no | — |
| 179 | #376 | — | 2 | 1 | no | — |
| 180 | #376 | — | 2 | 1 | no | — |
| 181 | C43 | — | 2 | 1 | no | — |
| 182 | #340 | C22×C53 | 1 | 4 | yes | — |
| 183 | #340 | C22×C53 | 1 | 4 | yes | — |
| 184 | C53 | #21360 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 185 | C23×C4 | D4×C53 | 1 | 1 | yes | characteristic |
| 186 | C23×C4 | D4×C53 | 1 | 1 | yes | characteristic |
| 187 | C23×C4 | C23×C53 | 1 | 1 | yes | characteristic |
| 188 | C22×C8 | C23×C53 | 1 | 2 | yes | — |
| 189 | C22×C8 | C23×C53 | 1 | 2 | yes | — |
| 190 | #117 | — | 2 | 8 | no | — |
| 191 | C2×C4⋊C4 | C23×C53 | 1 | 4 | yes | — |
| 192 | C2×C4⋊C4 | C23×C53 | 1 | 4 | yes | — |
| 193 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 194 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 195 | C2×C42 | — | 2 | 2 | no | — |
| 196 | C2×C42 | — | 2 | 2 | no | — |
| 197 | C2×C42 | — | 2 | 2 | no | — |
| 198 | C2×C42 | — | 2 | 2 | no | — |
| 199 | C2×C42 | C2×C4×C53 | 1 | 4 | yes | — |
| 200 | C2×C42 | D4×C53 | 1 | 1 | yes | characteristic |
| 201 | C2×C42 | D4×C53 | 1 | 1 | yes | characteristic |
| 202 | C2×C42 | Q8×C53 | 1 | 4 | yes | — |
| 203 | C2×C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 204 | C2×C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 205 | #107 | — | 2 | 8 | no | — |
| 206 | #106 | — | 2 | 8 | no | — |
| 207 | C4⋊C8 | C2×C4×C53 | 1 | 8 | yes | — |
| 208 | C22⋊C8 | C2×C4×C53 | 1 | 8 | yes | — |
| 209 | C4×C8 | C2×C4×C53 | 1 | 4 | yes | — |
| 210 | C4×C8 | C2×C4×C53 | 1 | 4 | yes | — |
| 211 | #95 | — | 2 | 2 | no | — |
| 212 | #95 | — | 2 | 2 | no | — |
| 213 | #95 | — | 2 | 2 | no | — |
| 214 | #95 | — | 2 | 2 | no | — |
| 215 | #95 | — | 2 | 2 | no | — |
| 216 | #95 | — | 2 | 2 | no | — |
| 217 | C24 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 218 | C22×C4 | — | 2 | 4 | no | — |
| 219 | C22×C4 | — | 2 | 4 | no | — |
| 220 | C22×C4 | — | 2 | 4 | no | — |
| 221 | C22×C4 | — | 2 | 4 | no | — |
| 222 | C22×C4 | — | 4 | 2 | no | — |
| 223 | C22×C4 | C4⋊C4×C53 | 1 | 4 | yes | — |
| 224 | C22×C4 | C4⋊C4×C53 | 1 | 4 | yes | — |
| 225 | C22×C4 | C22×C4×C53 | 1 | 2 | yes | — |
| 226 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 228 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 229 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 230 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 231 | C22×C4 | C2×Q8×C53 | 1 | 2 | yes | — |
| 232 | C22×C4 | C24×C53 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | — | 2 | 4 | no | — |
| 234 | C2×C8 | — | 2 | 4 | no | — |
| 235 | C2×C8 | C22×C4×C53 | 1 | 4 | yes | — |
| 236 | C2×C8 | C22×C4×C53 | 1 | 4 | yes | — |
| 237 | C4⋊C4 | — | 2 | 4 | no | — |
| 238 | C4⋊C4 | — | 2 | 4 | no | — |
| 239 | C4⋊C4 | — | 2 | 4 | no | — |
| 240 | C4⋊C4 | #41×C53 | 1 | 8 | yes | — |
| 241 | C22⋊C4 | — | 2 | 8 | no | — |
| 242 | C22⋊C4 | — | 4 | 4 | no | — |
| 243 | C42 | — | 2 | 8 | no | — |
| 244 | C42 | — | 4 | 4 | no | — |
| 245 | C42 | — | 4 | 4 | no | — |
| 246 | C42 | — | 4 | 4 | no | — |
| 247 | C42 | C4⋊C4×C53 | 1 | 4 | yes | — |
| 248 | C42 | C4⋊C4×C53 | 1 | 4 | yes | — |
| 249 | C42 | C22×C4×C53 | 1 | 2 | yes | — |
| 250 | C42 | C22×C4×C53 | 1 | 2 | yes | — |
| 251 | C42 | C2×Q8×C53 | 1 | 2 | yes | — |
| 252 | C42 | C2×Q8×C53 | 1 | 2 | yes | — |
| 253 | C23 | — | 2 | 2 | no | — |
| 254 | C23 | — | 2 | 2 | no | — |
| 255 | C23 | — | 2 | 2 | no | — |
| 256 | C23 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 257 | C23 | C22×D4×C53 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | — | 2 | 2 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 2 | 2 | no | — |
| 262 | C2×C4 | — | 2 | 4 | no | — |
| 263 | C2×C4 | — | 2 | 4 | no | — |
| 264 | C2×C4 | — | 2 | 8 | no | — |
| 265 | C2×C4 | — | 2 | 8 | no | — |
| 266 | C2×C4 | — | 4 | 4 | no | — |
| 267 | C2×C4 | — | 4 | 4 | no | — |
| 268 | C2×C4 | — | 4 | 4 | no | — |
| 269 | C2×C4 | — | 4 | 4 | no | — |
| 270 | C2×C4 | — | 4 | 4 | no | — |
| 271 | C2×C4 | — | 4 | 4 | no | — |
| 272 | C2×C4 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 273 | C2×C4 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 274 | C2×C4 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 275 | C2×C4 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 276 | C2×C4 | C2×C4⋊C4×C53 | 1 | 2 | yes | — |
| 277 | C2×C4 | #135×C53 | 1 | 4 | yes | — |
| 278 | C2×C4 | C23×C4×C53 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | C22×Q8×C53 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C2×#41×C53 | 1 | 4 | yes | — |
| 281 | C8 | — | 2 | 4 | no | — |
| 282 | C8 | — | 2 | 4 | no | — |
| 283 | C22 | — | 2 | 2 | no | — |
| 284 | C22 | — | 2 | 2 | no | — |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 4 | 4 | no | — |
| 287 | C22 | C22×C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 288 | C22 | C53×#525 | 1 | 2 | yes | — |
| 289 | C22 | C2×#135×C53 | 1 | 2 | yes | — |
| 290 | C22 | C53×#575 | 1 | 2 | yes | — |
| 291 | C4 | — | 2 | 2 | no | — |
| 292 | C4 | — | 2 | 2 | no | — |
| 293 | C4 | — | 2 | 2 | no | — |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 4 | 8 | no | — |
| 296 | C4 | — | 8 | 4 | no | — |
| 297 | C4 | C53×#427 | 1 | 4 | yes | — |
| 298 | C4 | C4×#41×C53 | 1 | 2 | yes | — |
| 299 | C4 | C53×#533 | 1 | 2 | yes | — |
| 300 | C2 | — | 4 | 2 | no | — |
| 301 | C2 | C53×#2296 | 1 | 1 | yes | characteristic |
| 302 | C2 | C2×C53×#427 | 1 | 1 | yes | characteristic |
| 303 | C2 | C53×#2911 | 1 | 1 | yes | characteristic |
| 304 | C2 | C53×#3102 | 1 | 4 | yes | — |
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