← C53×#21781 #27229293
Subgroups — C53×#21781 #27229293
Order 13568.
1258 conjugacy classes of subgroups.
The subgroup lattice has 318 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#3487 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#2889 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C53×#1914 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C53×#1891 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C53×#1830 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C53×#1785 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×2−1+4×C53 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×#41×C53 | C22 | 1 | 1 | yes | characteristic |
| 9 | C53×#568 | C22 | 1 | 4 | yes | — |
| 10 | C53×#550 | — | 2 | 4 | no | — |
| 11 | C53×#548 | — | 2 | 4 | no | — |
| 12 | C53×#535 | C4 | 1 | 4 | yes | — |
| 13 | C53×#535 | C22 | 1 | 2 | yes | — |
| 14 | C53×#525 | C4 | 1 | 4 | yes | — |
| 15 | C53×#525 | C22 | 1 | 2 | yes | — |
| 16 | C53×#519 | C22 | 1 | 1 | yes | characteristic |
| 17 | C53×#413 | — | 2 | 4 | no | — |
| 18 | C53×#413 | C22 | 1 | 4 | yes | — |
| 19 | C2×#101×C53 | — | 2 | 4 | no | — |
| 20 | C2×#101×C53 | C22 | 1 | 4 | yes | — |
| 21 | C2×#100×C53 | C22 | 1 | 4 | yes | — |
| 22 | C53×#407 | C22 | 1 | 4 | yes | — |
| 23 | C53×#344 | C22 | 1 | 8 | yes | — |
| 24 | 2−1+4×C53 | — | 4 | 4 | no | — |
| 25 | C2×#41×C53 | — | 2 | 4 | no | — |
| 26 | C2×#41×C53 | — | 4 | 4 | no | — |
| 27 | C2×#41×C53 | C2×C4 | 1 | 4 | yes | — |
| 28 | C2×#41×C53 | D4 | 1 | 4 | yes | — |
| 29 | C2×#41×C53 | C23 | 1 | 2 | yes | — |
| 30 | C22×Q8×C53 | D4 | 1 | 4 | yes | — |
| 31 | C22×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×D4×C53 | C2×C4 | 1 | 2 | yes | — |
| 33 | C22×D4×C53 | C23 | 1 | 1 | yes | characteristic |
| 34 | C23×C4×C53 | C2×C4 | 1 | 2 | yes | — |
| 35 | C23×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 36 | #131×C53 | — | 4 | 8 | no | — |
| 37 | C2×M16×C53 | — | 2 | 8 | no | — |
| 38 | C2×M16×C53 | C23 | 1 | 8 | yes | — |
| 39 | #127×C53 | — | 2 | 4 | no | — |
| 40 | #127×C53 | — | 2 | 2 | no | — |
| 41 | #124×C53 | — | 2 | 4 | no | — |
| 42 | #124×C53 | — | 2 | 2 | no | — |
| 43 | #123×C53 | C2×C4 | 1 | 4 | yes | — |
| 44 | C22⋊Q8×C53 | — | 2 | 4 | no | — |
| 45 | C22⋊Q8×C53 | — | 2 | 2 | no | — |
| 46 | #121×C53 | — | 2 | 4 | no | — |
| 47 | #121×C53 | — | 2 | 2 | no | — |
| 48 | #121×C53 | — | 2 | 8 | no | — |
| 49 | #121×C53 | — | 2 | 4 | no | — |
| 50 | #120×C53 | C2×C4 | 1 | 8 | yes | — |
| 51 | #120×C53 | C2×C4 | 1 | 4 | yes | — |
| 52 | C4×Q8×C53 | — | 2 | 4 | no | — |
| 53 | C4×D4×C53 | — | 2 | 8 | no | — |
| 54 | C4×D4×C53 | — | 2 | 4 | no | — |
| 55 | #117×C53 | C2×C4 | 1 | 4 | yes | — |
| 56 | #117×C53 | C23 | 1 | 2 | yes | — |
| 57 | #101×C53 | — | 2 | 4 | no | — |
| 58 | #101×C53 | — | 2 | 8 | no | — |
| 59 | #100×C53 | — | 2 | 4 | no | — |
| 60 | #100×C53 | — | 4 | 4 | no | — |
| 61 | C22⋊C8×C53 | — | 4 | 4 | no | — |
| 62 | C24×C53 | C22×C4 | 1 | 2 | yes | — |
| 63 | C24×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 64 | #41×C53 | — | 2 | 8 | no | — |
| 65 | #41×C53 | — | 4 | 8 | no | — |
| 66 | #41×C53 | — | 4 | 8 | no | — |
| 67 | #41×C53 | — | 4 | 16 | no | — |
| 68 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 69 | C2×Q8×C53 | — | 4 | 8 | no | — |
| 70 | C2×Q8×C53 | C22⋊C4 | 1 | 4 | yes | — |
| 71 | C2×Q8×C53 | C2×D4 | 1 | 4 | yes | — |
| 72 | C2×Q8×C53 | C2×D4 | 1 | 2 | yes | — |
| 73 | C2×D4×C53 | — | 2 | 8 | no | — |
| 74 | C2×D4×C53 | — | 2 | 4 | no | — |
| 75 | C2×D4×C53 | — | 2 | 4 | no | — |
| 76 | C2×D4×C53 | — | 2 | 2 | no | — |
| 77 | C2×D4×C53 | — | 2 | 4 | no | — |
| 78 | C2×D4×C53 | — | 4 | 4 | no | — |
| 79 | C2×D4×C53 | C22⋊C4 | 1 | 4 | yes | — |
| 80 | C2×D4×C53 | C22×C4 | 1 | 4 | yes | — |
| 81 | C2×D4×C53 | C2×D4 | 1 | 2 | yes | — |
| 82 | C22×C4×C53 | — | 2 | 4 | no | — |
| 83 | C22×C4×C53 | — | 2 | 2 | no | — |
| 84 | C22×C4×C53 | — | 2 | 4 | no | — |
| 85 | C22×C4×C53 | — | 4 | 4 | no | — |
| 86 | C22×C4×C53 | C22⋊C4 | 1 | 8 | yes | — |
| 87 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C53 | C22×C4 | 1 | 2 | yes | — |
| 89 | C22×C4×C53 | C2×D4 | 1 | 4 | yes | — |
| 90 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 91 | M16×C53 | — | 2 | 8 | no | — |
| 92 | M16×C53 | — | 2 | 8 | no | — |
| 93 | M16×C53 | — | 4 | 8 | no | — |
| 94 | C2×C8×C53 | — | 2 | 8 | no | — |
| 95 | C2×C8×C53 | — | 4 | 8 | no | — |
| 96 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 97 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 98 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 99 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 100 | C22⋊C4×C53 | — | 2 | 8 | no | — |
| 101 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 102 | C22⋊C4×C53 | C22×C4 | 1 | 4 | yes | — |
| 103 | C42×C53 | — | 2 | 4 | no | — |
| 104 | C42×C53 | — | 2 | 2 | no | — |
| 105 | C23×C53 | — | 2 | 2 | no | — |
| 106 | C23×C53 | — | 2 | 1 | no | — |
| 107 | C23×C53 | — | 2 | 2 | no | — |
| 108 | C23×C53 | — | 2 | 4 | no | — |
| 109 | C23×C53 | — | 8 | 1 | no | — |
| 110 | C23×C53 | — | 8 | 2 | no | — |
| 111 | C23×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 112 | C23×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 113 | Q8×C53 | — | 2 | 8 | no | — |
| 114 | Q8×C53 | — | 2 | 4 | no | — |
| 115 | Q8×C53 | — | 4 | 4 | no | — |
| 116 | D4×C53 | — | 2 | 8 | no | — |
| 117 | D4×C53 | — | 2 | 4 | no | — |
| 118 | D4×C53 | — | 4 | 4 | no | — |
| 119 | D4×C53 | — | 4 | 8 | no | — |
| 120 | D4×C53 | — | 4 | 4 | no | — |
| 121 | D4×C53 | — | 8 | 4 | no | — |
| 122 | D4×C53 | — | 8 | 8 | no | — |
| 123 | C2×C4×C53 | — | 2 | 4 | no | — |
| 124 | C2×C4×C53 | — | 2 | 8 | no | — |
| 125 | C2×C4×C53 | — | 2 | 1 | no | — |
| 126 | C2×C4×C53 | — | 2 | 4 | no | — |
| 127 | C2×C4×C53 | — | 2 | 2 | no | — |
| 128 | C2×C4×C53 | — | 4 | 8 | no | — |
| 129 | C2×C4×C53 | — | 4 | 8 | no | — |
| 130 | C2×C4×C53 | — | 4 | 4 | no | — |
| 131 | C2×C4×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 132 | C2×C4×C53 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 133 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 134 | C2×C4×C53 | H(5,𝔽2) | 1 | 4 | yes | — |
| 135 | C8×C53 | — | 4 | 8 | no | — |
| 136 | #21781 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 137 | C22×C53 | — | 2 | 4 | no | — |
| 138 | C22×C53 | — | 4 | 1 | no | — |
| 139 | C22×C53 | — | 4 | 4 | no | — |
| 140 | C22×C53 | — | 4 | 2 | no | — |
| 141 | C22×C53 | — | 4 | 4 | no | — |
| 142 | C22×C53 | — | 4 | 4 | no | — |
| 143 | C22×C53 | — | 4 | 1 | no | — |
| 144 | C22×C53 | — | 4 | 2 | no | — |
| 145 | C22×C53 | — | 8 | 4 | no | — |
| 146 | C22×C53 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C22×C53 | #518 | 1 | 2 | yes | — |
| 148 | C4×C53 | — | 2 | 4 | no | — |
| 149 | C4×C53 | — | 2 | 4 | no | — |
| 150 | C4×C53 | — | 4 | 4 | no | — |
| 151 | C4×C53 | — | 4 | 1 | no | — |
| 152 | C4×C53 | #534 | 1 | 4 | yes | — |
| 153 | #3487 | C2×C53 | 1 | 1 | yes | characteristic |
| 154 | #2889 | C2×C53 | 1 | 2 | yes | complemented |
| 155 | #1914 | C2×C53 | 1 | 4 | yes | complemented |
| 156 | #1891 | C2×C53 | 1 | 4 | yes | complemented |
| 157 | #1830 | C2×C53 | 1 | 2 | yes | complemented |
| 158 | #1785 | C2×C53 | 1 | 2 | yes | complemented |
| 159 | C2×C53 | — | 2 | 1 | no | — |
| 160 | C2×C53 | — | 2 | 2 | no | — |
| 161 | C2×C53 | — | 4 | 4 | no | — |
| 162 | C2×C53 | — | 8 | 1 | no | — |
| 163 | C2×C53 | — | 8 | 2 | no | — |
| 164 | C2×C53 | #2308 | 1 | 1 | yes | characteristic |
| 165 | C2×2−1+4 | C22×C53 | 1 | 1 | yes | characteristic |
| 166 | C22×#41 | C22×C53 | 1 | 1 | yes | characteristic |
| 167 | #568 | C22×C53 | 1 | 4 | yes | — |
| 168 | #550 | — | 2 | 4 | no | — |
| 169 | #548 | — | 2 | 4 | no | — |
| 170 | #535 | C4×C53 | 1 | 4 | yes | — |
| 171 | #535 | C22×C53 | 1 | 2 | yes | — |
| 172 | #525 | C4×C53 | 1 | 4 | yes | — |
| 173 | #525 | C22×C53 | 1 | 2 | yes | — |
| 174 | #519 | C22×C53 | 1 | 1 | yes | characteristic |
| 175 | #413 | — | 2 | 4 | no | — |
| 176 | #413 | C22×C53 | 1 | 4 | yes | — |
| 177 | C2×#101 | — | 2 | 4 | no | — |
| 178 | C2×#101 | C22×C53 | 1 | 4 | yes | — |
| 179 | C2×#100 | C22×C53 | 1 | 4 | yes | — |
| 180 | #407 | C22×C53 | 1 | 4 | yes | — |
| 181 | #344 | C22×C53 | 1 | 8 | yes | — |
| 182 | C53 | #21781 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 183 | 2−1+4 | — | 4 | 4 | no | — |
| 184 | C2×#41 | — | 2 | 4 | no | — |
| 185 | C2×#41 | — | 4 | 4 | no | — |
| 186 | C2×#41 | C2×C4×C53 | 1 | 4 | yes | — |
| 187 | C2×#41 | D4×C53 | 1 | 4 | yes | — |
| 188 | C2×#41 | C23×C53 | 1 | 2 | yes | — |
| 189 | C22×Q8 | D4×C53 | 1 | 4 | yes | — |
| 190 | C22×Q8 | C23×C53 | 1 | 1 | yes | characteristic |
| 191 | C22×D4 | C2×C4×C53 | 1 | 2 | yes | — |
| 192 | C22×D4 | C23×C53 | 1 | 1 | yes | characteristic |
| 193 | C23×C4 | C2×C4×C53 | 1 | 2 | yes | — |
| 194 | C23×C4 | C23×C53 | 1 | 1 | yes | characteristic |
| 195 | #131 | — | 4 | 8 | no | — |
| 196 | C2×M16 | — | 2 | 8 | no | — |
| 197 | C2×M16 | C23×C53 | 1 | 8 | yes | — |
| 198 | #127 | — | 2 | 2 | no | — |
| 199 | #127 | — | 2 | 4 | no | — |
| 200 | #124 | — | 2 | 2 | no | — |
| 201 | #124 | — | 2 | 4 | no | — |
| 202 | #123 | C2×C4×C53 | 1 | 4 | yes | — |
| 203 | C22⋊Q8 | — | 2 | 2 | no | — |
| 204 | C22⋊Q8 | — | 2 | 4 | no | — |
| 205 | #121 | — | 2 | 2 | no | — |
| 206 | #121 | — | 2 | 4 | no | — |
| 207 | #121 | — | 2 | 4 | no | — |
| 208 | #121 | — | 2 | 8 | no | — |
| 209 | #120 | C2×C4×C53 | 1 | 8 | yes | — |
| 210 | #120 | C2×C4×C53 | 1 | 4 | yes | — |
| 211 | C4×Q8 | — | 2 | 4 | no | — |
| 212 | C4×D4 | — | 2 | 4 | no | — |
| 213 | C4×D4 | — | 2 | 8 | no | — |
| 214 | #117 | C2×C4×C53 | 1 | 4 | yes | — |
| 215 | #117 | C23×C53 | 1 | 2 | yes | — |
| 216 | #101 | — | 2 | 4 | no | — |
| 217 | #101 | — | 2 | 8 | no | — |
| 218 | #100 | — | 2 | 4 | no | — |
| 219 | #100 | — | 4 | 4 | no | — |
| 220 | C22⋊C8 | — | 4 | 4 | no | — |
| 221 | C24 | C22×C4×C53 | 1 | 2 | yes | — |
| 222 | C24 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 223 | #41 | — | 2 | 8 | no | — |
| 224 | #41 | — | 4 | 8 | no | — |
| 225 | #41 | — | 4 | 8 | no | — |
| 226 | #41 | — | 4 | 16 | no | — |
| 227 | C2×Q8 | — | 2 | 4 | no | — |
| 228 | C2×Q8 | — | 4 | 8 | no | — |
| 229 | C2×Q8 | C22⋊C4×C53 | 1 | 4 | yes | — |
| 230 | C2×Q8 | C2×D4×C53 | 1 | 4 | yes | — |
| 231 | C2×Q8 | C2×D4×C53 | 1 | 2 | yes | — |
| 232 | C2×D4 | — | 2 | 4 | no | — |
| 233 | C2×D4 | — | 2 | 8 | no | — |
| 234 | C2×D4 | — | 2 | 4 | no | — |
| 235 | C2×D4 | — | 2 | 2 | no | — |
| 236 | C2×D4 | — | 2 | 4 | no | — |
| 237 | C2×D4 | — | 4 | 4 | no | — |
| 238 | C2×D4 | C22⋊C4×C53 | 1 | 4 | yes | — |
| 239 | C2×D4 | C22×C4×C53 | 1 | 4 | yes | — |
| 240 | C2×D4 | C2×D4×C53 | 1 | 2 | yes | — |
| 241 | C22×C4 | — | 2 | 4 | no | — |
| 242 | C22×C4 | — | 2 | 2 | no | — |
| 243 | C22×C4 | — | 2 | 4 | no | — |
| 244 | C22×C4 | — | 4 | 4 | no | — |
| 245 | C22×C4 | C22⋊C4×C53 | 1 | 8 | yes | — |
| 246 | C22×C4 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 247 | C22×C4 | C22×C4×C53 | 1 | 2 | yes | — |
| 248 | C22×C4 | C2×D4×C53 | 1 | 4 | yes | — |
| 249 | C22×C4 | C24×C53 | 1 | 1 | yes | characteristic |
| 250 | M16 | — | 2 | 8 | no | — |
| 251 | M16 | — | 2 | 8 | no | — |
| 252 | M16 | — | 4 | 8 | no | — |
| 253 | C2×C8 | — | 2 | 8 | no | — |
| 254 | C2×C8 | — | 4 | 8 | no | — |
| 255 | C4⋊C4 | — | 2 | 4 | no | — |
| 256 | C4⋊C4 | — | 2 | 4 | no | — |
| 257 | C4⋊C4 | — | 2 | 2 | no | — |
| 258 | C22⋊C4 | — | 2 | 4 | no | — |
| 259 | C22⋊C4 | — | 2 | 4 | no | — |
| 260 | C22⋊C4 | — | 2 | 8 | no | — |
| 261 | C22⋊C4 | C22×C4×C53 | 1 | 4 | yes | — |
| 262 | C42 | — | 2 | 4 | no | — |
| 263 | C42 | — | 2 | 2 | no | — |
| 264 | C23 | — | 2 | 2 | no | — |
| 265 | C23 | — | 2 | 1 | no | — |
| 266 | C23 | — | 2 | 2 | no | — |
| 267 | C23 | — | 2 | 4 | no | — |
| 268 | C23 | — | 8 | 1 | no | — |
| 269 | C23 | — | 8 | 2 | no | — |
| 270 | C23 | C2×C22⋊C4×C53 | 1 | 4 | yes | — |
| 271 | C23 | C23×C4×C53 | 1 | 1 | yes | characteristic |
| 272 | Q8 | — | 2 | 8 | no | — |
| 273 | Q8 | — | 2 | 4 | no | — |
| 274 | Q8 | — | 4 | 4 | no | — |
| 275 | D4 | — | 2 | 8 | no | — |
| 276 | D4 | — | 2 | 4 | no | — |
| 277 | D4 | — | 4 | 4 | no | — |
| 278 | D4 | — | 4 | 8 | no | — |
| 279 | D4 | — | 4 | 4 | no | — |
| 280 | D4 | — | 8 | 4 | no | — |
| 281 | D4 | — | 8 | 8 | no | — |
| 282 | C2×C4 | — | 2 | 4 | no | — |
| 283 | C2×C4 | — | 2 | 8 | no | — |
| 284 | C2×C4 | — | 2 | 4 | no | — |
| 285 | C2×C4 | — | 2 | 1 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 8 | no | — |
| 288 | C2×C4 | — | 4 | 8 | no | — |
| 289 | C2×C4 | — | 4 | 4 | no | — |
| 290 | C2×C4 | C2×C22⋊C4×C53 | 1 | 4 | yes | — |
| 291 | C2×C4 | C2×C22⋊C4×C53 | 1 | 4 | yes | — |
| 292 | C2×C4 | C22×D4×C53 | 1 | 2 | yes | — |
| 293 | C2×C4 | H(5,𝔽2)×C53 | 1 | 4 | yes | — |
| 294 | C8 | — | 4 | 8 | no | — |
| 295 | C22 | — | 2 | 4 | no | — |
| 296 | C22 | — | 4 | 1 | no | — |
| 297 | C22 | — | 4 | 4 | no | — |
| 298 | C22 | — | 4 | 2 | no | — |
| 299 | C22 | — | 4 | 4 | no | — |
| 300 | C22 | — | 4 | 4 | no | — |
| 301 | C22 | — | 4 | 1 | no | — |
| 302 | C22 | — | 4 | 2 | no | — |
| 303 | C22 | — | 8 | 4 | no | — |
| 304 | C22 | C22×C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 305 | C22 | C53×#518 | 1 | 2 | yes | — |
| 306 | C4 | — | 2 | 4 | no | — |
| 307 | C4 | — | 2 | 4 | no | — |
| 308 | C4 | — | 4 | 4 | no | — |
| 309 | C4 | — | 4 | 1 | no | — |
| 310 | C4 | C53×#534 | 1 | 4 | yes | — |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 2 | 2 | no | — |
| 313 | C2 | — | 4 | 4 | no | — |
| 314 | C2 | — | 8 | 1 | no | — |
| 315 | C2 | — | 8 | 2 | no | — |
| 316 | C2 | C53×#2308 | 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