← #27301529
Subgroups — #27301529
Order 13568.
458 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16811040 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810439 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#16810427 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20274779 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20272825 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272808 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×C53×#473 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811040 | — | 2 | 2 | no | — |
| 9 | #16811040 | — | 2 | 2 | no | — |
| 10 | C2×#13133157 | — | 2 | 1 | no | — |
| 11 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×#11719908 | — | 2 | 1 | no | — |
| 13 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16810624 | — | 2 | 1 | no | — |
| 17 | #16810439 | — | 2 | 2 | no | — |
| 18 | #16810427 | — | 2 | 1 | no | — |
| 19 | #16810427 | — | 2 | 1 | no | — |
| 20 | #16810391 | — | 2 | 1 | no | — |
| 21 | C2×#137×C53 | — | 2 | 1 | no | — |
| 22 | C22×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#473 | — | 2 | 2 | no | — |
| 25 | C53×#473 | C4 | 1 | 4 | yes | complemented |
| 26 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#101×C53 | — | 2 | 1 | no | — |
| 28 | C2×#11719917 | — | 2 | 1 | no | — |
| 29 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 30 | #13133157 | — | 2 | 4 | no | — |
| 31 | #13133157 | — | 2 | 1 | no | — |
| 32 | #13133157 | D4 | 1 | 2 | yes | — |
| 33 | C2×#11719908 | — | 2 | 4 | no | — |
| 34 | C2×#11719908 | — | 2 | 1 | no | — |
| 35 | C2×#11719908 | — | 2 | 1 | no | — |
| 36 | C2×#11719908 | — | 2 | 1 | no | — |
| 37 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 39 | #13133153 | — | 2 | 1 | no | — |
| 40 | #13133153 | — | 2 | 1 | no | — |
| 41 | #13133151 | — | 2 | 1 | no | — |
| 42 | #13133151 | — | 2 | 1 | no | — |
| 43 | #13133141 | — | 4 | 1 | no | — |
| 44 | #13133138 | — | 4 | 1 | no | — |
| 45 | C2×#41×C53 | D4 | 1 | 1 | yes | characteristic |
| 46 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 47 | #137×C53 | — | 2 | 4 | no | — |
| 48 | #137×C53 | — | 4 | 2 | no | — |
| 49 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 50 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 51 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 52 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 53 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 54 | C2×Q16×C53 | — | 4 | 1 | no | — |
| 55 | C2×Q16×C53 | C2×C4 | 1 | 2 | yes | — |
| 56 | C2×SD16×C53 | — | 4 | 1 | no | — |
| 57 | #131×C53 | — | 2 | 1 | no | — |
| 58 | #131×C53 | — | 2 | 1 | no | — |
| 59 | #131×C53 | C2×C4 | 1 | 2 | yes | — |
| 60 | #131×C53 | D4 | 1 | 2 | yes | — |
| 61 | C2×M16×C53 | — | 2 | 1 | no | — |
| 62 | C2×M16×C53 | — | 2 | 1 | no | — |
| 63 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 64 | C22×C8×C53 | — | 2 | 1 | no | — |
| 65 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 66 | #108×C53 | — | 2 | 1 | no | — |
| 67 | #108×C53 | C2×C4 | 1 | 2 | yes | — |
| 68 | #101×C53 | — | 2 | 4 | no | — |
| 69 | #41×C53 | — | 2 | 1 | no | — |
| 70 | #41×C53 | — | 2 | 1 | no | — |
| 71 | #41×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 72 | #41×C53 | SD16 | 1 | 2 | yes | — |
| 73 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 74 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 75 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 76 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 77 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 78 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 79 | C2×Q8×C53 | D8 | 1 | 1 | yes | characteristic |
| 80 | C2×D4×C53 | — | 2 | 1 | no | — |
| 81 | C2×D4×C53 | D8 | 1 | 1 | yes | characteristic |
| 82 | C22×C4×C53 | — | 2 | 1 | no | — |
| 83 | C22×C4×C53 | — | 4 | 1 | no | — |
| 84 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C22×Dic53 | — | 4 | 1 | no | — |
| 86 | Q16×C53 | — | 2 | 2 | no | — |
| 87 | Q16×C53 | — | 2 | 2 | no | — |
| 88 | Q16×C53 | — | 4 | 4 | no | — |
| 89 | Q16×C53 | — | 4 | 1 | no | — |
| 90 | Q16×C53 | — | 4 | 1 | no | — |
| 91 | SD16×C53 | — | 4 | 4 | no | — |
| 92 | M16×C53 | — | 2 | 1 | no | — |
| 93 | M16×C53 | — | 2 | 2 | no | — |
| 94 | M16×C53 | — | 2 | 2 | no | — |
| 95 | M16×C53 | — | 2 | 2 | no | — |
| 96 | M16×C53 | — | 4 | 1 | no | — |
| 97 | M16×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 98 | C2×C8×C53 | — | 2 | 1 | no | — |
| 99 | C2×C8×C53 | — | 2 | 2 | no | — |
| 100 | C2×C8×C53 | — | 2 | 2 | no | — |
| 101 | C2×C8×C53 | — | 4 | 1 | no | — |
| 102 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 107 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 108 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 109 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 110 | #11719917 | — | 4 | 1 | no | — |
| 111 | #11719917 | — | 4 | 1 | no | — |
| 112 | C2×#88322 | — | 4 | 1 | no | — |
| 113 | #11719908 | — | 2 | 1 | no | — |
| 114 | #11719908 | — | 2 | 2 | no | — |
| 115 | #11719908 | — | 2 | 1 | no | — |
| 116 | C23×C53 | — | 2 | 1 | no | — |
| 117 | C23×C53 | #135 | 1 | 1 | yes | characteristic |
| 118 | Q8×C53 | — | 2 | 1 | no | — |
| 119 | Q8×C53 | — | 4 | 2 | no | — |
| 120 | Q8×C53 | — | 4 | 2 | no | — |
| 121 | Q8×C53 | — | 8 | 1 | no | — |
| 122 | Q8×C53 | — | 8 | 1 | no | — |
| 123 | Q8×C53 | #102 | 1 | 2 | yes | — |
| 124 | D4×C53 | — | 2 | 2 | no | — |
| 125 | D4×C53 | — | 2 | 2 | no | — |
| 126 | D4×C53 | — | 2 | 1 | no | — |
| 127 | D4×C53 | #102 | 1 | 2 | yes | — |
| 128 | C2×C4×C53 | — | 2 | 1 | no | — |
| 129 | C2×C4×C53 | — | 2 | 2 | no | — |
| 130 | C2×C4×C53 | — | 2 | 2 | no | — |
| 131 | C2×C4×C53 | — | 4 | 2 | no | — |
| 132 | C2×C4×C53 | — | 8 | 1 | no | — |
| 133 | C2×C4×C53 | — | 8 | 1 | no | — |
| 134 | C2×C4×C53 | C2×C22⋊C4 | 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 | C2×D8 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C53 | C2×SD16 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 140 | C2×Dic53 | — | 8 | 1 | no | — |
| 141 | C2×Dic53 | — | 16 | 1 | no | — |
| 142 | C8×C53 | — | 2 | 2 | no | — |
| 143 | C8×C53 | — | 4 | 2 | no | — |
| 144 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 145 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 146 | C8×C53 | #120 | 1 | 1 | yes | characteristic |
| 147 | C8×C53 | #123 | 1 | 1 | yes | characteristic |
| 148 | #88322 | — | 8 | 1 | no | — |
| 149 | #13072 | — | 53 | 1 | no | maximal, Sylow |
| 150 | C22×C53 | — | 2 | 1 | no | — |
| 151 | C22×C53 | — | 2 | 1 | no | — |
| 152 | C22×C53 | — | 2 | 2 | no | — |
| 153 | C22×C53 | — | 4 | 1 | no | — |
| 154 | C22×C53 | — | 4 | 1 | no | — |
| 155 | C22×C53 | #416 | 1 | 1 | yes | characteristic |
| 156 | C22×C53 | #465 | 1 | 1 | yes | characteristic |
| 157 | C22×C53 | #466 | 1 | 1 | yes | characteristic |
| 158 | C4×C53 | — | 2 | 2 | no | — |
| 159 | C4×C53 | — | 8 | 2 | no | — |
| 160 | C4×C53 | #386 | 1 | 1 | yes | characteristic |
| 161 | C4×C53 | C2×#102 | 1 | 1 | yes | characteristic |
| 162 | C4×C53 | #465 | 1 | 1 | yes | characteristic |
| 163 | C4×C53 | #466 | 1 | 1 | yes | characteristic |
| 164 | Dic53 | — | 16 | 1 | no | — |
| 165 | C2×#504 | — | 53 | 1 | no | — |
| 166 | C2×#473 | D53 | 1 | 1 | yes | characteristic |
| 167 | C2×#362 | — | 53 | 1 | no | — |
| 168 | C2×#358 | — | 53 | 1 | no | — |
| 169 | #1816 | — | 53 | 1 | no | — |
| 170 | #1387 | — | 53 | 1 | no | — |
| 171 | #1381 | — | 53 | 1 | no | — |
| 172 | C2×C53 | — | 2 | 1 | no | — |
| 173 | C2×C53 | — | 2 | 1 | no | — |
| 174 | C2×C53 | — | 4 | 2 | no | — |
| 175 | C2×C53 | #1893 | 1 | 1 | yes | characteristic |
| 176 | C2×C53 | #2222 | 1 | 1 | yes | characteristic |
| 177 | C2×C53 | #2223 | 1 | 1 | yes | characteristic |
| 178 | C2×#137 | — | 2 | 1 | no | — |
| 179 | C22×Q16 | C2×D53 | 1 | 1 | yes | characteristic |
| 180 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 181 | #504 | — | 106 | 2 | no | — |
| 182 | #504 | — | 106 | 2 | no | — |
| 183 | C2×M32 | — | 53 | 1 | no | — |
| 184 | C2×M32 | — | 106 | 1 | no | — |
| 185 | C22×C16 | — | 53 | 1 | no | — |
| 186 | C22×C16 | — | 106 | 1 | no | — |
| 187 | #473 | — | 2 | 2 | no | — |
| 188 | #473 | Dic53 | 1 | 4 | yes | complemented |
| 189 | C2×#108 | — | 53 | 1 | no | — |
| 190 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 191 | C2×#107 | — | 53 | 1 | no | — |
| 192 | #417 | — | 106 | 1 | no | — |
| 193 | C2×#101 | — | 2 | 1 | no | — |
| 194 | #362 | — | 106 | 2 | no | — |
| 195 | #358 | — | 106 | 1 | no | — |
| 196 | #358 | — | 106 | 1 | no | — |
| 197 | #343 | — | 106 | 1 | no | — |
| 198 | C53 | #13072 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 199 | C2×#41 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 200 | C22×Q8 | — | 2 | 1 | no | — |
| 201 | #137 | — | 2 | 4 | no | — |
| 202 | #137 | — | 4 | 2 | no | — |
| 203 | C2×Q16 | — | 2 | 1 | no | — |
| 204 | C2×Q16 | — | 2 | 1 | no | — |
| 205 | C2×Q16 | — | 2 | 1 | no | — |
| 206 | C2×Q16 | — | 2 | 1 | no | — |
| 207 | C2×Q16 | — | 2 | 1 | no | — |
| 208 | C2×Q16 | — | 4 | 1 | no | — |
| 209 | C2×Q16 | C2×Dic53 | 1 | 2 | yes | — |
| 210 | C2×SD16 | — | 4 | 1 | no | — |
| 211 | #131 | — | 2 | 1 | no | — |
| 212 | #131 | — | 2 | 1 | no | — |
| 213 | #131 | C2×Dic53 | 1 | 2 | yes | — |
| 214 | #131 | C22⋊D53 | 1 | 2 | yes | — |
| 215 | C2×M16 | — | 2 | 1 | no | — |
| 216 | C2×M16 | — | 2 | 1 | no | — |
| 217 | C2×M16 | — | 106 | 1 | no | — |
| 218 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 219 | C22×C8 | — | 2 | 1 | no | — |
| 220 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 221 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 222 | M32 | — | 53 | 2 | no | — |
| 223 | M32 | — | 106 | 4 | no | — |
| 224 | M32 | — | 106 | 1 | no | — |
| 225 | C2×C16 | — | 53 | 1 | no | — |
| 226 | C2×C16 | — | 53 | 1 | no | — |
| 227 | C2×C16 | — | 106 | 4 | no | — |
| 228 | C2×C16 | — | 106 | 1 | no | — |
| 229 | C2×C16 | — | 106 | 1 | no | — |
| 230 | C2×C16 | — | 106 | 1 | no | — |
| 231 | #108 | — | 2 | 1 | no | — |
| 232 | #108 | — | 106 | 1 | no | — |
| 233 | #108 | — | 106 | 1 | no | — |
| 234 | #108 | C2×Dic53 | 1 | 2 | yes | — |
| 235 | #107 | — | 106 | 1 | no | — |
| 236 | #107 | — | 106 | 1 | no | — |
| 237 | #103 | — | 212 | 1 | no | — |
| 238 | #102 | — | 212 | 1 | no | — |
| 239 | #101 | — | 2 | 4 | no | — |
| 240 | #41 | — | 2 | 1 | no | — |
| 241 | #41 | — | 2 | 1 | no | — |
| 242 | #41 | #11719923 | 1 | 2 | yes | — |
| 243 | #41 | C22⋊Dic53 | 1 | 2 | yes | — |
| 244 | C2×Q8 | — | 2 | 2 | no | — |
| 245 | C2×Q8 | — | 4 | 1 | no | — |
| 246 | C2×Q8 | — | 4 | 1 | no | — |
| 247 | C2×Q8 | — | 4 | 1 | no | — |
| 248 | C2×Q8 | — | 4 | 1 | no | — |
| 249 | C2×Q8 | — | 4 | 1 | no | — |
| 250 | C2×Q8 | #11719922 | 1 | 1 | yes | characteristic |
| 251 | C2×D4 | — | 2 | 1 | no | — |
| 252 | C2×D4 | #11719922 | 1 | 1 | yes | characteristic |
| 253 | C22×C4 | — | 2 | 1 | no | — |
| 254 | C22×C4 | — | 4 | 1 | no | — |
| 255 | C22×C4 | — | 212 | 1 | no | — |
| 256 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 257 | Q16 | — | 2 | 2 | no | — |
| 258 | Q16 | — | 2 | 2 | no | — |
| 259 | Q16 | — | 4 | 4 | no | — |
| 260 | Q16 | — | 4 | 1 | no | — |
| 261 | Q16 | — | 4 | 1 | no | — |
| 262 | SD16 | — | 4 | 4 | no | — |
| 263 | M16 | — | 2 | 1 | no | — |
| 264 | M16 | — | 2 | 2 | no | — |
| 265 | M16 | — | 2 | 2 | no | — |
| 266 | M16 | — | 2 | 2 | no | — |
| 267 | M16 | — | 4 | 1 | no | — |
| 268 | M16 | — | 212 | 1 | no | — |
| 269 | M16 | — | 212 | 1 | no | — |
| 270 | M16 | C22⋊Dic53 | 1 | 2 | yes | — |
| 271 | C2×C8 | — | 2 | 1 | no | — |
| 272 | C2×C8 | — | 2 | 2 | no | — |
| 273 | C2×C8 | — | 2 | 2 | no | — |
| 274 | C2×C8 | — | 4 | 1 | no | — |
| 275 | C2×C8 | — | 212 | 1 | no | — |
| 276 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 277 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 278 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 279 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 280 | C2×C8 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 281 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 282 | C4⋊C4 | — | 212 | 1 | no | — |
| 283 | C4⋊C4 | — | 212 | 1 | no | — |
| 284 | C16 | — | 106 | 1 | no | — |
| 285 | C16 | — | 106 | 1 | no | — |
| 286 | C16 | — | 106 | 2 | no | — |
| 287 | C23 | — | 2 | 1 | no | — |
| 288 | C23 | #13133237 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 2 | 1 | no | — |
| 290 | Q8 | — | 4 | 2 | no | — |
| 291 | Q8 | — | 4 | 2 | no | — |
| 292 | Q8 | — | 8 | 1 | no | — |
| 293 | Q8 | — | 8 | 1 | no | — |
| 294 | Q8 | #13133138 | 1 | 2 | yes | — |
| 295 | D4 | — | 2 | 2 | no | — |
| 296 | D4 | — | 2 | 2 | no | — |
| 297 | D4 | — | 2 | 1 | no | — |
| 298 | D4 | #13133138 | 1 | 2 | yes | — |
| 299 | C2×C4 | — | 2 | 1 | no | — |
| 300 | C2×C4 | — | 2 | 2 | no | — |
| 301 | C2×C4 | — | 2 | 2 | no | — |
| 302 | C2×C4 | — | 4 | 2 | no | — |
| 303 | C2×C4 | — | 8 | 1 | no | — |
| 304 | C2×C4 | — | 8 | 1 | no | — |
| 305 | C2×C4 | — | 848 | 1 | no | — |
| 306 | C2×C4 | — | 424 | 1 | no | — |
| 307 | C2×C4 | C2×C22⋊Dic53 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | #13133189 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | #13133189 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | C2×#11719923 | 1 | 1 | yes | characteristic |
| 312 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 313 | C8 | — | 2 | 2 | no | — |
| 314 | C8 | — | 4 | 2 | no | — |
| 315 | C8 | — | 424 | 1 | no | — |
| 316 | C8 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 317 | C8 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 318 | C8 | #13133185 | 1 | 1 | yes | characteristic |
| 319 | C8 | #13133198 | 1 | 1 | yes | characteristic |
| 320 | C22 | — | 2 | 1 | no | — |
| 321 | C22 | — | 2 | 1 | no | — |
| 322 | C22 | — | 2 | 2 | no | — |
| 323 | C22 | — | 4 | 1 | no | — |
| 324 | C22 | — | 4 | 1 | no | — |
| 325 | C22 | #16810619 | 1 | 1 | yes | characteristic |
| 326 | C22 | #16810850 | 1 | 1 | yes | characteristic |
| 327 | C22 | #16810857 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 2 | 2 | no | — |
| 329 | C4 | — | 8 | 2 | no | — |
| 330 | C4 | — | 848 | 1 | no | — |
| 331 | C4 | #16810503 | 1 | 1 | yes | characteristic |
| 332 | C4 | C2×#13133138 | 1 | 1 | yes | characteristic |
| 333 | C4 | #16810850 | 1 | 1 | yes | characteristic |
| 334 | C4 | #16810857 | 1 | 1 | yes | characteristic |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 2 | 1 | no | — |
| 337 | C2 | — | 4 | 2 | no | — |
| 338 | C2 | #20275144 | 1 | 1 | yes | characteristic |
| 339 | C2 | #20276838 | 1 | 1 | yes | characteristic |
| 340 | C2 | #20276847 | 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