← #32911316
Subgroups — #32911316
Order 18176.
10796 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22603151 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22603127 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | #22603119 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | #22603111 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C2×#17986807 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | #22603052 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C71×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #17986836 | — | 2 | 6 | no | — |
| 9 | #17986835 | C22 | 1 | 15 | yes | complemented |
| 10 | C2×#14556174 | C22 | 1 | 15 | yes | complemented |
| 11 | H(5,𝔽2)×D71 | — | 2 | 10 | no | — |
| 12 | #17986814 | C22 | 1 | 60 | yes | complemented |
| 13 | #17986810 | C22 | 1 | 20 | yes | complemented |
| 14 | #17986807 | C22 | 1 | 60 | yes | complemented |
| 15 | #17986807 | C22 | 1 | 120 | yes | complemented |
| 16 | #17986802 | C22 | 1 | 60 | yes | complemented |
| 17 | #17986797 | C22 | 1 | 90 | yes | complemented |
| 18 | C2×#14556159 | C22 | 1 | 45 | yes | complemented |
| 19 | C2×#14556153 | C22 | 1 | 45 | yes | complemented |
| 20 | C2×#14556148 | C22 | 1 | 45 | yes | complemented |
| 21 | C22×#11765852 | C22 | 1 | 15 | yes | complemented |
| 22 | #17986766 | C22 | 1 | 15 | yes | complemented |
| 23 | C2×#14556135 | C22 | 1 | 15 | yes | complemented |
| 24 | #585×C71 | — | 2 | 10 | no | — |
| 25 | #585×C71 | C22 | 1 | 1 | yes | characteristic |
| 26 | #585×C71 | C22 | 1 | 15 | yes | complemented |
| 27 | C2×2−1+4×C71 | — | 2 | 6 | no | — |
| 28 | C2×2−1+4×C71 | C22 | 1 | 15 | yes | complemented |
| 29 | #14556176 | — | 2 | 15 | no | — |
| 30 | #14556175 | C23 | 1 | 20 | yes | — |
| 31 | #14556174 | — | 2 | 60 | no | — |
| 32 | #14556174 | C23 | 1 | 60 | yes | — |
| 33 | #14556173 | — | 2 | 45 | no | — |
| 34 | C2×#11765866 | — | 2 | 15 | no | — |
| 35 | C2×#11765865 | C23 | 1 | 45 | yes | — |
| 36 | #41×D71 | — | 2 | 60 | no | — |
| 37 | C22×D284 | C23 | 1 | 15 | yes | — |
| 38 | C2×D4×D71 | — | 2 | 45 | no | — |
| 39 | #14556159 | C23 | 1 | 180 | yes | complemented |
| 40 | #14556158 | C23 | 1 | 60 | yes | complemented |
| 41 | #14556154 | C23 | 1 | 60 | yes | complemented |
| 42 | #14556153 | C23 | 1 | 180 | yes | complemented |
| 43 | #14556148 | C23 | 1 | 60 | yes | complemented |
| 44 | #14556148 | C23 | 1 | 180 | yes | complemented |
| 45 | C2×#11765855 | C23 | 1 | 45 | yes | complemented |
| 46 | C2×#11765853 | C23 | 1 | 15 | yes | complemented |
| 47 | C2×#11765852 | C23 | 1 | 45 | yes | complemented |
| 48 | C2×#11765852 | C23 | 1 | 90 | yes | complemented |
| 49 | C2×#11765850 | C23 | 1 | 45 | yes | complemented |
| 50 | #14556135 | C23 | 1 | 20 | yes | complemented |
| 51 | #14556135 | C23 | 1 | 60 | yes | complemented |
| 52 | C2×#11765849 | C23 | 1 | 45 | yes | complemented |
| 53 | C22×C71⋊C8 | C23 | 1 | 15 | yes | complemented |
| 54 | 2−1+4×C71 | — | 2 | 60 | no | — |
| 55 | 2−1+4×C71 | — | 2 | 15 | no | — |
| 56 | 2−1+4×C71 | D4 | 1 | 6 | yes | — |
| 57 | 2−1+4×C71 | C23 | 1 | 60 | yes | complemented |
| 58 | H(5,𝔽2)×C71 | — | 2 | 45 | no | — |
| 59 | H(5,𝔽2)×C71 | D4 | 1 | 10 | yes | — |
| 60 | H(5,𝔽2)×C71 | C23 | 1 | 20 | yes | complemented |
| 61 | C2×#41×C71 | — | 2 | 60 | no | — |
| 62 | C2×#41×C71 | — | 2 | 45 | no | — |
| 63 | C2×#41×C71 | C23 | 1 | 15 | yes | — |
| 64 | C2×#41×C71 | C23 | 1 | 45 | yes | complemented |
| 65 | C22×Q8×C71 | — | 2 | 15 | no | — |
| 66 | C22×Q8×C71 | C23 | 1 | 15 | yes | complemented |
| 67 | C23×D71 | — | 2 | 15 | no | — |
| 68 | #11765866 | — | 2 | 60 | no | — |
| 69 | #11765865 | — | 2 | 90 | no | — |
| 70 | #11765865 | C24 | 1 | 60 | yes | — |
| 71 | #11765864 | — | 2 | 60 | no | — |
| 72 | C2×Dic142 | C24 | 1 | 15 | yes | — |
| 73 | Q8×D71 | — | 2 | 20 | no | — |
| 74 | C2×D284 | — | 2 | 45 | no | — |
| 75 | C2×D284 | C24 | 1 | 45 | yes | — |
| 76 | C2×C22⋊D71 | — | 2 | 45 | no | — |
| 77 | D4×D71 | — | 2 | 180 | no | — |
| 78 | C2×C4×D71 | — | 2 | 45 | no | — |
| 79 | #11765855 | C24 | 1 | 60 | yes | — |
| 80 | #11765853 | C24 | 1 | 6 | yes | — |
| 81 | #11765853 | C24 | 1 | 30 | yes | — |
| 82 | #11765852 | C24 | 1 | 10 | yes | — |
| 83 | #11765852 | C24 | 1 | 90 | yes | — |
| 84 | #11765850 | C24 | 1 | 60 | yes | — |
| 85 | #11765849 | C24 | 1 | 60 | yes | — |
| 86 | C2×C71⋊C8 | C24 | 1 | 15 | yes | — |
| 87 | C2×C71⋊C8 | C24 | 1 | 45 | yes | — |
| 88 | #41×C71 | — | 2 | 20 | no | — |
| 89 | #41×C71 | — | 2 | 180 | no | — |
| 90 | #41×C71 | — | 2 | 90 | no | — |
| 91 | #41×C71 | C2×D4 | 1 | 60 | yes | — |
| 92 | #41×C71 | C24 | 1 | 60 | yes | — |
| 93 | #41×C71 | C24 | 1 | 20 | yes | — |
| 94 | C2×Q8×C71 | — | 2 | 60 | no | — |
| 95 | C2×Q8×C71 | — | 2 | 45 | no | — |
| 96 | C2×Q8×C71 | C2×D4 | 1 | 15 | yes | — |
| 97 | C2×Q8×C71 | C24 | 1 | 45 | yes | — |
| 98 | C2×D4×C71 | — | 2 | 45 | no | — |
| 99 | C2×D4×C71 | — | 2 | 60 | no | — |
| 100 | C2×D4×C71 | C2×D4 | 1 | 45 | yes | — |
| 101 | C2×D4×C71 | C24 | 1 | 15 | yes | — |
| 102 | C22×C4×C71 | — | 2 | 15 | no | — |
| 103 | C22×C4×C71 | — | 2 | 45 | no | — |
| 104 | C22×C4×C71 | C24 | 1 | 15 | yes | — |
| 105 | C23×C71 | — | 2 | 15 | no | — |
| 106 | C23×C71 | C22×D4 | 1 | 15 | yes | — |
| 107 | C22×D71 | — | 2 | 45 | no | — |
| 108 | C22×D71 | — | 16 | 15 | no | — |
| 109 | Q8×C71 | — | 2 | 45 | no | — |
| 110 | Q8×C71 | C22×D4 | 1 | 20 | yes | — |
| 111 | Q8×C71 | C25 | 1 | 10 | yes | — |
| 112 | D4×C71 | — | 2 | 60 | no | — |
| 113 | D4×C71 | — | 2 | 15 | no | — |
| 114 | D4×C71 | C22×D4 | 1 | 60 | yes | — |
| 115 | D4×C71 | C25 | 1 | 6 | yes | — |
| 116 | C2×C4×C71 | — | 2 | 45 | no | — |
| 117 | C2×C4×C71 | — | 2 | 60 | no | — |
| 118 | C2×C4×C71 | C22×D4 | 1 | 45 | yes | — |
| 119 | C2×C4×C71 | C25 | 1 | 15 | yes | — |
| 120 | C22⋊D71 | — | 2 | 60 | no | — |
| 121 | C2×Dic71 | — | 2 | 15 | no | — |
| 122 | D284 | — | 2 | 45 | no | — |
| 123 | D284 | C25 | 1 | 10 | yes | — |
| 124 | C4×D71 | — | 2 | 60 | no | — |
| 125 | Dic142 | — | 2 | 15 | no | — |
| 126 | Dic142 | C25 | 1 | 6 | yes | — |
| 127 | C71⋊C8 | C25 | 1 | 1 | yes | characteristic |
| 128 | C71⋊C8 | C25 | 1 | 15 | yes | — |
| 129 | C22×C71 | — | 2 | 6 | no | — |
| 130 | C22×C71 | — | 4 | 15 | no | — |
| 131 | C22×C71 | — | 8 | 15 | no | — |
| 132 | C22×C71 | C23×D4 | 1 | 15 | yes | — |
| 133 | C2×D71 | — | 2 | 10 | no | — |
| 134 | C2×D71 | — | 8 | 45 | no | — |
| 135 | C4×C71 | — | 2 | 10 | no | — |
| 136 | C4×C71 | C23×D4 | 1 | 15 | yes | — |
| 137 | C4×C71 | C26 | 1 | 1 | yes | characteristic |
| 138 | Dic71 | — | 2 | 6 | no | — |
| 139 | #63092 | — | 71 | 1 | no | maximal, Sylow |
| 140 | C2×C71 | — | 2 | 15 | no | — |
| 141 | C2×C71 | — | 4 | 6 | no | — |
| 142 | C2×C71 | C24×D4 | 1 | 1 | yes | characteristic |
| 143 | D71 | — | 4 | 10 | no | — |
| 144 | 2−1+6 | D71 | 1 | 1 | yes | characteristic, complemented |
| 145 | H(7,𝔽2) | — | 71 | 1 | no | — |
| 146 | #3586 | — | 71 | 15 | no | — |
| 147 | #3585 | — | 71 | 15 | no | — |
| 148 | #3584 | — | 71 | 15 | no | — |
| 149 | C2×#577 | — | 71 | 15 | no | — |
| 150 | #3573 | — | 71 | 1 | no | — |
| 151 | C71 | #63092 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 152 | #585 | — | 2 | 10 | no | — |
| 153 | #585 | — | 71 | 15 | no | — |
| 154 | #585 | — | 142 | 6 | no | — |
| 155 | #585 | C2×D71 | 1 | 1 | yes | characteristic |
| 156 | #585 | C2×D71 | 1 | 15 | yes | complemented |
| 157 | C2×2−1+4 | — | 2 | 6 | no | — |
| 158 | C2×2−1+4 | C2×D71 | 1 | 15 | yes | complemented |
| 159 | C2×H(5,𝔽2) | — | 71 | 15 | no | — |
| 160 | C2×H(5,𝔽2) | — | 142 | 10 | no | — |
| 161 | #578 | — | 71 | 60 | no | — |
| 162 | #577 | — | 71 | 60 | no | — |
| 163 | #577 | — | 71 | 120 | no | — |
| 164 | #577 | — | 71 | 20 | no | — |
| 165 | #576 | — | 71 | 60 | no | — |
| 166 | #575 | — | 71 | 90 | no | — |
| 167 | C2×#137 | — | 71 | 45 | no | — |
| 168 | C2×AGL(1,ℤ/8) | — | 71 | 45 | no | — |
| 169 | C2×#135 | — | 71 | 45 | no | — |
| 170 | C22×SD16 | — | 71 | 15 | no | — |
| 171 | #568 | — | 71 | 15 | no | — |
| 172 | C2×#131 | — | 71 | 15 | no | — |
| 173 | 2−1+4 | — | 2 | 60 | no | — |
| 174 | 2−1+4 | — | 2 | 15 | no | — |
| 175 | 2−1+4 | — | 71 | 20 | no | — |
| 176 | 2−1+4 | — | 142 | 15 | no | — |
| 177 | 2−1+4 | C22⋊D71 | 1 | 6 | yes | — |
| 178 | 2−1+4 | C22×D71 | 1 | 60 | yes | complemented |
| 179 | H(5,𝔽2) | — | 2 | 45 | no | — |
| 180 | H(5,𝔽2) | — | 71 | 60 | no | — |
| 181 | H(5,𝔽2) | — | 142 | 60 | no | — |
| 182 | H(5,𝔽2) | — | 142 | 45 | no | — |
| 183 | H(5,𝔽2) | C22⋊D71 | 1 | 10 | yes | — |
| 184 | H(5,𝔽2) | C22×D71 | 1 | 20 | yes | complemented |
| 185 | C2×#41 | — | 2 | 60 | no | — |
| 186 | C2×#41 | — | 2 | 45 | no | — |
| 187 | C2×#41 | — | 71 | 45 | no | — |
| 188 | C2×#41 | — | 142 | 60 | no | — |
| 189 | C2×#41 | — | 142 | 15 | no | — |
| 190 | C2×#41 | C22×D71 | 1 | 15 | yes | — |
| 191 | C2×#41 | C22×D71 | 1 | 45 | yes | complemented |
| 192 | C22×Q8 | — | 2 | 15 | no | — |
| 193 | C22×Q8 | C22×D71 | 1 | 15 | yes | complemented |
| 194 | C22×D4 | — | 71 | 15 | no | — |
| 195 | C22×D4 | — | 142 | 45 | no | — |
| 196 | #137 | — | 71 | 60 | no | — |
| 197 | #137 | — | 71 | 180 | no | — |
| 198 | AGL(1,ℤ/8) | — | 71 | 180 | no | — |
| 199 | AGL(1,ℤ/8) | — | 71 | 60 | no | — |
| 200 | #135 | — | 71 | 60 | no | — |
| 201 | #135 | — | 71 | 180 | no | — |
| 202 | C2×Q16 | — | 71 | 45 | no | — |
| 203 | C2×SD16 | — | 71 | 45 | no | — |
| 204 | C2×SD16 | — | 71 | 90 | no | — |
| 205 | C2×SD16 | — | 71 | 15 | no | — |
| 206 | C2×D8 | — | 71 | 45 | no | — |
| 207 | #131 | — | 71 | 20 | no | — |
| 208 | #131 | — | 71 | 60 | no | — |
| 209 | C2×M16 | — | 71 | 45 | no | — |
| 210 | C22×C8 | — | 71 | 15 | no | — |
| 211 | C24 | — | 142 | 15 | no | — |
| 212 | #41 | — | 2 | 20 | no | — |
| 213 | #41 | — | 2 | 180 | no | — |
| 214 | #41 | — | 2 | 90 | no | — |
| 215 | #41 | — | 71 | 60 | no | — |
| 216 | #41 | — | 142 | 60 | no | — |
| 217 | #41 | — | 142 | 60 | no | — |
| 218 | #41 | — | 142 | 90 | no | — |
| 219 | #41 | C2×C22⋊D71 | 1 | 60 | yes | — |
| 220 | #41 | C23×D71 | 1 | 60 | yes | — |
| 221 | #41 | C23×D71 | 1 | 20 | yes | — |
| 222 | C2×Q8 | — | 2 | 60 | no | — |
| 223 | C2×Q8 | — | 2 | 45 | no | — |
| 224 | C2×Q8 | — | 71 | 15 | no | — |
| 225 | C2×Q8 | — | 142 | 20 | no | — |
| 226 | C2×Q8 | C2×C22⋊D71 | 1 | 15 | yes | — |
| 227 | C2×Q8 | C23×D71 | 1 | 45 | yes | — |
| 228 | C2×D4 | — | 2 | 45 | no | — |
| 229 | C2×D4 | — | 2 | 60 | no | — |
| 230 | C2×D4 | — | 71 | 45 | no | — |
| 231 | C2×D4 | — | 142 | 45 | no | — |
| 232 | C2×D4 | — | 142 | 180 | no | — |
| 233 | C2×D4 | — | 142 | 45 | no | — |
| 234 | C2×D4 | C2×C22⋊D71 | 1 | 45 | yes | — |
| 235 | C2×D4 | C23×D71 | 1 | 15 | yes | — |
| 236 | C22×C4 | — | 2 | 15 | no | — |
| 237 | C22×C4 | — | 2 | 45 | no | — |
| 238 | C22×C4 | — | 142 | 45 | no | — |
| 239 | C22×C4 | C23×D71 | 1 | 15 | yes | — |
| 240 | Q16 | — | 71 | 60 | no | — |
| 241 | SD16 | — | 71 | 10 | no | — |
| 242 | SD16 | — | 71 | 90 | no | — |
| 243 | SD16 | — | 71 | 6 | no | — |
| 244 | SD16 | — | 71 | 30 | no | — |
| 245 | D8 | — | 71 | 60 | no | — |
| 246 | M16 | — | 71 | 60 | no | — |
| 247 | C2×C8 | — | 71 | 15 | no | — |
| 248 | C2×C8 | — | 71 | 45 | no | — |
| 249 | C23 | — | 2 | 15 | no | — |
| 250 | C23 | — | 1136 | 15 | no | — |
| 251 | C23 | — | 142 | 45 | no | — |
| 252 | C23 | C22×C22⋊D71 | 1 | 15 | yes | — |
| 253 | Q8 | — | 2 | 45 | no | — |
| 254 | Q8 | — | 71 | 6 | no | — |
| 255 | Q8 | — | 142 | 15 | no | — |
| 256 | Q8 | C22×C22⋊D71 | 1 | 20 | yes | — |
| 257 | Q8 | C24×D71 | 1 | 10 | yes | — |
| 258 | D4 | — | 2 | 60 | no | — |
| 259 | D4 | — | 2 | 15 | no | — |
| 260 | D4 | — | 71 | 10 | no | — |
| 261 | D4 | — | 142 | 60 | no | — |
| 262 | D4 | — | 142 | 45 | no | — |
| 263 | D4 | C22×C22⋊D71 | 1 | 60 | yes | — |
| 264 | D4 | C24×D71 | 1 | 6 | yes | — |
| 265 | C2×C4 | — | 2 | 45 | no | — |
| 266 | C2×C4 | — | 2 | 60 | no | — |
| 267 | C2×C4 | — | 142 | 15 | no | — |
| 268 | C2×C4 | — | 142 | 60 | no | — |
| 269 | C2×C4 | C22×C22⋊D71 | 1 | 45 | yes | — |
| 270 | C2×C4 | C24×D71 | 1 | 15 | yes | — |
| 271 | C8 | — | 71 | 1 | no | — |
| 272 | C8 | — | 71 | 15 | no | — |
| 273 | C22 | — | 2 | 6 | no | — |
| 274 | C22 | — | 4 | 15 | no | — |
| 275 | C22 | — | 8 | 15 | no | — |
| 276 | C22 | — | 568 | 45 | no | — |
| 277 | C22 | — | 142 | 10 | no | — |
| 278 | C22 | C23×C22⋊D71 | 1 | 15 | yes | — |
| 279 | C4 | — | 2 | 10 | no | — |
| 280 | C4 | — | 142 | 6 | no | — |
| 281 | C4 | C23×C22⋊D71 | 1 | 15 | yes | — |
| 282 | C4 | C25×D71 | 1 | 1 | yes | characteristic |
| 283 | C2 | — | 2 | 15 | no | — |
| 284 | C2 | — | 4 | 6 | no | — |
| 285 | C2 | — | 284 | 10 | no | — |
| 286 | C2 | C24×C22⋊D71 | 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