← C71×#60665 #31881537
Subgroups — C71×#60665 #31881537
Order 18176.
1834 conjugacy classes of subgroups.
The subgroup lattice has 250 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C71×#3503 | C2 | 1 | 2 | yes | maximal |
| 2 | C71×#3165 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C71×#3163 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | C71×#3160 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#498×C71 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C71×#2912 | C2 | 1 | 4 | yes | maximal |
| 7 | C4×#131×C71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#131×C71 | C22 | 1 | 2 | yes | — |
| 9 | Q82×C71 | — | 2 | 2 | no | — |
| 10 | #557×C71 | — | 2 | 2 | no | — |
| 11 | #554×C71 | — | 2 | 6 | no | — |
| 12 | D4×Q8×C71 | — | 2 | 6 | no | — |
| 13 | #544×C71 | C22 | 1 | 12 | yes | — |
| 14 | #533×C71 | C22 | 1 | 6 | yes | — |
| 15 | C2×C4⋊Q8×C71 | C22 | 1 | 6 | yes | — |
| 16 | #520×C71 | C22 | 1 | 4 | yes | — |
| 17 | C4×#41×C71 | C22 | 1 | 1 | yes | characteristic |
| 18 | #501×C71 | C22 | 1 | 24 | yes | — |
| 19 | #500×C71 | C22 | 1 | 6 | yes | — |
| 20 | #499×C71 | C22 | 1 | 12 | yes | — |
| 21 | #498×C71 | C22 | 1 | 3 | yes | — |
| 22 | #498×C71 | C22 | 1 | 12 | yes | — |
| 23 | #498×C71 | C22 | 1 | 6 | yes | — |
| 24 | #498×C71 | C22 | 1 | 1 | yes | characteristic |
| 25 | #428×C71 | C22 | 1 | 24 | yes | — |
| 26 | #427×C71 | C22 | 1 | 12 | yes | — |
| 27 | C2×#106×C71 | C22 | 1 | 12 | yes | — |
| 28 | #405×C71 | C22 | 1 | 6 | yes | — |
| 29 | C4×M16×C71 | C22 | 1 | 3 | yes | — |
| 30 | C2×C4×C8×C71 | C22 | 1 | 3 | yes | — |
| 31 | C2×#41×C71 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×Q8×C71 | — | 2 | 6 | no | — |
| 33 | #131×C71 | Q8 | 1 | 16 | yes | complemented |
| 34 | C2×M16×C71 | C23 | 1 | 6 | yes | — |
| 35 | C22×C8×C71 | C23 | 1 | 6 | yes | — |
| 36 | C4⋊Q8×C71 | — | 2 | 12 | no | — |
| 37 | C4⋊Q8×C71 | — | 2 | 6 | no | — |
| 38 | C4⋊Q8×C71 | — | 2 | 6 | no | — |
| 39 | C4⋊Q8×C71 | C23 | 1 | 6 | yes | — |
| 40 | C4⋊Q8×C71 | C23 | 1 | 12 | yes | — |
| 41 | C4⋊Q8×C71 | C23 | 1 | 2 | yes | — |
| 42 | #125×C71 | — | 2 | 12 | no | — |
| 43 | #125×C71 | C23 | 1 | 12 | yes | — |
| 44 | C22⋊Q8×C71 | — | 2 | 6 | no | — |
| 45 | C22⋊Q8×C71 | — | 2 | 6 | no | — |
| 46 | C22⋊Q8×C71 | — | 2 | 24 | no | — |
| 47 | C4×Q8×C71 | — | 2 | 4 | no | — |
| 48 | C4×Q8×C71 | — | 2 | 6 | no | — |
| 49 | C4×Q8×C71 | — | 2 | 6 | no | — |
| 50 | C4×Q8×C71 | D4 | 1 | 2 | yes | — |
| 51 | C4×D4×C71 | — | 2 | 12 | no | — |
| 52 | C4×D4×C71 | D4 | 1 | 6 | yes | — |
| 53 | #117×C71 | C23 | 1 | 12 | yes | — |
| 54 | #117×C71 | C23 | 1 | 3 | yes | — |
| 55 | C2×C4⋊C4×C71 | — | 2 | 6 | no | — |
| 56 | C2×C4⋊C4×C71 | C23 | 1 | 12 | yes | — |
| 57 | C2×C42×C71 | C23 | 1 | 3 | yes | — |
| 58 | #107×C71 | C23 | 1 | 24 | yes | — |
| 59 | #106×C71 | C23 | 1 | 12 | yes | — |
| 60 | #106×C71 | C23 | 1 | 24 | yes | — |
| 61 | #106×C71 | C23 | 1 | 4 | yes | — |
| 62 | #97×C71 | C23 | 1 | 6 | yes | — |
| 63 | C4×C8×C71 | C23 | 1 | 6 | yes | — |
| 64 | C4×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4×C8×C71 | C23 | 1 | 3 | yes | — |
| 66 | #41×C71 | C2×Q8 | 1 | 8 | yes | — |
| 67 | C2×Q8×C71 | — | 2 | 2 | no | — |
| 68 | C2×Q8×C71 | — | 2 | 6 | no | — |
| 69 | C2×Q8×C71 | — | 2 | 12 | no | — |
| 70 | C2×Q8×C71 | — | 8 | 12 | no | — |
| 71 | C2×Q8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×D4×C71 | C2×D4 | 1 | 3 | yes | — |
| 73 | C22×C4×C71 | — | 2 | 12 | no | — |
| 74 | C22×C4×C71 | C2×D4 | 1 | 6 | yes | — |
| 75 | C22×C4×C71 | C24 | 1 | 3 | yes | — |
| 76 | M16×C71 | C2×Q8 | 1 | 24 | yes | complemented |
| 77 | C2×C8×C71 | C2×Q8 | 1 | 24 | yes | complemented |
| 78 | C2×C8×C71 | C24 | 1 | 2 | yes | — |
| 79 | C2×C8×C71 | C24 | 1 | 6 | yes | — |
| 80 | C4⋊C4×C71 | — | 2 | 6 | no | — |
| 81 | C4⋊C4×C71 | — | 2 | 12 | no | — |
| 82 | C4⋊C4×C71 | — | 2 | 12 | no | — |
| 83 | C4⋊C4×C71 | — | 2 | 2 | no | — |
| 84 | C4⋊C4×C71 | — | 2 | 12 | no | — |
| 85 | C4⋊C4×C71 | — | 2 | 12 | no | — |
| 86 | C4⋊C4×C71 | C2×D4 | 1 | 6 | yes | — |
| 87 | C4⋊C4×C71 | C24 | 1 | 12 | yes | — |
| 88 | C4⋊C4×C71 | C24 | 1 | 4 | yes | — |
| 89 | C22⋊C4×C71 | — | 2 | 12 | no | — |
| 90 | C22⋊C4×C71 | C2×D4 | 1 | 6 | yes | — |
| 91 | C42×C71 | — | 2 | 12 | no | — |
| 92 | C42×C71 | C2×D4 | 1 | 6 | yes | — |
| 93 | C42×C71 | C24 | 1 | 1 | yes | characteristic |
| 94 | C42×C71 | C24 | 1 | 3 | yes | — |
| 95 | C23×C71 | C22×D4 | 1 | 3 | yes | — |
| 96 | Q8×C71 | — | 4 | 4 | no | — |
| 97 | Q8×C71 | — | 4 | 12 | no | — |
| 98 | Q8×C71 | — | 8 | 12 | no | — |
| 99 | Q8×C71 | C4⋊Q8 | 1 | 4 | yes | — |
| 100 | D4×C71 | C4⋊Q8 | 1 | 12 | yes | — |
| 101 | C2×C4×C71 | — | 2 | 12 | no | — |
| 102 | C2×C4×C71 | — | 2 | 4 | no | — |
| 103 | C2×C4×C71 | — | 2 | 12 | no | — |
| 104 | C2×C4×C71 | — | 8 | 12 | no | — |
| 105 | C2×C4×C71 | C22×D4 | 1 | 6 | yes | — |
| 106 | C2×C4×C71 | C22×D4 | 1 | 3 | yes | — |
| 107 | C2×C4×C71 | C22×D4 | 1 | 2 | yes | — |
| 108 | C2×C4×C71 | C22×Q8 | 1 | 12 | yes | — |
| 109 | C2×C4×C71 | C25 | 1 | 1 | yes | characteristic |
| 110 | C8×C71 | C22×Q8 | 1 | 4 | yes | — |
| 111 | C8×C71 | C22×Q8 | 1 | 12 | yes | — |
| 112 | C22×C71 | — | 2 | 6 | no | — |
| 113 | C22×C71 | C2×C4⋊Q8 | 1 | 6 | yes | — |
| 114 | C22×C71 | C23×D4 | 1 | 1 | yes | characteristic |
| 115 | C4×C71 | — | 2 | 6 | no | — |
| 116 | C4×C71 | — | 4 | 12 | no | — |
| 117 | C4×C71 | — | 4 | 4 | no | — |
| 118 | C4×C71 | C2×C4⋊Q8 | 1 | 6 | yes | — |
| 119 | C4×C71 | #577 | 1 | 4 | yes | — |
| 120 | C4×C71 | C23×Q8 | 1 | 2 | yes | — |
| 121 | #60665 | C71 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 122 | C2×C71 | — | 2 | 6 | no | — |
| 123 | C2×C71 | C22×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 124 | C2×C71 | C2×#577 | 1 | 2 | yes | — |
| 125 | #3503 | C2×C71 | 1 | 2 | yes | — |
| 126 | #3165 | C2×C71 | 1 | 6 | yes | complemented |
| 127 | #3163 | C2×C71 | 1 | 12 | yes | complemented |
| 128 | #3160 | C2×C71 | 1 | 3 | yes | complemented |
| 129 | C2×#498 | C2×C71 | 1 | 3 | yes | complemented |
| 130 | #2912 | C2×C71 | 1 | 4 | yes | — |
| 131 | C4×#131 | C2×C71 | 1 | 1 | yes | characteristic |
| 132 | C71 | #60665 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 133 | C2×#131 | C22×C71 | 1 | 2 | yes | — |
| 134 | Q82 | — | 2 | 2 | no | — |
| 135 | #557 | — | 2 | 2 | no | — |
| 136 | #554 | — | 2 | 6 | no | — |
| 137 | D4×Q8 | — | 2 | 6 | no | — |
| 138 | #544 | C22×C71 | 1 | 12 | yes | — |
| 139 | #533 | C22×C71 | 1 | 6 | yes | — |
| 140 | C2×C4⋊Q8 | C22×C71 | 1 | 6 | yes | — |
| 141 | #520 | C22×C71 | 1 | 4 | yes | — |
| 142 | C4×#41 | C22×C71 | 1 | 1 | yes | characteristic |
| 143 | #501 | C22×C71 | 1 | 24 | yes | — |
| 144 | #500 | C22×C71 | 1 | 6 | yes | — |
| 145 | #499 | C22×C71 | 1 | 12 | yes | — |
| 146 | #498 | C22×C71 | 1 | 3 | yes | — |
| 147 | #498 | C22×C71 | 1 | 12 | yes | — |
| 148 | #498 | C22×C71 | 1 | 6 | yes | — |
| 149 | #498 | C22×C71 | 1 | 1 | yes | characteristic |
| 150 | #428 | C22×C71 | 1 | 24 | yes | — |
| 151 | #427 | C22×C71 | 1 | 12 | yes | — |
| 152 | C2×#106 | C22×C71 | 1 | 12 | yes | — |
| 153 | #405 | C22×C71 | 1 | 6 | yes | — |
| 154 | C4×M16 | C22×C71 | 1 | 3 | yes | — |
| 155 | C2×C4×C8 | C22×C71 | 1 | 3 | yes | — |
| 156 | C2×#41 | C23×C71 | 1 | 1 | yes | characteristic |
| 157 | C22×Q8 | — | 2 | 6 | no | — |
| 158 | #131 | Q8×C71 | 1 | 16 | yes | complemented |
| 159 | C2×M16 | C23×C71 | 1 | 6 | yes | — |
| 160 | C22×C8 | C23×C71 | 1 | 6 | yes | — |
| 161 | C4⋊Q8 | — | 2 | 12 | no | — |
| 162 | C4⋊Q8 | — | 2 | 6 | no | — |
| 163 | C4⋊Q8 | — | 2 | 6 | no | — |
| 164 | C4⋊Q8 | C23×C71 | 1 | 6 | yes | — |
| 165 | C4⋊Q8 | C23×C71 | 1 | 12 | yes | — |
| 166 | C4⋊Q8 | C23×C71 | 1 | 2 | yes | — |
| 167 | #125 | — | 2 | 12 | no | — |
| 168 | #125 | C23×C71 | 1 | 12 | yes | — |
| 169 | C22⋊Q8 | — | 2 | 6 | no | — |
| 170 | C22⋊Q8 | — | 2 | 6 | no | — |
| 171 | C22⋊Q8 | — | 2 | 24 | no | — |
| 172 | C4×Q8 | — | 2 | 4 | no | — |
| 173 | C4×Q8 | — | 2 | 6 | no | — |
| 174 | C4×Q8 | — | 2 | 6 | no | — |
| 175 | C4×Q8 | D4×C71 | 1 | 2 | yes | — |
| 176 | C4×D4 | — | 2 | 12 | no | — |
| 177 | C4×D4 | D4×C71 | 1 | 6 | yes | — |
| 178 | #117 | C23×C71 | 1 | 12 | yes | — |
| 179 | #117 | C23×C71 | 1 | 3 | yes | — |
| 180 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 181 | C2×C4⋊C4 | C23×C71 | 1 | 12 | yes | — |
| 182 | C2×C42 | C23×C71 | 1 | 3 | yes | — |
| 183 | #107 | C23×C71 | 1 | 24 | yes | — |
| 184 | #106 | C23×C71 | 1 | 12 | yes | — |
| 185 | #106 | C23×C71 | 1 | 24 | yes | — |
| 186 | #106 | C23×C71 | 1 | 4 | yes | — |
| 187 | #97 | C23×C71 | 1 | 6 | yes | — |
| 188 | C4×C8 | C23×C71 | 1 | 1 | yes | characteristic |
| 189 | C4×C8 | C23×C71 | 1 | 3 | yes | — |
| 190 | C4×C8 | C23×C71 | 1 | 6 | yes | — |
| 191 | #41 | C2×Q8×C71 | 1 | 8 | yes | — |
| 192 | C2×Q8 | — | 2 | 2 | no | — |
| 193 | C2×Q8 | — | 2 | 6 | no | — |
| 194 | C2×Q8 | — | 2 | 12 | no | — |
| 195 | C2×Q8 | — | 8 | 12 | no | — |
| 196 | C2×Q8 | C2×D4×C71 | 1 | 1 | yes | characteristic |
| 197 | C2×D4 | C2×D4×C71 | 1 | 3 | yes | — |
| 198 | C22×C4 | — | 2 | 12 | no | — |
| 199 | C22×C4 | C2×D4×C71 | 1 | 6 | yes | — |
| 200 | C22×C4 | C24×C71 | 1 | 3 | yes | — |
| 201 | M16 | C2×Q8×C71 | 1 | 24 | yes | complemented |
| 202 | C2×C8 | C2×Q8×C71 | 1 | 24 | yes | complemented |
| 203 | C2×C8 | C24×C71 | 1 | 2 | yes | — |
| 204 | C2×C8 | C24×C71 | 1 | 6 | yes | — |
| 205 | C4⋊C4 | — | 2 | 6 | no | — |
| 206 | C4⋊C4 | — | 2 | 12 | no | — |
| 207 | C4⋊C4 | — | 2 | 12 | no | — |
| 208 | C4⋊C4 | — | 2 | 2 | no | — |
| 209 | C4⋊C4 | — | 2 | 12 | no | — |
| 210 | C4⋊C4 | — | 2 | 12 | no | — |
| 211 | C4⋊C4 | C2×D4×C71 | 1 | 6 | yes | — |
| 212 | C4⋊C4 | C24×C71 | 1 | 12 | yes | — |
| 213 | C4⋊C4 | C24×C71 | 1 | 4 | yes | — |
| 214 | C22⋊C4 | — | 2 | 12 | no | — |
| 215 | C22⋊C4 | C2×D4×C71 | 1 | 6 | yes | — |
| 216 | C42 | — | 2 | 12 | no | — |
| 217 | C42 | C2×D4×C71 | 1 | 6 | yes | — |
| 218 | C42 | C24×C71 | 1 | 1 | yes | characteristic |
| 219 | C42 | C24×C71 | 1 | 3 | yes | — |
| 220 | C23 | C22×D4×C71 | 1 | 3 | yes | — |
| 221 | Q8 | — | 4 | 12 | no | — |
| 222 | Q8 | — | 4 | 4 | no | — |
| 223 | Q8 | — | 8 | 12 | no | — |
| 224 | Q8 | C4⋊Q8×C71 | 1 | 4 | yes | — |
| 225 | D4 | C4⋊Q8×C71 | 1 | 12 | yes | — |
| 226 | C2×C4 | — | 2 | 12 | no | — |
| 227 | C2×C4 | — | 2 | 4 | no | — |
| 228 | C2×C4 | — | 2 | 12 | no | — |
| 229 | C2×C4 | — | 8 | 12 | no | — |
| 230 | C2×C4 | C22×D4×C71 | 1 | 6 | yes | — |
| 231 | C2×C4 | C22×D4×C71 | 1 | 3 | yes | — |
| 232 | C2×C4 | C22×D4×C71 | 1 | 2 | yes | — |
| 233 | C2×C4 | C22×Q8×C71 | 1 | 12 | yes | — |
| 234 | C2×C4 | C25×C71 | 1 | 1 | yes | characteristic |
| 235 | C8 | C22×Q8×C71 | 1 | 4 | yes | — |
| 236 | C8 | C22×Q8×C71 | 1 | 12 | yes | — |
| 237 | C22 | — | 2 | 6 | no | — |
| 238 | C22 | C2×C4⋊Q8×C71 | 1 | 6 | yes | — |
| 239 | C22 | C23×D4×C71 | 1 | 1 | yes | characteristic |
| 240 | C4 | — | 2 | 6 | no | — |
| 241 | C4 | — | 4 | 12 | no | — |
| 242 | C4 | — | 4 | 4 | no | — |
| 243 | C4 | C2×C4⋊Q8×C71 | 1 | 6 | yes | — |
| 244 | C4 | #577×C71 | 1 | 4 | yes | — |
| 245 | C4 | C23×Q8×C71 | 1 | 2 | yes | — |
| 246 | C2 | — | 2 | 6 | no | — |
| 247 | C2 | C22×C4⋊Q8×C71 | 1 | 1 | yes | characteristic |
| 248 | C2 | C2×#577×C71 | 1 | 2 | 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