← C5×#70138 #14303076
Subgroups — C5×#70138 #14303076
Order 1920.
616 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#5825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#5819 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×C5×#413 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×C5×#824 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C5×#4556 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C5×#1915 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C3×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×C5×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×C5×M16 | C22 | 1 | 2 | yes | — |
| 10 | C5×#948 | — | 2 | 4 | no | — |
| 11 | C5×#944 | — | 2 | 1 | no | — |
| 12 | C5×#936 | — | 2 | 1 | no | — |
| 13 | C5×#935 | C4 | 1 | 4 | yes | — |
| 14 | C5×#926 | C22 | 1 | 1 | yes | characteristic |
| 15 | C5×#924 | — | 2 | 2 | no | — |
| 16 | C2×C5×#208 | — | 2 | 2 | no | — |
| 17 | C2×C5×#208 | C22 | 1 | 2 | yes | — |
| 18 | C3×C5×#101 | — | 2 | 1 | no | — |
| 19 | C3×C5×#100 | — | 2 | 1 | no | — |
| 20 | C5×#824 | — | 2 | 4 | no | — |
| 21 | #70138 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 22 | C5×#568 | — | 3 | 1 | no | — |
| 23 | C5×#536 | — | 3 | 1 | no | — |
| 24 | C5×#413 | D3 | 1 | 1 | yes | characteristic |
| 25 | C2×C5×#101 | — | 3 | 2 | no | — |
| 26 | C5×#344 | — | 3 | 2 | no | — |
| 27 | C3×C5×#41 | — | 2 | 2 | no | — |
| 28 | C3×C5×#41 | — | 2 | 2 | no | — |
| 29 | C2×C3×C5×Q8 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C3×C5×D4 | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×C3×C5×D4 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C3×C4×C5 | D4 | 1 | 2 | yes | — |
| 33 | C22×C3×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×C5×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×C5×Dic6 | — | 2 | 1 | no | — |
| 36 | C2×C5×Dic6 | — | 2 | 1 | no | — |
| 37 | C2×C5×Dic6 | — | 2 | 2 | no | — |
| 38 | C2×C5×Dic6 | — | 2 | 2 | no | — |
| 39 | C3×C5×M16 | — | 2 | 2 | no | — |
| 40 | C3×C5×M16 | — | 2 | 2 | no | — |
| 41 | C2×C3×C5×C8 | — | 2 | 2 | no | — |
| 42 | C5×#217 | — | 2 | 4 | no | — |
| 43 | C5×#217 | C2×C4 | 1 | 2 | yes | — |
| 44 | C5×C4⋊Dic3 | — | 2 | 1 | no | — |
| 45 | C5×Dic3⋊C4 | — | 2 | 4 | no | — |
| 46 | C4×C5×Dic3 | — | 2 | 1 | no | — |
| 47 | C5×#208 | — | 2 | 4 | no | — |
| 48 | C5×#208 | — | 2 | 2 | no | — |
| 49 | C5×#208 | D4 | 1 | 4 | yes | — |
| 50 | C2×C5×C3⋊C8 | — | 2 | 4 | no | — |
| 51 | C2×C5×C3⋊C8 | — | 2 | 2 | no | — |
| 52 | #5825 | C2×C5 | 1 | 1 | yes | characteristic |
| 53 | #5819 | C2×C5 | 1 | 1 | yes | characteristic |
| 54 | C3×#413 | C2×C5 | 1 | 1 | yes | characteristic |
| 55 | C2×#824 | C2×C5 | 1 | 2 | yes | complemented |
| 56 | #4556 | C2×C5 | 1 | 2 | yes | complemented |
| 57 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | C22×C5×Q8 | — | 3 | 1 | no | — |
| 59 | C5×#131 | — | 6 | 4 | no | — |
| 60 | C2×C5×M16 | — | 3 | 2 | no | — |
| 61 | C2×C5×M16 | — | 6 | 2 | no | — |
| 62 | C2×C5×M16 | C2×D3 | 1 | 2 | yes | — |
| 63 | C5×C4⋊Q8 | — | 6 | 1 | no | — |
| 64 | C5×#124 | — | 6 | 1 | no | — |
| 65 | C5×#123 | — | 3 | 4 | no | — |
| 66 | C5×C22⋊Q8 | — | 6 | 2 | no | — |
| 67 | C5×#117 | — | 3 | 1 | no | — |
| 68 | C5×#101 | — | 2 | 1 | no | — |
| 69 | C5×#101 | — | 6 | 4 | no | — |
| 70 | C5×#100 | — | 2 | 1 | no | — |
| 71 | #1915 | — | 3 | 1 | no | Sylow |
| 72 | C23×C3×C5 | C22⋊C4 | 1 | 2 | yes | — |
| 73 | C23×C3×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C3×C5×Q8 | — | 2 | 2 | no | — |
| 75 | C3×C5×D4 | — | 2 | 4 | no | — |
| 76 | C3×C5×D4 | — | 2 | 2 | no | — |
| 77 | C2×C3×C4×C5 | — | 2 | 2 | no | — |
| 78 | C2×C3×C4×C5 | — | 2 | 2 | no | — |
| 79 | C2×C3×C4×C5 | C22⋊C4 | 1 | 2 | yes | — |
| 80 | C2×C3×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 81 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×C4×C5 | #41 | 1 | 2 | yes | — |
| 84 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 85 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 86 | C2×C5×Dic3 | — | 2 | 4 | no | — |
| 87 | C5×Dic6 | — | 4 | 1 | no | — |
| 88 | C5×Dic6 | — | 4 | 2 | no | — |
| 89 | C5×Dic6 | — | 4 | 1 | no | — |
| 90 | C3×C5×C8 | — | 4 | 2 | no | — |
| 91 | C5×C3⋊C8 | — | 2 | 4 | no | — |
| 92 | C2×C3×#41 | C22×C5 | 1 | 1 | yes | characteristic |
| 93 | C22×Dic6 | C22×C5 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×M16 | C22×C5 | 1 | 2 | yes | — |
| 95 | #948 | — | 2 | 4 | no | — |
| 96 | #944 | — | 2 | 1 | no | — |
| 97 | #936 | — | 2 | 1 | no | — |
| 98 | #935 | C4×C5 | 1 | 4 | yes | — |
| 99 | #926 | C22×C5 | 1 | 1 | yes | characteristic |
| 100 | #924 | — | 2 | 2 | no | — |
| 101 | C2×#208 | — | 2 | 2 | no | — |
| 102 | C2×#208 | C22×C5 | 1 | 2 | yes | — |
| 103 | C3×#101 | — | 2 | 1 | no | — |
| 104 | C3×#100 | — | 2 | 1 | no | — |
| 105 | #824 | — | 2 | 4 | no | — |
| 106 | C5×#41 | — | 2 | 2 | no | — |
| 107 | C5×#41 | — | 2 | 2 | no | — |
| 108 | C2×C5×Q8 | — | 6 | 1 | no | — |
| 109 | C2×C5×Q8 | — | 6 | 2 | no | — |
| 110 | C2×C5×Q8 | — | 6 | 2 | no | — |
| 111 | C2×C5×Q8 | — | 6 | 1 | no | — |
| 112 | C2×C5×Q8 | #67 | 1 | 1 | yes | characteristic |
| 113 | C2×C5×D4 | C4×D3 | 1 | 2 | yes | — |
| 114 | C2×C5×D4 | #67 | 1 | 1 | yes | characteristic |
| 115 | C22×C4×C5 | — | 3 | 2 | no | — |
| 116 | C22×C4×C5 | D12 | 1 | 2 | yes | — |
| 117 | C22×C4×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 118 | C5×M16 | — | 2 | 2 | no | — |
| 119 | C5×M16 | — | 2 | 2 | no | — |
| 120 | C5×M16 | — | 3 | 4 | no | — |
| 121 | C5×M16 | — | 6 | 4 | no | — |
| 122 | C5×M16 | — | 6 | 2 | no | — |
| 123 | C2×C5×C8 | — | 2 | 2 | no | — |
| 124 | C2×C5×C8 | — | 6 | 4 | no | — |
| 125 | C2×C5×C8 | — | 6 | 2 | no | — |
| 126 | C5×C4⋊C4 | — | 6 | 4 | no | — |
| 127 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 128 | C5×C22⋊C4 | — | 3 | 2 | no | — |
| 129 | C5×C22⋊C4 | — | 6 | 4 | no | — |
| 130 | C42×C5 | — | 6 | 1 | no | — |
| 131 | #568 | — | 3 | 1 | no | — |
| 132 | #536 | — | 3 | 1 | no | — |
| 133 | #413 | C5×D3 | 1 | 1 | yes | characteristic |
| 134 | C2×#101 | — | 3 | 2 | no | — |
| 135 | #344 | — | 3 | 2 | no | — |
| 136 | C22×C3×C5 | — | 2 | 2 | no | — |
| 137 | C22×C3×C5 | — | 4 | 1 | no | — |
| 138 | C22×C3×C5 | — | 4 | 2 | no | — |
| 139 | C22×C3×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 140 | C22×C3×C5 | C4×D4 | 1 | 2 | yes | — |
| 141 | C3×C4×C5 | — | 2 | 2 | no | — |
| 142 | C3×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 143 | C3×C4×C5 | #121 | 1 | 2 | yes | — |
| 144 | C3×C4×C5 | #123 | 1 | 1 | yes | characteristic |
| 145 | C5×Dic3 | — | 4 | 2 | no | — |
| 146 | C5×Dic3 | — | 4 | 2 | no | — |
| 147 | C3×#41 | — | 2 | 2 | no | — |
| 148 | C3×#41 | — | 2 | 2 | no | — |
| 149 | C2×C3×Q8 | C5×D4 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×D4 | C2×C4×C5 | 1 | 2 | yes | — |
| 151 | C2×C3×D4 | C5×D4 | 1 | 1 | yes | characteristic |
| 152 | C22×C3×C4 | C5×D4 | 1 | 2 | yes | — |
| 153 | C22×C3×C4 | C23×C5 | 1 | 1 | yes | characteristic |
| 154 | C22×Dic3 | C2×C4×C5 | 1 | 2 | yes | — |
| 155 | C2×Dic6 | — | 2 | 2 | no | — |
| 156 | C2×Dic6 | — | 2 | 1 | no | — |
| 157 | C2×Dic6 | — | 2 | 1 | no | — |
| 158 | C2×Dic6 | — | 2 | 2 | no | — |
| 159 | C3×M16 | — | 2 | 2 | no | — |
| 160 | C3×M16 | — | 2 | 2 | no | — |
| 161 | C2×C3×C8 | — | 2 | 2 | no | — |
| 162 | #217 | — | 2 | 4 | no | — |
| 163 | #217 | C2×C4×C5 | 1 | 2 | yes | — |
| 164 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 165 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 166 | C4×Dic3 | — | 2 | 1 | no | — |
| 167 | #208 | — | 2 | 4 | no | — |
| 168 | #208 | — | 2 | 2 | no | — |
| 169 | #208 | C5×D4 | 1 | 4 | yes | — |
| 170 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 171 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 172 | C23×C5 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 173 | C23×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 174 | C5×Q8 | — | 2 | 2 | no | — |
| 175 | C5×Q8 | — | 12 | 1 | no | — |
| 176 | C5×Q8 | — | 12 | 1 | no | — |
| 177 | C5×Q8 | — | 12 | 2 | no | — |
| 178 | C5×D4 | — | 2 | 4 | no | — |
| 179 | C5×D4 | — | 2 | 2 | no | — |
| 180 | C2×C4×C5 | — | 2 | 2 | no | — |
| 181 | C2×C4×C5 | — | 2 | 2 | no | — |
| 182 | C2×C4×C5 | — | 6 | 4 | no | — |
| 183 | C2×C4×C5 | — | 6 | 2 | no | — |
| 184 | C2×C4×C5 | — | 6 | 2 | no | — |
| 185 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 186 | C2×C4×C5 | C2×D12 | 1 | 1 | yes | characteristic |
| 187 | C2×C4×C5 | D3×D4 | 1 | 2 | yes | — |
| 188 | C2×C4×C5 | #237 | 1 | 2 | yes | — |
| 189 | C2×C4×C5 | C2×#67 | 1 | 1 | yes | characteristic |
| 190 | C5×C8 | — | 4 | 2 | no | — |
| 191 | C5×C8 | — | 6 | 4 | no | — |
| 192 | C2×#41 | C2×C5×D3 | 1 | 1 | yes | characteristic |
| 193 | C22×Q8 | — | 3 | 1 | no | — |
| 194 | #131 | — | 6 | 4 | no | — |
| 195 | C2×M16 | — | 3 | 2 | no | — |
| 196 | C2×M16 | — | 6 | 2 | no | — |
| 197 | C2×M16 | C2×C5×D3 | 1 | 2 | yes | — |
| 198 | C4⋊Q8 | — | 6 | 1 | no | — |
| 199 | #124 | — | 6 | 1 | no | — |
| 200 | #123 | — | 3 | 4 | no | — |
| 201 | C22⋊Q8 | — | 6 | 2 | no | — |
| 202 | #117 | — | 3 | 1 | no | — |
| 203 | #101 | — | 2 | 1 | no | — |
| 204 | #101 | — | 6 | 4 | no | — |
| 205 | #100 | — | 2 | 1 | no | — |
| 206 | C2×C3×C5 | — | 2 | 1 | no | — |
| 207 | C2×C3×C5 | — | 2 | 2 | no | — |
| 208 | C2×C3×C5 | — | 4 | 2 | no | — |
| 209 | C2×C3×C5 | #386 | 1 | 1 | yes | characteristic |
| 210 | C23×C3 | C5×C22⋊C4 | 1 | 2 | yes | — |
| 211 | C23×C3 | C22×C4×C5 | 1 | 1 | yes | characteristic |
| 212 | C3×Q8 | — | 2 | 2 | no | — |
| 213 | C3×D4 | — | 2 | 4 | no | — |
| 214 | C3×D4 | — | 2 | 2 | no | — |
| 215 | C2×C3×C4 | — | 2 | 2 | no | — |
| 216 | C2×C3×C4 | — | 2 | 2 | no | — |
| 217 | C2×C3×C4 | C5×C22⋊C4 | 1 | 2 | yes | — |
| 218 | C2×C3×C4 | C2×C5×D4 | 1 | 1 | yes | characteristic |
| 219 | C2×C3×C4 | C2×C5×D4 | 1 | 2 | yes | — |
| 220 | C2×C3×C4 | C2×C5×D4 | 1 | 1 | yes | characteristic |
| 221 | C2×C3×C4 | C5×#41 | 1 | 2 | yes | — |
| 222 | C2×Dic3 | — | 2 | 2 | no | — |
| 223 | C2×Dic3 | — | 2 | 2 | no | — |
| 224 | C2×Dic3 | — | 2 | 4 | no | — |
| 225 | Dic6 | — | 4 | 1 | no | — |
| 226 | Dic6 | — | 4 | 2 | no | — |
| 227 | Dic6 | — | 4 | 1 | no | — |
| 228 | C3×C8 | — | 4 | 2 | no | — |
| 229 | C3⋊C8 | — | 2 | 4 | no | — |
| 230 | C22×C5 | — | 2 | 2 | no | — |
| 231 | C22×C5 | — | 4 | 1 | no | — |
| 232 | C22×C5 | — | 4 | 2 | no | — |
| 233 | C22×C5 | #67⋊C4 | 1 | 2 | yes | — |
| 234 | C22×C5 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 235 | C4×C5 | — | 2 | 2 | no | — |
| 236 | C4×C5 | — | 12 | 2 | no | — |
| 237 | C4×C5 | — | 12 | 2 | no | — |
| 238 | C4×C5 | #882 | 1 | 1 | yes | characteristic |
| 239 | C4×C5 | #886 | 1 | 1 | yes | characteristic |
| 240 | C4×C5 | #939 | 1 | 2 | yes | — |
| 241 | #41 | — | 2 | 2 | no | — |
| 242 | #41 | — | 2 | 2 | no | — |
| 243 | C2×Q8 | — | 6 | 1 | no | — |
| 244 | C2×Q8 | — | 6 | 2 | no | — |
| 245 | C2×Q8 | — | 6 | 2 | no | — |
| 246 | C2×Q8 | — | 6 | 1 | no | — |
| 247 | C2×Q8 | C5×#67 | 1 | 1 | yes | characteristic |
| 248 | C2×D4 | C4×C5×D3 | 1 | 2 | yes | — |
| 249 | C2×D4 | C5×#67 | 1 | 1 | yes | characteristic |
| 250 | C22×C4 | — | 3 | 2 | no | — |
| 251 | C22×C4 | C5×D12 | 1 | 2 | yes | — |
| 252 | C22×C4 | C22×C5×D3 | 1 | 1 | yes | characteristic |
| 253 | M16 | — | 2 | 2 | no | — |
| 254 | M16 | — | 2 | 2 | no | — |
| 255 | M16 | — | 3 | 4 | no | — |
| 256 | M16 | — | 6 | 4 | no | — |
| 257 | M16 | — | 6 | 2 | no | — |
| 258 | C2×C8 | — | 2 | 2 | no | — |
| 259 | C2×C8 | — | 6 | 4 | no | — |
| 260 | C2×C8 | — | 6 | 2 | no | — |
| 261 | C4⋊C4 | — | 6 | 4 | no | — |
| 262 | C4⋊C4 | — | 6 | 1 | no | — |
| 263 | C22⋊C4 | — | 3 | 2 | no | — |
| 264 | C22⋊C4 | — | 6 | 4 | no | — |
| 265 | C42 | — | 6 | 1 | no | — |
| 266 | C3×C5 | #1915 | 1 | 1 | yes | characteristic, complemented, Hall |
| 267 | C22×C3 | — | 2 | 2 | no | — |
| 268 | C22×C3 | — | 4 | 1 | no | — |
| 269 | C22×C3 | — | 4 | 2 | no | — |
| 270 | C22×C3 | C2×C5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 271 | C22×C3 | C4×C5×D4 | 1 | 2 | yes | — |
| 272 | C3×C4 | — | 2 | 2 | no | — |
| 273 | C3×C4 | C5×#120 | 1 | 1 | yes | characteristic |
| 274 | C3×C4 | C5×#121 | 1 | 2 | yes | — |
| 275 | C3×C4 | C5×#123 | 1 | 1 | yes | characteristic |
| 276 | Dic3 | — | 4 | 2 | no | — |
| 277 | Dic3 | — | 4 | 2 | no | — |
| 278 | C2×C5 | — | 2 | 1 | no | — |
| 279 | C2×C5 | — | 2 | 2 | no | — |
| 280 | C2×C5 | — | 4 | 2 | no | — |
| 281 | C2×C5 | #4958 | 1 | 1 | yes | characteristic |
| 282 | C23 | C5×(C2×D3)⋊C4 | 1 | 2 | yes | — |
| 283 | C23 | C2×C4×C5×D3 | 1 | 1 | yes | characteristic |
| 284 | Q8 | — | 2 | 2 | no | — |
| 285 | Q8 | — | 12 | 1 | no | — |
| 286 | Q8 | — | 12 | 1 | no | — |
| 287 | Q8 | — | 12 | 2 | no | — |
| 288 | D4 | — | 2 | 4 | no | — |
| 289 | D4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 2 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 6 | 4 | no | — |
| 293 | C2×C4 | — | 6 | 2 | no | — |
| 294 | C2×C4 | — | 6 | 2 | no | — |
| 295 | C2×C4 | C5×(C2×D3)⋊C4 | 1 | 2 | yes | — |
| 296 | C2×C4 | C2×C5×D12 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | C5×D3×D4 | 1 | 2 | yes | — |
| 298 | C2×C4 | C5×#237 | 1 | 2 | yes | — |
| 299 | C2×C4 | C2×C5×#67 | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 4 | 2 | no | — |
| 301 | C8 | — | 6 | 4 | no | — |
| 302 | C2×C3 | — | 2 | 1 | no | — |
| 303 | C2×C3 | — | 2 | 2 | no | — |
| 304 | C2×C3 | — | 4 | 2 | no | — |
| 305 | C2×C3 | C5×#386 | 1 | 1 | yes | characteristic |
| 306 | C5 | #70138 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 307 | C22 | — | 2 | 2 | no | — |
| 308 | C22 | — | 4 | 1 | no | — |
| 309 | C22 | — | 4 | 2 | no | — |
| 310 | C22 | C5×#67⋊C4 | 1 | 2 | yes | — |
| 311 | C22 | C2×C5×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 2 | 2 | no | — |
| 313 | C4 | — | 12 | 2 | no | — |
| 314 | C4 | — | 12 | 2 | no | — |
| 315 | C4 | C5×#882 | 1 | 1 | yes | characteristic |
| 316 | C4 | C5×#886 | 1 | 1 | yes | characteristic |
| 317 | C4 | C5×#939 | 1 | 2 | yes | — |
| 318 | C3 | C5×#1915 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 319 | C2 | — | 2 | 1 | no | — |
| 320 | C2 | — | 2 | 2 | no | — |
| 321 | C2 | — | 4 | 2 | no | — |
| 322 | C2 | C5×#4958 | 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