← #14286233
Subgroups — #14286233
Order 1920.
328 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 | Q8×C3⋊#170 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11752362 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#65079 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11746801 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11746755 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C8×C3⋊#170 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #11746743 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10603968 | — | 3 | 1 | no | maximal, Hall |
| 9 | C3×Q8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×#689 | C4 | 1 | 1 | yes | characteristic |
| 11 | C3×#4033 | C22 | 1 | 1 | yes | characteristic |
| 12 | #91233 | — | 2 | 1 | no | — |
| 13 | C4⋊(C3⋊#170) | — | 2 | 1 | no | — |
| 14 | C4⋊(C3⋊#170) | C22 | 1 | 1 | yes | characteristic |
| 15 | C4⋊(C3⋊#170) | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×C3⋊#170 | — | 2 | 1 | no | — |
| 17 | C4×C3⋊#170 | — | 2 | 1 | no | — |
| 18 | C4×C3⋊#170 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×C3⋊#170 | C22 | 1 | 1 | yes | characteristic |
| 20 | C3×C5×#103 | C4 | 1 | 1 | yes | characteristic |
| 21 | C3×#3948 | C4 | 1 | 1 | yes | characteristic |
| 22 | C3×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 23 | C3×#3939 | C4 | 1 | 1 | yes | characteristic |
| 24 | Q16×C3⋊C8 | — | 5 | 1 | no | maximal, Hall |
| 25 | Q8×#170 | — | 3 | 1 | no | — |
| 26 | #65703 | — | 3 | 1 | no | — |
| 27 | #65079 | D3 | 1 | 1 | yes | characteristic |
| 28 | Q8⋊#170 | — | 3 | 1 | no | — |
| 29 | #64884 | — | 3 | 1 | no | — |
| 30 | C8×#170 | — | 3 | 1 | no | — |
| 31 | #64872 | — | 3 | 1 | no | — |
| 32 | C2×C3×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×Dic10 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×C3⋊#170 | — | 2 | 1 | no | — |
| 35 | C2×C3⋊#170 | — | 2 | 1 | no | — |
| 36 | C2×C3⋊#170 | — | 2 | 1 | no | — |
| 37 | C2×C3⋊#170 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×C3⋊#170 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C3×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C3×C5×C4⋊C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C3×#689 | C8 | 1 | 4 | yes | complemented |
| 42 | C3×C4⋊Dic5 | — | 2 | 1 | no | — |
| 43 | C3×C4⋊Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C3×#683 | — | 2 | 1 | no | — |
| 45 | C3×C4×Dic5 | — | 2 | 1 | no | — |
| 46 | C3×C4×Dic5 | — | 2 | 1 | no | — |
| 47 | C3×C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×C3×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C3×C4×Q16 | — | 5 | 1 | no | — |
| 50 | Q8×C3⋊C8 | — | 5 | 1 | no | — |
| 51 | Q8×C3⋊C8 | — | 5 | 1 | no | — |
| 52 | #4535 | — | 5 | 1 | no | — |
| 53 | #4535 | — | 5 | 1 | no | — |
| 54 | #4458 | — | 5 | 1 | no | — |
| 55 | C8×C3⋊C8 | — | 5 | 1 | no | — |
| 56 | Q8×Dic5 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C2×#689 | Dic3 | 1 | 1 | yes | characteristic |
| 58 | #4033 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | #4004 | — | 6 | 1 | no | — |
| 60 | C4⋊#170 | — | 3 | 1 | no | — |
| 61 | C4⋊#170 | — | 3 | 1 | no | — |
| 62 | C4⋊#170 | — | 6 | 1 | no | — |
| 63 | C4×#170 | — | 3 | 1 | no | — |
| 64 | C4×#170 | — | 3 | 1 | no | — |
| 65 | C4×#170 | — | 6 | 1 | no | — |
| 66 | C4×#170 | — | 6 | 1 | no | — |
| 67 | C5×#103 | Dic3 | 1 | 1 | yes | characteristic |
| 68 | #3948 | Dic3 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic5 | C2×D3 | 1 | 1 | yes | characteristic |
| 70 | #3939 | Dic3 | 1 | 1 | yes | characteristic |
| 71 | C8×Q16 | — | 15 | 1 | no | Sylow |
| 72 | C3×C5×Q8 | — | 2 | 1 | no | — |
| 73 | C3×C5×Q8 | C2×C8 | 1 | 2 | yes | — |
| 74 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 75 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 76 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×Dic5 | — | 2 | 1 | no | — |
| 78 | C2×C3×Dic5 | — | 2 | 1 | no | — |
| 79 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 81 | C3×Dic10 | — | 2 | 1 | no | — |
| 82 | C3×Dic10 | C2×C8 | 1 | 2 | yes | — |
| 83 | C3⋊#170 | — | 2 | 2 | no | — |
| 84 | C3⋊#170 | — | 2 | 1 | no | — |
| 85 | C3⋊#170 | — | 4 | 1 | no | — |
| 86 | C3⋊#170 | — | 4 | 1 | no | — |
| 87 | C3⋊#170 | Q16 | 1 | 2 | yes | complemented |
| 88 | C3×C5×C8 | — | 2 | 1 | no | — |
| 89 | C3×#168 | C2×C8 | 1 | 2 | yes | — |
| 90 | C2×C3×Q16 | — | 5 | 1 | no | — |
| 91 | C3×C4×Q8 | — | 5 | 1 | no | — |
| 92 | C3×C4×Q8 | — | 5 | 1 | no | — |
| 93 | C3×#107 | — | 5 | 1 | no | — |
| 94 | C3×#103 | — | 5 | 1 | no | — |
| 95 | C3×#103 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 96 | C3×C4×C8 | — | 5 | 1 | no | — |
| 97 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 98 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 99 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 100 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 101 | C4×C3⋊C8 | — | 5 | 1 | no | — |
| 102 | C4×C3⋊C8 | — | 5 | 1 | no | — |
| 103 | C4×C3⋊C8 | — | 10 | 1 | no | — |
| 104 | C4×C3⋊C8 | — | 10 | 1 | no | — |
| 105 | C2×C5×Q8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 106 | C2×Dic10 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 107 | C2×#170 | — | 3 | 1 | no | — |
| 108 | C2×#170 | — | 3 | 1 | no | — |
| 109 | C2×#170 | — | 6 | 1 | no | — |
| 110 | C2×#170 | — | 6 | 1 | no | — |
| 111 | C2×#170 | — | 6 | 1 | no | — |
| 112 | C2×C5×C8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 113 | C5×C4⋊C4 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 114 | #689 | C3⋊C8 | 1 | 4 | yes | complemented |
| 115 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 116 | C4⋊Dic5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 117 | #683 | — | 2 | 1 | no | — |
| 118 | C4×Dic5 | — | 2 | 1 | no | — |
| 119 | C4×Dic5 | — | 2 | 1 | no | — |
| 120 | C4×Dic5 | C22×D3 | 1 | 1 | yes | characteristic |
| 121 | C2×#168 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 122 | C8×Q8 | — | 15 | 1 | no | — |
| 123 | C8×Q8 | — | 15 | 1 | no | — |
| 124 | C4×Q16 | — | 5 | 1 | no | — |
| 125 | #335 | — | 15 | 1 | no | — |
| 126 | Q8⋊C8 | — | 15 | 1 | no | — |
| 127 | Q8⋊C8 | — | 15 | 1 | no | — |
| 128 | C82 | — | 15 | 1 | no | — |
| 129 | C22×C3×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 130 | C3×C4×C5 | — | 2 | 2 | no | — |
| 131 | C3×C4×C5 | — | 4 | 1 | no | — |
| 132 | C3×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 133 | C3×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 134 | C3×Dic5 | — | 2 | 2 | no | — |
| 135 | C3×Dic5 | — | 2 | 1 | no | — |
| 136 | C3×Dic5 | — | 4 | 1 | no | — |
| 137 | C3×Dic5 | C2×Q16 | 1 | 1 | yes | characteristic |
| 138 | C3×Dic5 | #135 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×Q8 | — | 5 | 1 | no | — |
| 140 | C2×C3×Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 141 | C3×Q16 | — | 5 | 4 | no | — |
| 142 | C2×C3×C8 | — | 5 | 1 | no | — |
| 143 | C2×C3×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 144 | C3×C4⋊C4 | — | 5 | 1 | no | — |
| 145 | C3×C4⋊C4 | — | 10 | 1 | no | — |
| 146 | C3×C4⋊C4 | — | 10 | 1 | no | — |
| 147 | C3×C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 148 | C3×C42 | — | 5 | 1 | no | — |
| 149 | C3×C42 | — | 10 | 1 | no | — |
| 150 | C3×C42 | — | 10 | 1 | no | — |
| 151 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 152 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 153 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 154 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 155 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 156 | C5×Q8 | — | 2 | 1 | no | — |
| 157 | C5×Q8 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 158 | C2×C4×C5 | — | 2 | 1 | no | — |
| 159 | C2×C4×C5 | — | 2 | 1 | no | — |
| 160 | C2×C4×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 161 | C2×Dic5 | — | 2 | 1 | no | — |
| 162 | C2×Dic5 | — | 2 | 1 | no | — |
| 163 | C2×Dic5 | D3×D4 | 1 | 1 | yes | characteristic |
| 164 | C2×Dic5 | #237 | 1 | 1 | yes | characteristic |
| 165 | Dic10 | — | 2 | 1 | no | — |
| 166 | Dic10 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 167 | #170 | — | 3 | 2 | no | — |
| 168 | #170 | — | 6 | 2 | no | — |
| 169 | #170 | — | 6 | 1 | no | — |
| 170 | #170 | — | 12 | 1 | no | — |
| 171 | #170 | — | 12 | 1 | no | — |
| 172 | C5×C8 | — | 2 | 1 | no | — |
| 173 | #168 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 174 | C2×Q16 | — | 5 | 1 | no | — |
| 175 | C4×Q8 | — | 5 | 1 | no | — |
| 176 | C4×Q8 | — | 5 | 1 | no | — |
| 177 | #107 | — | 5 | 1 | no | — |
| 178 | C4⋊C8 | — | 15 | 1 | no | — |
| 179 | C4⋊C8 | — | 15 | 1 | no | — |
| 180 | C4⋊C8 | — | 30 | 1 | no | — |
| 181 | C4⋊C8 | — | 30 | 1 | no | — |
| 182 | #103 | — | 5 | 1 | no | — |
| 183 | #103 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 184 | C4×C8 | — | 5 | 1 | no | — |
| 185 | C4×C8 | — | 15 | 1 | no | — |
| 186 | C4×C8 | — | 15 | 1 | no | — |
| 187 | C4×C8 | — | 30 | 1 | no | — |
| 188 | C4×C8 | — | 30 | 1 | no | — |
| 189 | C2×C3×C5 | C8×D4 | 1 | 1 | yes | characteristic |
| 190 | C2×C3×C5 | C4×Q16 | 1 | 1 | yes | characteristic |
| 191 | C2×C3×C5 | #443 | 1 | 1 | yes | characteristic |
| 192 | C3×Q8 | — | 2 | 1 | no | — |
| 193 | C3×Q8 | — | 5 | 2 | no | — |
| 194 | C3×Q8 | — | 10 | 1 | no | — |
| 195 | C3×Q8 | D5⋊C8 | 1 | 2 | yes | — |
| 196 | C2×C3×C4 | — | 2 | 1 | no | — |
| 197 | C2×C3×C4 | — | 2 | 1 | no | — |
| 198 | C2×C3×C4 | — | 5 | 1 | no | — |
| 199 | C2×C3×C4 | — | 5 | 1 | no | — |
| 200 | C2×C3×C4 | — | 10 | 1 | no | — |
| 201 | C2×C3×C4 | — | 10 | 1 | no | — |
| 202 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 203 | C3×C8 | — | 2 | 1 | no | — |
| 204 | C3×C8 | — | 5 | 2 | no | — |
| 205 | C3⋊C8 | — | 5 | 2 | no | — |
| 206 | C3⋊C8 | — | 10 | 1 | no | — |
| 207 | C3⋊C8 | — | 10 | 2 | no | — |
| 208 | C3⋊C8 | — | 20 | 1 | no | — |
| 209 | C3⋊C8 | — | 20 | 1 | no | — |
| 210 | C22×C5 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 211 | C4×C5 | — | 2 | 2 | no | — |
| 212 | C4×C5 | — | 4 | 1 | no | — |
| 213 | C4×C5 | C22×C3⋊C8 | 1 | 1 | yes | characteristic |
| 214 | C4×C5 | #948 | 1 | 1 | yes | characteristic |
| 215 | Dic5 | — | 2 | 1 | no | — |
| 216 | Dic5 | — | 2 | 2 | no | — |
| 217 | Dic5 | — | 4 | 1 | no | — |
| 218 | Dic5 | D3×Q16 | 1 | 1 | yes | characteristic |
| 219 | Dic5 | #919 | 1 | 1 | yes | characteristic |
| 220 | C2×Q8 | — | 5 | 1 | no | — |
| 221 | C2×Q8 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 222 | Q16 | — | 5 | 4 | no | — |
| 223 | C2×C8 | — | 5 | 1 | no | — |
| 224 | C2×C8 | — | 15 | 1 | no | — |
| 225 | C2×C8 | — | 15 | 1 | no | — |
| 226 | C2×C8 | — | 30 | 1 | no | — |
| 227 | C2×C8 | — | 30 | 1 | no | — |
| 228 | C2×C8 | — | 30 | 1 | no | — |
| 229 | C2×C8 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 230 | C4⋊C4 | — | 5 | 1 | no | — |
| 231 | C4⋊C4 | — | 10 | 1 | no | — |
| 232 | C4⋊C4 | — | 10 | 1 | no | — |
| 233 | C4⋊C4 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 234 | C42 | — | 5 | 1 | no | — |
| 235 | C42 | — | 10 | 1 | no | — |
| 236 | C42 | — | 10 | 1 | no | — |
| 237 | C3×C5 | C8×Q16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 238 | C22×C3 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 239 | C3×C4 | — | 2 | 2 | no | — |
| 240 | C3×C4 | — | 4 | 1 | no | — |
| 241 | C3×C4 | — | 5 | 1 | no | — |
| 242 | C3×C4 | — | 5 | 1 | no | — |
| 243 | C3×C4 | — | 10 | 2 | no | — |
| 244 | C3×C4 | — | 10 | 1 | no | — |
| 245 | C3×C4 | — | 20 | 1 | no | — |
| 246 | C3×C4 | C2×D5⋊C8 | 1 | 1 | yes | characteristic |
| 247 | C3×C4 | #4131 | 1 | 1 | yes | characteristic |
| 248 | C2×C5 | D4×C3⋊C8 | 1 | 1 | yes | characteristic |
| 249 | C2×C5 | Dic3×Q16 | 1 | 1 | yes | characteristic |
| 250 | C2×C5 | #5196 | 1 | 1 | yes | characteristic |
| 251 | Q8 | — | 2 | 1 | no | — |
| 252 | Q8 | — | 5 | 2 | no | — |
| 253 | Q8 | — | 10 | 1 | no | — |
| 254 | Q8 | D5⋊(C3⋊C8) | 1 | 2 | yes | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 5 | 1 | no | — |
| 258 | C2×C4 | — | 5 | 1 | no | — |
| 259 | C2×C4 | — | 10 | 1 | no | — |
| 260 | C2×C4 | — | 10 | 1 | no | — |
| 261 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 262 | C8 | — | 2 | 1 | no | — |
| 263 | C8 | — | 5 | 2 | no | — |
| 264 | C8 | — | 15 | 2 | no | — |
| 265 | C8 | — | 30 | 1 | no | — |
| 266 | C8 | — | 30 | 2 | no | — |
| 267 | C8 | — | 60 | 1 | no | — |
| 268 | C8 | — | 60 | 1 | no | — |
| 269 | C2×C3 | #65692 | 1 | 1 | yes | characteristic |
| 270 | C2×C3 | #65736 | 1 | 1 | yes | characteristic |
| 271 | C2×C3 | Q16×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 272 | C5 | Q16×C3⋊C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 273 | C22 | D4×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 274 | C4 | — | 2 | 2 | no | — |
| 275 | C4 | — | 4 | 1 | no | — |
| 276 | C4 | — | 5 | 1 | no | — |
| 277 | C4 | — | 5 | 1 | no | — |
| 278 | C4 | — | 10 | 2 | no | — |
| 279 | C4 | — | 10 | 1 | no | — |
| 280 | C4 | — | 20 | 1 | no | — |
| 281 | C4 | C2×D5⋊(C3⋊C8) | 1 | 1 | yes | characteristic |
| 282 | C4 | #91990 | 1 | 1 | yes | characteristic |
| 283 | C3 | #10603968 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 284 | C2 | #11752351 | 1 | 1 | yes | characteristic |
| 285 | C2 | #11752395 | 1 | 1 | yes | characteristic |
| 286 | C2 | Q16×ΓL(1,𝔽16) | 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