← #14412170
Subgroups — #14412170
Order 1920.
1512 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×#4149 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756989 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D5⋊#833 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11752420 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #10617553 | — | 3 | 1 | no | maximal, Hall |
| 6 | C2×C3×C5×#41 | C4 | 1 | 1 | yes | characteristic |
| 7 | C3×#4149 | — | 2 | 8 | no | — |
| 8 | C2×C3×#713 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×#713 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×D4×D5 | C4 | 1 | 3 | yes | — |
| 11 | C2×C3×D4×D5 | C22 | 1 | 3 | yes | — |
| 12 | C2×C3×#709 | C22 | 1 | 3 | yes | — |
| 13 | C22×C3×D20 | C4 | 1 | 3 | yes | — |
| 14 | #91992 | — | 2 | 8 | no | — |
| 15 | D5⋊#208 | C22 | 1 | 12 | yes | complemented |
| 16 | #91242 | C22 | 1 | 12 | yes | complemented |
| 17 | #91234 | C22 | 1 | 4 | yes | complemented |
| 18 | #84100 | — | 5 | 1 | no | maximal, Hall |
| 19 | C2×#4149 | D3 | 1 | 1 | yes | characteristic |
| 20 | #66260 | — | 3 | 2 | no | — |
| 21 | D5⋊#100 | — | 3 | 6 | no | — |
| 22 | #65761 | — | 3 | 6 | no | — |
| 23 | C23×C3×D5 | C2×C4 | 1 | 6 | yes | — |
| 24 | C3×C5×#41 | — | 2 | 4 | no | — |
| 25 | C2×C3×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×C3×C5×D4 | C2×C4 | 1 | 3 | yes | — |
| 27 | C22×C3×C4×C5 | C2×C4 | 1 | 3 | yes | — |
| 28 | C2×C3×C22⋊D5 | C2×C4 | 1 | 3 | yes | — |
| 29 | C2×C3×C22⋊D5 | C23 | 1 | 3 | yes | — |
| 30 | C3×#713 | — | 2 | 4 | no | — |
| 31 | C3×#713 | D4 | 1 | 8 | yes | — |
| 32 | C3×D4×D5 | — | 2 | 12 | no | — |
| 33 | C3×D4×D5 | — | 2 | 12 | no | — |
| 34 | C3×#709 | — | 2 | 12 | no | — |
| 35 | C2×C3×D20 | — | 2 | 12 | no | — |
| 36 | C2×C3×D20 | C2×C4 | 1 | 3 | yes | — |
| 37 | C2×C3×D20 | C2×C4 | 1 | 3 | yes | — |
| 38 | C2×C3×D20 | C2×C4 | 1 | 3 | yes | — |
| 39 | C2×C3×C4×D5 | C2×C4 | 1 | 3 | yes | — |
| 40 | C2×C3×C4×D5 | C23 | 1 | 3 | yes | — |
| 41 | C2×C3×Dic10 | C23 | 1 | 1 | yes | characteristic |
| 42 | #6762 | C23 | 1 | 8 | yes | — |
| 43 | #6758 | — | 2 | 12 | no | — |
| 44 | D5⋊(C3⋊C8) | — | 2 | 12 | no | — |
| 45 | C2×C3×H(5,𝔽2) | — | 5 | 1 | no | — |
| 46 | #5819 | — | 5 | 2 | no | — |
| 47 | #5237 | — | 5 | 6 | no | — |
| 48 | C2×#833 | — | 5 | 6 | no | — |
| 49 | C2×C5×#41 | Dic3 | 1 | 1 | yes | characteristic |
| 50 | #4149 | — | 2 | 8 | no | — |
| 51 | C2×#713 | Dic3 | 1 | 1 | yes | characteristic |
| 52 | C2×#713 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | C2×D4×D5 | Dic3 | 1 | 3 | yes | — |
| 54 | C2×D4×D5 | C2×D3 | 1 | 3 | yes | — |
| 55 | C2×#709 | C2×D3 | 1 | 3 | yes | — |
| 56 | C22×D20 | Dic3 | 1 | 3 | yes | — |
| 57 | #4133 | — | 6 | 8 | no | — |
| 58 | D5⋊M16 | — | 3 | 12 | no | — |
| 59 | #4013 | — | 3 | 12 | no | — |
| 60 | #4005 | — | 3 | 4 | no | — |
| 61 | #2888 | — | 15 | 1 | no | Sylow |
| 62 | C23×C3×C5 | C22×C4 | 1 | 3 | yes | — |
| 63 | C22×C3×D5 | — | 2 | 12 | no | — |
| 64 | C22×C3×D5 | — | 8 | 6 | no | — |
| 65 | C22×C3×D5 | C22×C4 | 1 | 3 | yes | — |
| 66 | C22×C3×D5 | C22×C4 | 1 | 3 | yes | — |
| 67 | C3×C5×Q8 | C22⋊C4 | 1 | 4 | yes | — |
| 68 | C3×C5×D4 | — | 2 | 6 | no | — |
| 69 | C2×C3×C4×C5 | — | 2 | 6 | no | — |
| 70 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C3×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 72 | C3×C22⋊D5 | — | 2 | 6 | no | — |
| 73 | C3×C22⋊D5 | — | 2 | 6 | no | — |
| 74 | C2×C3×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 76 | C3×D20 | — | 2 | 6 | no | — |
| 77 | C3×D20 | — | 2 | 6 | no | — |
| 78 | C3×D20 | C22⋊C4 | 1 | 12 | yes | — |
| 79 | C3×C4×D5 | — | 2 | 6 | no | — |
| 80 | C3×C4×D5 | C2×D4 | 1 | 12 | yes | — |
| 81 | C3×Dic10 | — | 2 | 2 | no | — |
| 82 | C3⋊#170 | — | 2 | 8 | no | — |
| 83 | C3×H(5,𝔽2) | — | 10 | 8 | no | — |
| 84 | C2×C3×#41 | — | 5 | 1 | no | — |
| 85 | C2×C3×#41 | — | 5 | 1 | no | — |
| 86 | C2×C3×#41 | — | 5 | 3 | no | — |
| 87 | C2×C3×#41 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 88 | C22×C3×D4 | — | 5 | 3 | no | — |
| 89 | C22×C3×D4 | — | 5 | 3 | no | — |
| 90 | C22×C3×D4 | — | 5 | 3 | no | — |
| 91 | #948 | — | 10 | 8 | no | — |
| 92 | C2×#208 | — | 5 | 12 | no | — |
| 93 | #836 | — | 5 | 4 | no | — |
| 94 | #833 | — | 5 | 12 | no | — |
| 95 | C23×D5 | C2×Dic3 | 1 | 6 | yes | — |
| 96 | C5×#41 | — | 2 | 4 | no | — |
| 97 | C2×C5×Q8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 98 | C2×C5×D4 | C2×Dic3 | 1 | 3 | yes | — |
| 99 | C22×C4×C5 | C2×Dic3 | 1 | 3 | yes | — |
| 100 | C2×C22⋊D5 | C2×Dic3 | 1 | 3 | yes | — |
| 101 | C2×C22⋊D5 | C22×D3 | 1 | 3 | yes | — |
| 102 | #713 | — | 2 | 4 | no | — |
| 103 | #713 | #67 | 1 | 8 | yes | — |
| 104 | D4×D5 | — | 2 | 12 | no | — |
| 105 | D4×D5 | — | 2 | 12 | no | — |
| 106 | #709 | — | 2 | 12 | no | — |
| 107 | C2×D20 | — | 2 | 12 | no | — |
| 108 | C2×D20 | C2×Dic3 | 1 | 3 | yes | — |
| 109 | C2×D20 | C2×Dic3 | 1 | 3 | yes | — |
| 110 | C2×D20 | C2×Dic3 | 1 | 3 | yes | — |
| 111 | C2×C4×D5 | C2×Dic3 | 1 | 3 | yes | — |
| 112 | C2×C4×D5 | C22×D3 | 1 | 3 | yes | — |
| 113 | C2×Dic10 | C22×D3 | 1 | 1 | yes | characteristic |
| 114 | #704 | — | 3 | 8 | no | — |
| 115 | #700 | — | 6 | 12 | no | — |
| 116 | D5⋊C8 | — | 6 | 12 | no | — |
| 117 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 118 | #568 | — | 15 | 2 | no | — |
| 119 | #413 | — | 15 | 6 | no | — |
| 120 | C2×#100 | — | 15 | 6 | no | — |
| 121 | C22×C3×C5 | — | 2 | 3 | no | — |
| 122 | C22×C3×C5 | — | 4 | 3 | no | — |
| 123 | C22×C3×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 124 | C2×C3×D5 | — | 2 | 3 | no | — |
| 125 | C2×C3×D5 | — | 4 | 3 | no | — |
| 126 | C2×C3×D5 | — | 4 | 3 | no | — |
| 127 | C2×C3×D5 | — | 8 | 12 | no | — |
| 128 | C2×C3×D5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 129 | C3×C4×C5 | — | 2 | 1 | no | — |
| 130 | C3×C4×C5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 131 | C3×Dic5 | — | 2 | 1 | no | — |
| 132 | C3×Dic5 | C22×D4 | 1 | 2 | yes | — |
| 133 | C24×C3 | — | 5 | 6 | no | — |
| 134 | C3×#41 | — | 2 | 4 | no | — |
| 135 | C3×#41 | — | 5 | 8 | no | — |
| 136 | C3×#41 | — | 10 | 12 | no | — |
| 137 | C3×#41 | — | 10 | 4 | no | — |
| 138 | C2×C3×Q8 | — | 5 | 1 | no | — |
| 139 | C2×C3×Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 140 | C2×C3×D4 | — | 5 | 3 | no | — |
| 141 | C2×C3×D4 | — | 5 | 3 | no | — |
| 142 | C2×C3×D4 | — | 5 | 3 | no | — |
| 143 | C2×C3×D4 | — | 5 | 3 | no | — |
| 144 | C2×C3×D4 | — | 5 | 3 | no | — |
| 145 | C2×C3×D4 | — | 10 | 12 | no | — |
| 146 | C2×C3×D4 | — | 10 | 12 | no | — |
| 147 | C2×C3×D4 | — | 10 | 12 | no | — |
| 148 | C2×C3×D4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 149 | C22×C3×C4 | — | 5 | 3 | no | — |
| 150 | C22×C3×C4 | — | 5 | 3 | no | — |
| 151 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 152 | #208 | — | 5 | 8 | no | — |
| 153 | #208 | — | 10 | 12 | no | — |
| 154 | C2×C3⋊C8 | — | 10 | 12 | no | — |
| 155 | C23×C5 | C22×Dic3 | 1 | 3 | yes | — |
| 156 | C22×D5 | — | 2 | 12 | no | — |
| 157 | C22×D5 | — | 8 | 6 | no | — |
| 158 | C22×D5 | C22×Dic3 | 1 | 3 | yes | — |
| 159 | C22×D5 | C22×Dic3 | 1 | 3 | yes | — |
| 160 | C5×Q8 | #217 | 1 | 4 | yes | — |
| 161 | C5×D4 | — | 2 | 6 | no | — |
| 162 | C2×C4×C5 | — | 2 | 6 | no | — |
| 163 | C2×C4×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 164 | C2×C4×C5 | C22×Dic3 | 1 | 3 | yes | — |
| 165 | C22⋊D5 | — | 2 | 6 | no | — |
| 166 | C22⋊D5 | — | 2 | 6 | no | — |
| 167 | C2×Dic5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 168 | C2×Dic5 | C23×D3 | 1 | 1 | yes | characteristic |
| 169 | D20 | — | 2 | 6 | no | — |
| 170 | D20 | — | 2 | 6 | no | — |
| 171 | D20 | #217 | 1 | 12 | yes | — |
| 172 | C4×D5 | — | 2 | 6 | no | — |
| 173 | C4×D5 | C2×#67 | 1 | 12 | yes | — |
| 174 | Dic10 | — | 2 | 2 | no | — |
| 175 | #170 | — | 6 | 8 | no | — |
| 176 | H(5,𝔽2) | — | 10 | 8 | no | — |
| 177 | C2×#41 | — | 5 | 1 | no | — |
| 178 | C2×#41 | — | 5 | 1 | no | — |
| 179 | C2×#41 | — | 5 | 3 | no | — |
| 180 | C2×#41 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 181 | C22×D4 | — | 5 | 3 | no | — |
| 182 | C22×D4 | — | 5 | 3 | no | — |
| 183 | C22×D4 | — | 5 | 3 | no | — |
| 184 | #131 | — | 30 | 8 | no | — |
| 185 | C2×M16 | — | 15 | 12 | no | — |
| 186 | #101 | — | 15 | 4 | no | — |
| 187 | #100 | — | 15 | 12 | no | — |
| 188 | C2×C3×C5 | — | 2 | 1 | no | — |
| 189 | C2×C3×C5 | — | 4 | 3 | no | — |
| 190 | C2×C3×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C3×D5 | — | 2 | 6 | no | — |
| 192 | C3×D5 | — | 4 | 3 | no | — |
| 193 | C23×C3 | — | 5 | 3 | no | — |
| 194 | C23×C3 | — | 5 | 3 | no | — |
| 195 | C23×C3 | — | 10 | 12 | no | — |
| 196 | C23×C3 | — | 40 | 6 | no | — |
| 197 | C23×C3 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 198 | C3×Q8 | — | 10 | 2 | no | — |
| 199 | C3×Q8 | C22⋊AGL(1,𝔽5) | 1 | 4 | yes | — |
| 200 | C3×D4 | — | 2 | 6 | no | — |
| 201 | C3×D4 | — | 5 | 12 | no | — |
| 202 | C3×D4 | — | 10 | 6 | no | — |
| 203 | C3×D4 | — | 10 | 6 | no | — |
| 204 | C3×D4 | — | 10 | 6 | no | — |
| 205 | C3×D4 | — | 10 | 6 | no | — |
| 206 | C2×C3×C4 | — | 2 | 6 | no | — |
| 207 | C2×C3×C4 | — | 5 | 1 | no | — |
| 208 | C2×C3×C4 | — | 5 | 12 | no | — |
| 209 | C2×C3×C4 | — | 5 | 1 | no | — |
| 210 | C2×C3×C4 | — | 10 | 6 | no | — |
| 211 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 212 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 213 | C3⋊C8 | — | 10 | 8 | no | — |
| 214 | C22×C5 | — | 2 | 3 | no | — |
| 215 | C22×C5 | — | 4 | 3 | no | — |
| 216 | C22×C5 | C23×Dic3 | 1 | 1 | yes | characteristic |
| 217 | C2×D5 | — | 2 | 3 | no | — |
| 218 | C2×D5 | — | 4 | 3 | no | — |
| 219 | C2×D5 | — | 4 | 3 | no | — |
| 220 | C2×D5 | — | 8 | 12 | no | — |
| 221 | C2×D5 | C2×#217 | 1 | 6 | yes | — |
| 222 | C4×C5 | — | 2 | 1 | no | — |
| 223 | C4×C5 | C2×#217 | 1 | 6 | yes | — |
| 224 | Dic5 | — | 2 | 1 | no | — |
| 225 | Dic5 | C22×#67 | 1 | 2 | yes | — |
| 226 | C24 | — | 5 | 6 | no | — |
| 227 | #41 | — | 2 | 4 | no | — |
| 228 | #41 | — | 5 | 8 | no | — |
| 229 | #41 | — | 10 | 12 | no | — |
| 230 | #41 | — | 10 | 4 | no | — |
| 231 | C2×Q8 | — | 5 | 1 | no | — |
| 232 | C2×Q8 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 233 | C2×D4 | — | 5 | 3 | no | — |
| 234 | C2×D4 | — | 5 | 3 | no | — |
| 235 | C2×D4 | — | 5 | 3 | no | — |
| 236 | C2×D4 | — | 5 | 3 | no | — |
| 237 | C2×D4 | — | 5 | 3 | no | — |
| 238 | C2×D4 | — | 10 | 12 | no | — |
| 239 | C2×D4 | — | 10 | 12 | no | — |
| 240 | C2×D4 | — | 10 | 12 | no | — |
| 241 | C2×D4 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 242 | C22×C4 | — | 5 | 3 | no | — |
| 243 | C22×C4 | — | 5 | 3 | no | — |
| 244 | C22×C4 | C2×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 245 | M16 | — | 15 | 8 | no | — |
| 246 | M16 | — | 30 | 12 | no | — |
| 247 | C2×C8 | — | 30 | 12 | no | — |
| 248 | C3×C5 | #2888 | 1 | 1 | yes | characteristic, complemented, Hall |
| 249 | C22×C3 | — | 2 | 3 | no | — |
| 250 | C22×C3 | — | 4 | 3 | no | — |
| 251 | C22×C3 | — | 5 | 6 | no | — |
| 252 | C22×C3 | — | 10 | 3 | no | — |
| 253 | C22×C3 | — | 20 | 3 | no | — |
| 254 | C22×C3 | — | 20 | 3 | no | — |
| 255 | C22×C3 | — | 40 | 12 | no | — |
| 256 | C22×C3 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 257 | C3×C4 | — | 2 | 1 | no | — |
| 258 | C3×C4 | — | 5 | 2 | no | — |
| 259 | C3×C4 | — | 10 | 1 | no | — |
| 260 | C3×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 6 | yes | — |
| 261 | C2×C5 | — | 2 | 1 | no | — |
| 262 | C2×C5 | — | 4 | 3 | no | — |
| 263 | C2×C5 | C22×#217 | 1 | 1 | yes | characteristic |
| 264 | D5 | — | 2 | 6 | no | — |
| 265 | D5 | — | 4 | 3 | no | — |
| 266 | C23 | — | 5 | 3 | no | — |
| 267 | C23 | — | 5 | 3 | no | — |
| 268 | C23 | — | 10 | 12 | no | — |
| 269 | C23 | — | 40 | 6 | no | — |
| 270 | C23 | C22×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 271 | Q8 | — | 10 | 2 | no | — |
| 272 | Q8 | #6763 | 1 | 4 | yes | — |
| 273 | D4 | — | 2 | 6 | no | — |
| 274 | D4 | — | 5 | 12 | no | — |
| 275 | D4 | — | 10 | 6 | no | — |
| 276 | D4 | — | 10 | 6 | no | — |
| 277 | D4 | — | 10 | 6 | no | — |
| 278 | D4 | — | 10 | 6 | no | — |
| 279 | C2×C4 | — | 2 | 6 | no | — |
| 280 | C2×C4 | — | 5 | 1 | no | — |
| 281 | C2×C4 | — | 5 | 12 | no | — |
| 282 | C2×C4 | — | 5 | 1 | no | — |
| 283 | C2×C4 | — | 10 | 6 | no | — |
| 284 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 3 | yes | — |
| 286 | C8 | — | 30 | 8 | no | — |
| 287 | C2×C3 | — | 2 | 1 | no | — |
| 288 | C2×C3 | — | 4 | 3 | no | — |
| 289 | C2×C3 | — | 10 | 6 | no | — |
| 290 | C2×C3 | — | 20 | 3 | no | — |
| 291 | C2×C3 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 292 | C5 | #84100 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 293 | C22 | — | 2 | 3 | no | — |
| 294 | C22 | — | 4 | 3 | no | — |
| 295 | C22 | — | 5 | 6 | no | — |
| 296 | C22 | — | 10 | 3 | no | — |
| 297 | C22 | — | 20 | 3 | no | — |
| 298 | C22 | — | 20 | 3 | no | — |
| 299 | C22 | — | 40 | 12 | no | — |
| 300 | C22 | C23×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 5 | 2 | no | — |
| 303 | C4 | — | 10 | 1 | no | — |
| 304 | C4 | C2×#6763 | 1 | 6 | yes | — |
| 305 | C3 | #10617553 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 4 | 3 | no | — |
| 308 | C2 | — | 10 | 6 | no | — |
| 309 | C2 | — | 20 | 3 | no | — |
| 310 | C2 | C22×#6763 | 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