← #14296708
Subgroups — #14296708
Order 1920.
544 conjugacy classes of subgroups.
The subgroup lattice has 304 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#91223 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#3992 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | AGL(1,𝔽5)×C4⋊Dic3 | C2 | 1 | 2 | yes | maximal |
| 4 | D5×#817 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11746530 | C2 | 1 | 2 | yes | maximal |
| 6 | #10604048 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C4⋊ΓL(1,𝔽16) | C22 | 1 | 2 | yes | — |
| 8 | C2×C3×C4×AGL(1,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×Dic3×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 11 | C2×C3×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 12 | D5×C4⋊Dic3 | C22 | 1 | 2 | yes | — |
| 13 | #91223 | C4 | 1 | 4 | yes | complemented |
| 14 | C3×#3992 | C4 | 1 | 4 | yes | complemented |
| 15 | (C2×AGL(1,𝔽5))⋊Dic3 | — | 2 | 2 | no | — |
| 16 | #91101 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C5×#817 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | #90985 | C4 | 1 | 2 | yes | complemented |
| 19 | #69814 | — | 5 | 1 | no | maximal, Hall |
| 20 | C2×#3994 | — | 3 | 1 | no | — |
| 21 | C2×#3992 | D3 | 1 | 1 | yes | characteristic |
| 22 | C4⋊C4×AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 23 | D5×#106 | — | 3 | 1 | no | — |
| 24 | #64874 | — | 3 | 2 | no | — |
| 25 | C22×ΓL(1,𝔽16) | — | 2 | 2 | no | — |
| 26 | C22×C3×AGL(1,𝔽5) | D4 | 1 | 2 | yes | — |
| 27 | C2×C3×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×D5×Dic3 | — | 2 | 2 | no | — |
| 29 | C4⋊ΓL(1,𝔽16) | — | 2 | 2 | no | — |
| 30 | C4⋊ΓL(1,𝔽16) | C2×C4 | 1 | 4 | yes | — |
| 31 | C2×C3×#170 | — | 2 | 1 | no | — |
| 32 | C3×C4×AGL(1,𝔽5) | C2×C4 | 1 | 2 | yes | — |
| 33 | C3×C4×AGL(1,𝔽5) | Q8 | 1 | 2 | yes | — |
| 34 | C3×D5⋊C8 | — | 2 | 1 | no | — |
| 35 | C3×D5⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 36 | Dic3×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 37 | C2×C3×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4⋊Dic15 | C2×C4 | 1 | 2 | yes | — |
| 39 | C5×C4⋊Dic3 | C2×C4 | 1 | 2 | yes | — |
| 40 | C2×C3×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C3×C8×D5 | C2×C4 | 1 | 2 | yes | — |
| 42 | C3×C8×D5 | C2×C4 | 1 | 2 | yes | — |
| 43 | #6668 | — | 2 | 2 | no | — |
| 44 | C2×C3×#97 | — | 5 | 1 | no | — |
| 45 | C2×#818 | — | 5 | 1 | no | — |
| 46 | C2×#817 | — | 5 | 1 | no | — |
| 47 | C4×C4⋊Dic3 | — | 5 | 2 | no | — |
| 48 | #4551 | — | 5 | 2 | no | — |
| 49 | C2×C4⋊AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 50 | C2×C4×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 51 | C2×C4×AGL(1,𝔽5) | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | C2×D5⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | C2×C8×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | D5×C4⋊C4 | — | 3 | 2 | no | — |
| 55 | (C2×AGL(1,𝔽5))⋊C4 | — | 6 | 2 | no | — |
| 56 | #3994 | — | 3 | 4 | no | — |
| 57 | #3992 | Dic3 | 1 | 4 | yes | complemented |
| 58 | C5×#106 | — | 3 | 1 | no | — |
| 59 | #3949 | — | 3 | 1 | no | — |
| 60 | #3940 | — | 3 | 2 | no | — |
| 61 | #1776 | — | 15 | 1 | no | Sylow |
| 62 | C22×C3×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×ΓL(1,𝔽16) | — | 2 | 4 | no | — |
| 64 | C2×ΓL(1,𝔽16) | — | 4 | 2 | no | — |
| 65 | C2×ΓL(1,𝔽16) | — | 4 | 2 | no | — |
| 66 | C2×C3×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 67 | C2×C3×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 68 | C2×C3×AGL(1,𝔽5) | C4⋊C4 | 1 | 4 | yes | — |
| 69 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×Dic15 | — | 2 | 2 | no | — |
| 71 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 72 | C2×C3×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 73 | C3×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C3×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C3×C4×D5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 76 | C3×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 77 | D5×Dic3 | — | 4 | 2 | no | — |
| 78 | D5×Dic3 | — | 4 | 2 | no | — |
| 79 | C3×#170 | — | 2 | 2 | no | — |
| 80 | C3×C5×C8 | C42 | 1 | 2 | yes | complemented |
| 81 | C3×#168 | C42 | 1 | 2 | yes | complemented |
| 82 | C22×C3×C8 | — | 5 | 1 | no | — |
| 83 | C22×C3×C8 | — | 5 | 1 | no | — |
| 84 | C2×C3×C42 | — | 5 | 1 | no | — |
| 85 | C2×C4⋊Dic3 | — | 5 | 2 | no | — |
| 86 | C2×C4⋊Dic3 | — | 5 | 2 | no | — |
| 87 | C2×C4×Dic3 | — | 10 | 2 | no | — |
| 88 | C3×#97 | — | 5 | 4 | no | — |
| 89 | C3⋊#95 | — | 10 | 2 | no | — |
| 90 | #818 | — | 5 | 4 | no | — |
| 91 | #817 | — | 5 | 1 | no | — |
| 92 | #817 | — | 5 | 2 | no | — |
| 93 | #817 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 94 | C22×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 95 | C22×AGL(1,𝔽5) | D12 | 1 | 2 | yes | — |
| 96 | C2×C4×D5 | — | 6 | 2 | no | — |
| 97 | C2×C4×D5 | C22×D3 | 1 | 1 | yes | characteristic |
| 98 | C2×#170 | — | 2 | 1 | no | — |
| 99 | C4⋊AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 100 | C4⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 101 | C4×AGL(1,𝔽5) | — | 12 | 4 | no | — |
| 102 | C4×AGL(1,𝔽5) | Dic6 | 1 | 2 | yes | — |
| 103 | C4×AGL(1,𝔽5) | C2×Dic3 | 1 | 2 | yes | — |
| 104 | D5⋊C8 | — | 2 | 1 | no | — |
| 105 | D5⋊C8 | C2×Dic3 | 1 | 2 | yes | — |
| 106 | C2×C5×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 107 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 108 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 109 | #683 | — | 6 | 2 | no | — |
| 110 | C2×#168 | C4×D3 | 1 | 1 | yes | characteristic |
| 111 | C8×D5 | C4×D3 | 1 | 2 | yes | — |
| 112 | C8×D5 | C2×Dic3 | 1 | 2 | yes | — |
| 113 | C2×#107 | — | 15 | 1 | no | — |
| 114 | C2×#106 | — | 15 | 1 | no | — |
| 115 | C2×#97 | — | 5 | 1 | no | — |
| 116 | C4×C4⋊C4 | — | 15 | 2 | no | — |
| 117 | #340 | — | 15 | 2 | no | — |
| 118 | C22×C3×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×C3×D5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C3×D5 | C4×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×C3×D5 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 122 | C2×C3×D5 | #137 | 1 | 2 | yes | — |
| 123 | ΓL(1,𝔽16) | — | 4 | 4 | no | — |
| 124 | C3×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 125 | C3×C4×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 126 | C3×C4×C5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 127 | Dic15 | — | 4 | 2 | no | — |
| 128 | C3×Dic5 | #117 | 1 | 1 | yes | characteristic |
| 129 | C3×Dic5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 130 | C5×Dic3 | — | 4 | 2 | no | — |
| 131 | C22×C3×C4 | — | 5 | 1 | no | — |
| 132 | C22×C3×C4 | — | 5 | 2 | no | — |
| 133 | C22×Dic3 | — | 10 | 2 | no | — |
| 134 | C22×Dic3 | — | 10 | 2 | no | — |
| 135 | C2×C3×C8 | — | 5 | 1 | no | — |
| 136 | C2×C3×C8 | — | 5 | 2 | no | — |
| 137 | C2×C3×C8 | — | 5 | 2 | no | — |
| 138 | C2×C3×C8 | — | 5 | 2 | no | — |
| 139 | C2×C3×C8 | — | 10 | 1 | no | — |
| 140 | C2×C3×C8 | — | 10 | 1 | no | — |
| 141 | C2×C3×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 142 | C3×C42 | — | 5 | 2 | no | — |
| 143 | C3×C42 | — | 5 | 2 | no | — |
| 144 | C4⋊Dic3 | — | 5 | 4 | no | — |
| 145 | C4⋊Dic3 | — | 5 | 2 | no | — |
| 146 | C4⋊Dic3 | — | 10 | 2 | no | — |
| 147 | C4⋊Dic3 | — | 10 | 2 | no | — |
| 148 | C4⋊Dic3 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 149 | C4×Dic3 | — | 20 | 4 | no | — |
| 150 | C22×D5 | C2×D12 | 1 | 1 | yes | characteristic |
| 151 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 152 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 153 | C2×AGL(1,𝔽5) | — | 6 | 4 | no | — |
| 154 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 155 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 156 | C2×AGL(1,𝔽5) | C4⋊Dic3 | 1 | 4 | yes | — |
| 157 | C2×C4×C5 | — | 6 | 2 | no | — |
| 158 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 159 | C2×Dic5 | — | 6 | 2 | no | — |
| 160 | C2×Dic5 | #235 | 1 | 1 | yes | characteristic |
| 161 | C4×D5 | — | 12 | 2 | no | — |
| 162 | C4×D5 | — | 12 | 2 | no | — |
| 163 | C4×D5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 164 | C4×D5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 165 | C4×D5 | #235 | 1 | 1 | yes | characteristic |
| 166 | C4×D5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 167 | #170 | — | 2 | 2 | no | — |
| 168 | C5×C8 | C4×Dic3 | 1 | 2 | yes | complemented |
| 169 | #168 | C4×Dic3 | 1 | 2 | yes | complemented |
| 170 | C22×C8 | — | 5 | 1 | no | — |
| 171 | C22×C8 | — | 5 | 1 | no | — |
| 172 | C2×C4⋊C4 | — | 15 | 2 | no | — |
| 173 | C2×C4⋊C4 | — | 15 | 2 | no | — |
| 174 | C2×C42 | — | 5 | 1 | no | — |
| 175 | C2×C42 | — | 30 | 2 | no | — |
| 176 | #107 | — | 15 | 4 | no | — |
| 177 | #106 | — | 3 | 1 | no | — |
| 178 | #106 | — | 15 | 2 | no | — |
| 179 | #106 | — | 15 | 1 | no | — |
| 180 | #97 | — | 5 | 4 | no | — |
| 181 | #95 | — | 30 | 2 | no | — |
| 182 | C2×C3×C5 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 183 | C2×C3×C5 | #440 | 1 | 2 | yes | — |
| 184 | C3×D5 | #428 | 1 | 2 | yes | — |
| 185 | C3×D5 | #441 | 1 | 1 | yes | characteristic |
| 186 | C3×D5 | #442 | 1 | 1 | yes | characteristic |
| 187 | C23×C3 | — | 5 | 1 | no | — |
| 188 | C2×C3×C4 | — | 5 | 1 | no | — |
| 189 | C2×C3×C4 | — | 5 | 1 | no | — |
| 190 | C2×C3×C4 | — | 5 | 1 | no | — |
| 191 | C2×C3×C4 | — | 5 | 1 | no | — |
| 192 | C2×C3×C4 | — | 5 | 1 | no | — |
| 193 | C2×C3×C4 | — | 5 | 4 | no | — |
| 194 | C2×C3×C4 | — | 10 | 2 | no | — |
| 195 | C2×C3×C4 | — | 10 | 2 | no | — |
| 196 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 197 | C2×Dic3 | — | 2 | 2 | no | — |
| 198 | C2×Dic3 | — | 10 | 4 | no | — |
| 199 | C2×Dic3 | — | 10 | 2 | no | — |
| 200 | C2×Dic3 | — | 20 | 2 | no | — |
| 201 | C2×Dic3 | — | 20 | 2 | no | — |
| 202 | C2×Dic3 | — | 20 | 2 | no | — |
| 203 | C2×Dic3 | — | 20 | 2 | no | — |
| 204 | C3×C8 | — | 5 | 2 | no | — |
| 205 | C3×C8 | — | 10 | 2 | no | — |
| 206 | C3×C8 | C4×AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 207 | C22×C5 | C4×D12 | 1 | 1 | yes | characteristic |
| 208 | C2×D5 | C4×D12 | 1 | 1 | yes | characteristic |
| 209 | C2×D5 | #908 | 1 | 2 | yes | — |
| 210 | C2×D5 | #909 | 1 | 2 | yes | — |
| 211 | C2×D5 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 212 | AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 213 | AGL(1,𝔽5) | — | 12 | 4 | no | — |
| 214 | C4×C5 | — | 12 | 2 | no | — |
| 215 | C4×C5 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 216 | C4×C5 | C2×C4×Dic3 | 1 | 1 | yes | characteristic |
| 217 | Dic5 | — | 12 | 2 | no | — |
| 218 | Dic5 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 219 | Dic5 | #926 | 1 | 1 | yes | characteristic |
| 220 | C22×C4 | — | 5 | 1 | no | — |
| 221 | C22×C4 | — | 5 | 2 | no | — |
| 222 | C22×C4 | — | 30 | 2 | no | — |
| 223 | C22×C4 | — | 30 | 2 | no | — |
| 224 | C2×C8 | — | 5 | 1 | no | — |
| 225 | C2×C8 | — | 5 | 2 | no | — |
| 226 | C2×C8 | — | 5 | 2 | no | — |
| 227 | C2×C8 | — | 5 | 2 | no | — |
| 228 | C2×C8 | — | 10 | 1 | no | — |
| 229 | C2×C8 | — | 10 | 1 | no | — |
| 230 | C2×C8 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 231 | C4⋊C4 | — | 3 | 2 | no | — |
| 232 | C4⋊C4 | — | 15 | 4 | no | — |
| 233 | C4⋊C4 | — | 15 | 2 | no | — |
| 234 | C4⋊C4 | — | 30 | 2 | no | — |
| 235 | C4⋊C4 | — | 30 | 2 | no | — |
| 236 | C42 | — | 5 | 2 | no | — |
| 237 | C42 | — | 5 | 2 | no | — |
| 238 | C42 | — | 60 | 4 | no | — |
| 239 | C3×C5 | #1776 | 1 | 1 | yes | characteristic, complemented, Hall |
| 240 | C22×C3 | — | 5 | 1 | no | — |
| 241 | C22×C3 | — | 5 | 1 | no | — |
| 242 | C22×C3 | — | 5 | 2 | no | — |
| 243 | C22×C3 | — | 5 | 2 | no | — |
| 244 | C22×C3 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 245 | C3×C4 | — | 5 | 1 | no | — |
| 246 | C3×C4 | — | 5 | 1 | no | — |
| 247 | C3×C4 | — | 10 | 4 | no | — |
| 248 | C3×C4 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 249 | C3×C4 | Q8×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 250 | Dic3 | — | 4 | 2 | no | — |
| 251 | Dic3 | — | 20 | 4 | no | — |
| 252 | Dic3 | — | 20 | 2 | no | — |
| 253 | C2×C5 | #4710 | 1 | 2 | yes | — |
| 254 | C2×C5 | C4×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 255 | D5 | #4711 | 1 | 1 | yes | characteristic |
| 256 | D5 | #4716 | 1 | 1 | yes | characteristic |
| 257 | D5 | #5121 | 1 | 2 | yes | — |
| 258 | C23 | — | 5 | 1 | no | — |
| 259 | C2×C4 | — | 5 | 1 | no | — |
| 260 | C2×C4 | — | 5 | 1 | no | — |
| 261 | C2×C4 | — | 5 | 1 | no | — |
| 262 | C2×C4 | — | 5 | 1 | no | — |
| 263 | C2×C4 | — | 5 | 1 | no | — |
| 264 | C2×C4 | — | 5 | 4 | no | — |
| 265 | C2×C4 | — | 6 | 2 | no | — |
| 266 | C2×C4 | — | 10 | 2 | no | — |
| 267 | C2×C4 | — | 10 | 2 | no | — |
| 268 | C2×C4 | — | 30 | 4 | no | — |
| 269 | C2×C4 | — | 30 | 2 | no | — |
| 270 | C2×C4 | — | 60 | 2 | no | — |
| 271 | C2×C4 | — | 60 | 2 | no | — |
| 272 | C2×C4 | — | 60 | 2 | no | — |
| 273 | C2×C4 | — | 60 | 2 | no | — |
| 274 | C2×C4 | C2×D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 5 | 2 | no | — |
| 276 | C8 | — | 10 | 2 | no | — |
| 277 | C8 | Dic3×AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 278 | C2×C3 | — | 5 | 1 | no | — |
| 279 | C2×C3 | — | 5 | 2 | no | — |
| 280 | C2×C3 | — | 5 | 1 | no | — |
| 281 | C2×C3 | C4⋊C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 282 | C2×C3 | #65735 | 1 | 2 | yes | — |
| 283 | C5 | #69814 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 284 | C22 | — | 5 | 1 | no | — |
| 285 | C22 | — | 5 | 1 | no | — |
| 286 | C22 | — | 5 | 2 | no | — |
| 287 | C22 | — | 5 | 2 | no | — |
| 288 | C22 | AGL(1,𝔽5)×D12 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 5 | 1 | no | — |
| 290 | C4 | — | 5 | 1 | no | — |
| 291 | C4 | — | 10 | 4 | no | — |
| 292 | C4 | — | 12 | 2 | no | — |
| 293 | C4 | — | 60 | 4 | no | — |
| 294 | C4 | — | 60 | 2 | no | — |
| 295 | C4 | AGL(1,𝔽5)×Dic6 | 1 | 1 | yes | characteristic |
| 296 | C4 | C2×Dic3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 297 | C3 | #10604048 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 298 | C2 | — | 5 | 2 | no | — |
| 299 | C2 | — | 5 | 1 | no | — |
| 300 | C2 | — | 5 | 1 | no | — |
| 301 | C2 | #11751661 | 1 | 2 | yes | — |
| 302 | C2 | AGL(1,𝔽5)×C4⋊Dic3 | 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