← #14294227
Subgroups — #14294227
Order 1920.
720 conjugacy classes of subgroups.
The subgroup lattice has 260 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C42×ΓL(1,𝔽16) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#65676 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#91150 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×ΓL(1,𝔽16)⋊C8 | C2 | 1 | 2 | yes | maximal |
| 5 | D5×#803 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | AGL(1,𝔽5)×#97 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C4×ΓL(1,𝔽16) | C4 | 1 | 4 | yes | — |
| 8 | C2×C4×ΓL(1,𝔽16) | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×ΓL(1,𝔽16) | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×D5⋊C8 | C22 | 1 | 2 | yes | — |
| 11 | C3×C42×D5 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×D5×C3⋊C8 | C22 | 1 | 2 | yes | — |
| 13 | C3×#4002 | C4 | 1 | 4 | yes | complemented |
| 14 | #91150 | C4 | 1 | 8 | yes | complemented |
| 15 | ΓL(1,𝔽16)⋊C8 | C4 | 1 | 8 | yes | complemented |
| 16 | #91087 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C5×#803 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | #90952 | C4 | 1 | 2 | yes | complemented |
| 19 | Dic3×#97 | — | 5 | 1 | no | maximal, Hall |
| 20 | C2×#3992 | — | 3 | 2 | no | — |
| 21 | C2×C8×AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 22 | C42×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 23 | #65676 | D3 | 1 | 1 | yes | characteristic |
| 24 | D5×#97 | — | 3 | 1 | no | — |
| 25 | C22×ΓL(1,𝔽16) | C2×C4 | 1 | 2 | yes | — |
| 26 | C22×ΓL(1,𝔽16) | C2×C4 | 1 | 2 | yes | — |
| 27 | C2×C3×C4×D5 | C2×C4 | 1 | 2 | yes | — |
| 28 | C2×C3×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 29 | C4×ΓL(1,𝔽16) | — | 2 | 8 | no | — |
| 30 | C4×ΓL(1,𝔽16) | C2×C4 | 1 | 4 | yes | — |
| 31 | C4×ΓL(1,𝔽16) | C2×C4 | 1 | 4 | yes | — |
| 32 | C2×C3×#170 | C2×C4 | 1 | 4 | yes | — |
| 33 | C3×D5⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 34 | C3×D5⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 35 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×C3⋊#168 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×C5×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 38 | C3×C4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C3×C4×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 40 | D5×C3⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 41 | D5×C3⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 42 | C2×C3×#97 | — | 5 | 1 | no | — |
| 43 | C2×#815 | — | 5 | 2 | no | — |
| 44 | C2×C8×Dic3 | — | 5 | 2 | no | — |
| 45 | C42×Dic3 | — | 5 | 1 | no | — |
| 46 | C2×#803 | — | 5 | 1 | no | — |
| 47 | C2×C4×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 48 | C2×C4×AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 49 | C2×C4×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 50 | C2×D5⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 51 | C2×C8×D5 | — | 3 | 2 | no | — |
| 52 | C42×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | #4002 | Dic3 | 1 | 4 | yes | complemented |
| 54 | #3992 | — | 3 | 8 | no | — |
| 55 | C8×AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 56 | C5×#97 | — | 3 | 1 | no | — |
| 57 | #3947 | — | 3 | 2 | no | — |
| 58 | #3935 | — | 3 | 1 | no | — |
| 59 | C4×#97 | — | 15 | 1 | no | Sylow |
| 60 | C22×C3×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×ΓL(1,𝔽16) | — | 2 | 4 | no | — |
| 62 | C2×ΓL(1,𝔽16) | C42 | 1 | 4 | yes | — |
| 63 | C2×ΓL(1,𝔽16) | C42 | 1 | 4 | yes | — |
| 64 | C2×ΓL(1,𝔽16) | M16 | 1 | 8 | yes | — |
| 65 | C2×C3×C4×C5 | C42 | 1 | 2 | yes | — |
| 66 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C3×Dic5 | C42 | 1 | 2 | yes | — |
| 68 | C2×C3×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C3×C4×D5 | — | 2 | 4 | no | — |
| 70 | C3×C4×D5 | C22×C4 | 1 | 2 | yes | — |
| 71 | C3×C4×D5 | C22×C4 | 1 | 2 | yes | — |
| 72 | C3×#170 | C42 | 1 | 8 | yes | complemented |
| 73 | C3⋊#168 | C42 | 1 | 4 | yes | complemented |
| 74 | C5×C3⋊C8 | C42 | 1 | 4 | yes | complemented |
| 75 | C22×C3×C8 | — | 5 | 2 | no | — |
| 76 | C2×C3×C42 | — | 5 | 1 | no | — |
| 77 | C2×C4×Dic3 | — | 5 | 1 | no | — |
| 78 | C2×C4×Dic3 | — | 5 | 1 | no | — |
| 79 | C2×C4×Dic3 | — | 5 | 4 | no | — |
| 80 | C22×C3⋊C8 | — | 5 | 2 | no | — |
| 81 | C3×#97 | — | 5 | 4 | no | — |
| 82 | #815 | — | 5 | 8 | no | — |
| 83 | C8×Dic3 | — | 5 | 8 | no | — |
| 84 | #803 | — | 5 | 1 | no | — |
| 85 | #803 | — | 5 | 2 | no | — |
| 86 | #803 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 87 | C22×AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 88 | C22×AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 89 | C2×C4×D5 | C4×D3 | 1 | 2 | yes | — |
| 90 | C2×C4×D5 | C22×D3 | 1 | 1 | yes | characteristic |
| 91 | C2×#170 | C2×Dic3 | 1 | 4 | yes | — |
| 92 | C4×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 93 | C4×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 94 | C4×AGL(1,𝔽5) | — | 6 | 8 | no | — |
| 95 | D5⋊C8 | C4×D3 | 1 | 4 | yes | — |
| 96 | D5⋊C8 | C4×D3 | 1 | 4 | yes | — |
| 97 | C2×C5×C8 | — | 3 | 2 | no | — |
| 98 | C42×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 99 | C4×Dic5 | C4×D3 | 1 | 1 | yes | characteristic |
| 100 | C4×Dic5 | C2×Dic3 | 1 | 2 | yes | — |
| 101 | C2×#168 | — | 3 | 2 | no | — |
| 102 | C8×D5 | — | 3 | 4 | no | — |
| 103 | C8×D5 | — | 3 | 4 | no | — |
| 104 | C2×#97 | — | 5 | 1 | no | — |
| 105 | C2×#97 | — | 15 | 1 | no | — |
| 106 | C2×#97 | — | 15 | 2 | no | — |
| 107 | C2×C4×C8 | — | 15 | 2 | no | — |
| 108 | C43 | — | 15 | 1 | no | — |
| 109 | C22×C3×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×D5 | C2×C42 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×D5 | C2×C42 | 1 | 1 | yes | characteristic |
| 112 | C2×C3×D5 | C2×M16 | 1 | 4 | yes | — |
| 113 | ΓL(1,𝔽16) | — | 2 | 4 | no | — |
| 114 | ΓL(1,𝔽16) | #97 | 1 | 8 | yes | complemented |
| 115 | C3×C4×C5 | — | 2 | 2 | no | — |
| 116 | C3×C4×C5 | C2×C42 | 1 | 2 | yes | — |
| 117 | C3×Dic5 | — | 2 | 2 | no | — |
| 118 | C3×Dic5 | C2×C42 | 1 | 2 | yes | — |
| 119 | C22×C3×C4 | — | 5 | 1 | no | — |
| 120 | C22×C3×C4 | — | 5 | 2 | no | — |
| 121 | C22×Dic3 | — | 5 | 2 | no | — |
| 122 | C22×Dic3 | — | 5 | 2 | no | — |
| 123 | C2×C3×C8 | — | 5 | 4 | no | — |
| 124 | C2×C3×C8 | — | 5 | 4 | no | — |
| 125 | C2×C3×C8 | — | 5 | 4 | no | — |
| 126 | C3×C42 | — | 5 | 2 | no | — |
| 127 | C3×C42 | — | 5 | 1 | no | — |
| 128 | C3×C42 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 129 | C4×Dic3 | — | 5 | 4 | no | — |
| 130 | C4×Dic3 | — | 5 | 4 | no | — |
| 131 | C4×Dic3 | — | 10 | 8 | no | — |
| 132 | C2×C3⋊C8 | — | 5 | 2 | no | — |
| 133 | C2×C3⋊C8 | — | 5 | 4 | no | — |
| 134 | C2×C3⋊C8 | — | 5 | 4 | no | — |
| 135 | C2×C3⋊C8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 136 | C22×D5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 137 | C2×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 138 | C2×AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 139 | C2×AGL(1,𝔽5) | — | 3 | 4 | no | — |
| 140 | C2×AGL(1,𝔽5) | — | 6 | 4 | no | — |
| 141 | C2×C4×C5 | C4×Dic3 | 1 | 2 | yes | — |
| 142 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic5 | C4×Dic3 | 1 | 2 | yes | — |
| 144 | C2×Dic5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 145 | C4×D5 | — | 2 | 4 | no | — |
| 146 | C4×D5 | C2×C4×D3 | 1 | 2 | yes | — |
| 147 | C4×D5 | C2×C4×D3 | 1 | 2 | yes | — |
| 148 | #170 | C4×Dic3 | 1 | 8 | yes | complemented |
| 149 | C5×C8 | — | 3 | 4 | no | — |
| 150 | #168 | — | 3 | 4 | no | — |
| 151 | C22×C8 | — | 5 | 2 | no | — |
| 152 | C22×C8 | — | 15 | 2 | no | — |
| 153 | C2×C42 | — | 5 | 1 | no | — |
| 154 | C2×C42 | — | 15 | 1 | no | — |
| 155 | C2×C42 | — | 15 | 1 | no | — |
| 156 | C2×C42 | — | 15 | 4 | no | — |
| 157 | #97 | — | 3 | 1 | no | — |
| 158 | #97 | — | 5 | 4 | no | — |
| 159 | #97 | — | 15 | 2 | no | — |
| 160 | #97 | — | 15 | 8 | no | — |
| 161 | #97 | — | 15 | 1 | no | — |
| 162 | C4×C8 | — | 15 | 8 | no | — |
| 163 | C2×C3×C5 | C43 | 1 | 1 | yes | characteristic |
| 164 | C2×C3×C5 | C4×M16 | 1 | 2 | yes | — |
| 165 | C3×D5 | C2×#97 | 1 | 2 | yes | — |
| 166 | C3×D5 | C4×M16 | 1 | 2 | yes | — |
| 167 | C23×C3 | — | 5 | 1 | no | — |
| 168 | C2×C3×C4 | — | 5 | 1 | no | — |
| 169 | C2×C3×C4 | — | 5 | 2 | no | — |
| 170 | C2×C3×C4 | — | 5 | 2 | no | — |
| 171 | C2×C3×C4 | — | 5 | 2 | no | — |
| 172 | C2×C3×C4 | — | 10 | 4 | no | — |
| 173 | C2×C3×C4 | C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 174 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 175 | C2×Dic3 | — | 5 | 4 | no | — |
| 176 | C2×Dic3 | — | 5 | 8 | no | — |
| 177 | C2×Dic3 | — | 5 | 4 | no | — |
| 178 | C2×Dic3 | — | 10 | 4 | no | — |
| 179 | C3×C8 | — | 5 | 8 | no | — |
| 180 | C3⋊C8 | — | 5 | 4 | no | — |
| 181 | C3⋊C8 | C4×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 182 | C22×C5 | C2×C4×Dic3 | 1 | 1 | yes | characteristic |
| 183 | C2×D5 | C42×D3 | 1 | 1 | yes | characteristic |
| 184 | C2×D5 | C42×D3 | 1 | 1 | yes | characteristic |
| 185 | C2×D5 | D3×M16 | 1 | 4 | yes | — |
| 186 | AGL(1,𝔽5) | — | 3 | 8 | no | — |
| 187 | AGL(1,𝔽5) | — | 6 | 4 | no | — |
| 188 | C4×C5 | — | 2 | 2 | no | — |
| 189 | C4×C5 | C42×D3 | 1 | 2 | yes | — |
| 190 | Dic5 | — | 2 | 2 | no | — |
| 191 | Dic5 | C2×C4×Dic3 | 1 | 2 | yes | — |
| 192 | C22×C4 | — | 5 | 1 | no | — |
| 193 | C22×C4 | — | 5 | 2 | no | — |
| 194 | C22×C4 | — | 15 | 2 | no | — |
| 195 | C22×C4 | — | 15 | 2 | no | — |
| 196 | C2×C8 | — | 3 | 2 | no | — |
| 197 | C2×C8 | — | 5 | 4 | no | — |
| 198 | C2×C8 | — | 5 | 4 | no | — |
| 199 | C2×C8 | — | 5 | 4 | no | — |
| 200 | C2×C8 | — | 15 | 4 | no | — |
| 201 | C2×C8 | — | 15 | 4 | no | — |
| 202 | C2×C8 | — | 15 | 2 | no | — |
| 203 | C42 | — | 5 | 2 | no | — |
| 204 | C42 | — | 5 | 1 | no | — |
| 205 | C42 | — | 15 | 4 | no | — |
| 206 | C42 | — | 15 | 4 | no | — |
| 207 | C42 | — | 30 | 8 | no | — |
| 208 | C42 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 209 | C3×C5 | C4×#97 | 1 | 1 | yes | characteristic, complemented, Hall |
| 210 | C22×C3 | — | 5 | 1 | no | — |
| 211 | C22×C3 | — | 5 | 1 | no | — |
| 212 | C22×C3 | — | 5 | 4 | no | — |
| 213 | C22×C3 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 214 | C3×C4 | — | 2 | 2 | no | — |
| 215 | C3×C4 | — | 5 | 2 | no | — |
| 216 | C3×C4 | — | 10 | 2 | no | — |
| 217 | C3×C4 | C2×C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 218 | Dic3 | — | 5 | 8 | no | — |
| 219 | Dic3 | — | 10 | 4 | no | — |
| 220 | C2×C5 | C42×Dic3 | 1 | 1 | yes | characteristic |
| 221 | C2×C5 | Dic3×M16 | 1 | 2 | yes | — |
| 222 | D5 | D3×#97 | 1 | 2 | yes | — |
| 223 | D5 | #4707 | 1 | 2 | yes | — |
| 224 | C23 | — | 5 | 1 | no | — |
| 225 | C2×C4 | — | 5 | 1 | no | — |
| 226 | C2×C4 | — | 5 | 2 | no | — |
| 227 | C2×C4 | — | 5 | 2 | no | — |
| 228 | C2×C4 | — | 5 | 2 | no | — |
| 229 | C2×C4 | — | 10 | 4 | no | — |
| 230 | C2×C4 | — | 15 | 4 | no | — |
| 231 | C2×C4 | — | 15 | 8 | no | — |
| 232 | C2×C4 | — | 15 | 4 | no | — |
| 233 | C2×C4 | — | 30 | 4 | no | — |
| 234 | C2×C4 | Dic3×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 235 | C2×C4 | C2×D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 236 | C8 | — | 3 | 4 | no | — |
| 237 | C8 | — | 5 | 8 | no | — |
| 238 | C8 | — | 15 | 4 | no | — |
| 239 | C2×C3 | — | 5 | 2 | no | — |
| 240 | C2×C3 | — | 5 | 2 | no | — |
| 241 | C2×C3 | C42×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 242 | C2×C3 | M16×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 243 | C5 | Dic3×#97 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 244 | C22 | — | 5 | 4 | no | — |
| 245 | C22 | — | 5 | 1 | no | — |
| 246 | C22 | — | 5 | 1 | no | — |
| 247 | C22 | C2×Dic3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 248 | C4 | — | 2 | 2 | no | — |
| 249 | C4 | — | 5 | 2 | no | — |
| 250 | C4 | — | 10 | 2 | no | — |
| 251 | C4 | — | 15 | 8 | no | — |
| 252 | C4 | — | 30 | 4 | no | — |
| 253 | C4 | C4×D3×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 254 | C3 | AGL(1,𝔽5)×#97 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 255 | C2 | — | 5 | 2 | no | — |
| 256 | C2 | — | 5 | 2 | no | — |
| 257 | C2 | #11751698 | 1 | 2 | yes | — |
| 258 | C2 | C4×Dic3×AGL(1,𝔽5) | 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