← #14285827
Subgroups — #14285827
Order 1920.
336 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11752340 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D5⋊(C4⋊(C3⋊C8)) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#65122 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11746754 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11746752 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10604015 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×#6758 | — | 2 | 1 | no | — |
| 8 | C2×D5⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 9 | C3×C42×D5 | C22 | 1 | 1 | yes | characteristic |
| 10 | #91233 | — | 2 | 1 | no | — |
| 11 | C4⋊(C3⋊#170) | — | 2 | 1 | no | — |
| 12 | C4⋊(C3⋊#170) | C22 | 1 | 1 | yes | characteristic |
| 13 | C4⋊(C3⋊#170) | C22 | 1 | 1 | yes | characteristic |
| 14 | C4⋊(C3⋊#170) | C22 | 1 | 2 | yes | — |
| 15 | C3×C5×C4⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 16 | C3×#3952 | C4 | 1 | 2 | yes | — |
| 17 | C3×C8×Dic5 | C22 | 1 | 2 | yes | — |
| 18 | C3×C4⋊#168 | C4 | 1 | 1 | yes | characteristic |
| 19 | #70749 | — | 5 | 1 | no | maximal, Hall |
| 20 | #65681 | — | 3 | 1 | no | — |
| 21 | D5⋊(C4⋊C8) | — | 3 | 1 | no | — |
| 22 | #65122 | D3 | 1 | 1 | yes | characteristic |
| 23 | #64883 | — | 3 | 2 | no | — |
| 24 | #64881 | — | 3 | 2 | no | — |
| 25 | C2×C3×C4×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×C3×C4×D5 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C3×C4×D5 | Q8 | 1 | 1 | yes | characteristic |
| 28 | C2×C3⋊#170 | — | 2 | 1 | no | — |
| 29 | C2×C3⋊#170 | — | 2 | 1 | no | — |
| 30 | C2×C3⋊#170 | — | 2 | 2 | no | — |
| 31 | #6758 | — | 4 | 2 | no | — |
| 32 | D5⋊(C3⋊C8) | — | 4 | 2 | no | — |
| 33 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 34 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C3×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 36 | C3×C4×Dic5 | Q8 | 1 | 1 | yes | characteristic |
| 37 | C3×C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×C3×#168 | C2×C4 | 1 | 2 | yes | — |
| 39 | C3×#431 | — | 5 | 1 | no | — |
| 40 | #4924 | — | 5 | 1 | no | — |
| 41 | C2×C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 42 | #4457 | — | 5 | 2 | no | — |
| 43 | #4457 | — | 5 | 2 | no | — |
| 44 | C2×#700 | — | 6 | 1 | no | — |
| 45 | C2×D5⋊C8 | — | 6 | 1 | no | — |
| 46 | C42×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | #4004 | — | 6 | 1 | no | — |
| 48 | C4⋊#170 | — | 3 | 1 | no | — |
| 49 | C4⋊#170 | — | 3 | 2 | no | — |
| 50 | C4⋊#170 | — | 3 | 1 | no | — |
| 51 | C4⋊#170 | — | 6 | 1 | no | — |
| 52 | C5×C4⋊C8 | Dic3 | 1 | 1 | yes | characteristic |
| 53 | #3952 | Dic3 | 1 | 2 | yes | — |
| 54 | C8×Dic5 | C2×D3 | 1 | 2 | yes | — |
| 55 | C4⋊#168 | Dic3 | 1 | 1 | yes | characteristic |
| 56 | #1570 | — | 15 | 1 | no | Sylow |
| 57 | C22×C3×D5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 58 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C2×C3×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C3×Dic5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C3×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 64 | C3×C4×D5 | — | 2 | 1 | no | — |
| 65 | C3×C4×D5 | — | 2 | 1 | no | — |
| 66 | C3×C4×D5 | — | 4 | 1 | no | — |
| 67 | C3×C4×D5 | SD16 | 1 | 2 | yes | — |
| 68 | C3×C4×D5 | SD16 | 1 | 2 | yes | — |
| 69 | C3⋊#170 | — | 4 | 1 | no | — |
| 70 | C3⋊#170 | — | 4 | 1 | no | — |
| 71 | C3⋊#170 | — | 4 | 2 | no | — |
| 72 | C3×C5×C8 | — | 2 | 2 | no | — |
| 73 | C3×#168 | — | 2 | 2 | no | — |
| 74 | C2×C3×C42 | — | 5 | 1 | no | — |
| 75 | C2×#208 | — | 10 | 1 | no | — |
| 76 | C22×C3⋊C8 | — | 10 | 1 | no | — |
| 77 | C3×C4⋊C8 | — | 5 | 1 | no | — |
| 78 | C3×C4⋊C8 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 79 | C3×C22⋊C8 | — | 5 | 2 | no | — |
| 80 | C3×C4×C8 | — | 5 | 2 | no | — |
| 81 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 82 | C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 83 | C4⋊(C3⋊C8) | — | 5 | 2 | no | — |
| 84 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 85 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 86 | C2×C4×D5 | Dic6 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×D5 | D12 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×D5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 89 | C2×#170 | — | 6 | 1 | no | — |
| 90 | C2×#170 | — | 6 | 1 | no | — |
| 91 | C2×#170 | — | 6 | 2 | no | — |
| 92 | #700 | — | 12 | 2 | no | — |
| 93 | D5⋊C8 | — | 12 | 2 | no | — |
| 94 | C2×C5×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 95 | C42×C5 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 96 | C4×Dic5 | Dic6 | 1 | 1 | yes | characteristic |
| 97 | C4×Dic5 | D12 | 1 | 1 | yes | characteristic |
| 98 | C4×Dic5 | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | C2×#168 | C2×Dic3 | 1 | 2 | yes | — |
| 100 | #431 | — | 5 | 1 | no | — |
| 101 | #423 | — | 15 | 1 | no | — |
| 102 | C2×C4⋊C8 | — | 15 | 1 | no | — |
| 103 | #334 | — | 15 | 2 | no | — |
| 104 | #334 | — | 15 | 2 | no | — |
| 105 | C22×C3×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 106 | C2×C3×D5 | — | 2 | 1 | no | — |
| 107 | C2×C3×D5 | — | 2 | 1 | no | — |
| 108 | C2×C3×D5 | #106 | 1 | 2 | yes | — |
| 109 | C3×C4×C5 | — | 2 | 1 | no | — |
| 110 | C3×C4×C5 | #106 | 1 | 2 | yes | — |
| 111 | C3×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 112 | C3×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 113 | C3×Dic5 | — | 2 | 1 | no | — |
| 114 | C3×Dic5 | — | 2 | 1 | no | — |
| 115 | C3×Dic5 | C2×SD16 | 1 | 1 | yes | characteristic |
| 116 | C3×Dic5 | C2×SD16 | 1 | 1 | yes | characteristic |
| 117 | C22×C3×C4 | — | 5 | 1 | no | — |
| 118 | C22×C3×C4 | — | 5 | 1 | no | — |
| 119 | C22×C3×C4 | — | 5 | 1 | no | — |
| 120 | C2×C3×C8 | — | 5 | 2 | no | — |
| 121 | C2×C3×C8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 122 | C3×C42 | — | 5 | 1 | no | — |
| 123 | C3×C42 | — | 5 | 1 | no | — |
| 124 | C3×C42 | — | 5 | 1 | no | — |
| 125 | C3×C42 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 126 | #208 | — | 20 | 2 | no | — |
| 127 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 128 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 129 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 130 | C2×C3⋊C8 | — | 20 | 2 | no | — |
| 131 | C22×D5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 135 | C2×Dic5 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 136 | C2×Dic5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 137 | C2×Dic5 | C2×D12 | 1 | 1 | yes | characteristic |
| 138 | C4×D5 | — | 2 | 1 | no | — |
| 139 | C4×D5 | — | 2 | 1 | no | — |
| 140 | C4×D5 | — | 4 | 1 | no | — |
| 141 | C4×D5 | #204 | 1 | 2 | yes | — |
| 142 | C4×D5 | #204 | 1 | 2 | yes | — |
| 143 | #170 | — | 12 | 1 | no | — |
| 144 | #170 | — | 12 | 1 | no | — |
| 145 | #170 | — | 12 | 2 | no | — |
| 146 | C5×C8 | — | 2 | 2 | no | — |
| 147 | #168 | — | 2 | 2 | no | — |
| 148 | C2×M16 | — | 30 | 1 | no | — |
| 149 | C22×C8 | — | 30 | 1 | no | — |
| 150 | C2×C42 | — | 5 | 1 | no | — |
| 151 | C4⋊C8 | — | 5 | 1 | no | — |
| 152 | C4⋊C8 | — | 15 | 1 | no | — |
| 153 | C4⋊C8 | — | 15 | 1 | no | — |
| 154 | C4⋊C8 | — | 15 | 2 | no | — |
| 155 | C4⋊C8 | — | 30 | 1 | no | — |
| 156 | C4⋊C8 | — | 30 | 1 | no | — |
| 157 | C4⋊C8 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 158 | C22⋊C8 | — | 5 | 2 | no | — |
| 159 | C4×C8 | — | 5 | 2 | no | — |
| 160 | C2×C3×C5 | #424 | 1 | 1 | yes | characteristic |
| 161 | C2×C3×C5 | C2×#106 | 1 | 1 | yes | characteristic |
| 162 | C2×C3×C5 | #430 | 1 | 1 | yes | characteristic |
| 163 | C3×D5 | — | 2 | 2 | no | — |
| 164 | C23×C3 | — | 5 | 1 | no | — |
| 165 | C2×C3×C4 | — | 5 | 1 | no | — |
| 166 | C2×C3×C4 | — | 5 | 2 | no | — |
| 167 | C2×C3×C4 | — | 5 | 2 | no | — |
| 168 | C2×C3×C4 | — | 5 | 1 | no | — |
| 169 | C2×C3×C4 | — | 5 | 1 | no | — |
| 170 | C2×C3×C4 | — | 10 | 1 | no | — |
| 171 | C2×C3×C4 | — | 10 | 1 | no | — |
| 172 | C2×C3×C4 | — | 20 | 1 | no | — |
| 173 | C2×C3×C4 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 176 | C3×C8 | — | 2 | 2 | no | — |
| 177 | C3×C8 | — | 10 | 2 | no | — |
| 178 | C3⋊C8 | — | 20 | 1 | no | — |
| 179 | C3⋊C8 | — | 20 | 1 | no | — |
| 180 | C3⋊C8 | — | 20 | 2 | no | — |
| 181 | C22×C5 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 182 | C2×D5 | — | 2 | 1 | no | — |
| 183 | C2×D5 | — | 2 | 1 | no | — |
| 184 | C2×D5 | #817 | 1 | 2 | yes | — |
| 185 | C4×C5 | — | 2 | 1 | no | — |
| 186 | C4×C5 | #817 | 1 | 2 | yes | — |
| 187 | C4×C5 | #948 | 1 | 1 | yes | characteristic |
| 188 | C4×C5 | #948 | 1 | 1 | yes | characteristic |
| 189 | Dic5 | — | 2 | 1 | no | — |
| 190 | Dic5 | — | 2 | 1 | no | — |
| 191 | Dic5 | C2×#204 | 1 | 1 | yes | characteristic |
| 192 | Dic5 | C2×#204 | 1 | 1 | yes | characteristic |
| 193 | C22×C4 | — | 5 | 1 | no | — |
| 194 | C22×C4 | — | 5 | 1 | no | — |
| 195 | C22×C4 | — | 5 | 1 | no | — |
| 196 | M16 | — | 60 | 2 | no | — |
| 197 | C2×C8 | — | 5 | 2 | no | — |
| 198 | C2×C8 | — | 30 | 1 | no | — |
| 199 | C2×C8 | — | 30 | 1 | no | — |
| 200 | C2×C8 | — | 30 | 2 | no | — |
| 201 | C2×C8 | — | 60 | 2 | no | — |
| 202 | C2×C8 | C2×ΓL(1,𝔽16) | 1 | 2 | yes | — |
| 203 | C42 | — | 5 | 1 | no | — |
| 204 | C42 | — | 5 | 1 | no | — |
| 205 | C42 | — | 5 | 1 | no | — |
| 206 | C42 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 207 | C3×C5 | #1570 | 1 | 1 | yes | characteristic, complemented, Hall |
| 208 | C22×C3 | — | 5 | 2 | no | — |
| 209 | C22×C3 | — | 10 | 1 | no | — |
| 210 | C22×C3 | — | 10 | 1 | no | — |
| 211 | C22×C3 | C2×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 212 | C3×C4 | — | 2 | 1 | no | — |
| 213 | C3×C4 | — | 5 | 1 | no | — |
| 214 | C3×C4 | — | 5 | 1 | no | — |
| 215 | C3×C4 | — | 10 | 1 | no | — |
| 216 | C3×C4 | — | 10 | 1 | no | — |
| 217 | C3×C4 | #3993 | 1 | 2 | yes | — |
| 218 | C3×C4 | #4131 | 1 | 1 | yes | characteristic |
| 219 | C3×C4 | #4133 | 1 | 1 | yes | characteristic |
| 220 | C2×C5 | #4999 | 1 | 1 | yes | characteristic |
| 221 | C2×C5 | C2×#817 | 1 | 1 | yes | characteristic |
| 222 | C2×C5 | #5125 | 1 | 1 | yes | characteristic |
| 223 | D5 | — | 2 | 2 | no | — |
| 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 | 1 | no | — |
| 229 | C2×C4 | — | 5 | 1 | no | — |
| 230 | C2×C4 | — | 10 | 1 | no | — |
| 231 | C2×C4 | — | 10 | 1 | no | — |
| 232 | C2×C4 | — | 20 | 1 | no | — |
| 233 | C2×C4 | C4⋊ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | C4⋊ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | C22×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 236 | C8 | — | 2 | 2 | no | — |
| 237 | C8 | — | 10 | 2 | no | — |
| 238 | C8 | — | 60 | 1 | no | — |
| 239 | C8 | — | 60 | 1 | no | — |
| 240 | C8 | — | 60 | 2 | no | — |
| 241 | C2×C3 | — | 10 | 2 | no | — |
| 242 | C2×C3 | #65708 | 1 | 1 | yes | characteristic |
| 243 | C2×C3 | C2×#3993 | 1 | 1 | yes | characteristic |
| 244 | C2×C3 | #65729 | 1 | 1 | yes | characteristic |
| 245 | C5 | #70749 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 246 | C22 | — | 5 | 2 | no | — |
| 247 | C22 | — | 10 | 1 | no | — |
| 248 | C22 | — | 10 | 1 | no | — |
| 249 | C22 | C2×C4⋊ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 250 | C4 | — | 2 | 1 | no | — |
| 251 | C4 | — | 5 | 1 | no | — |
| 252 | C4 | — | 5 | 1 | no | — |
| 253 | C4 | — | 10 | 1 | no | — |
| 254 | C4 | — | 10 | 1 | no | — |
| 255 | C4 | #91222 | 1 | 2 | yes | — |
| 256 | C4 | #91990 | 1 | 1 | yes | characteristic |
| 257 | C4 | #91992 | 1 | 1 | yes | characteristic |
| 258 | C3 | #10604015 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 259 | C2 | — | 10 | 2 | no | — |
| 260 | C2 | #11752367 | 1 | 1 | yes | characteristic |
| 261 | C2 | C2×#91222 | 1 | 1 | yes | characteristic |
| 262 | C2 | #11752388 | 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