← #13043641
Subgroups — #13043641
Order 1280.
498 conjugacy classes of subgroups.
The subgroup lattice has 256 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10615059 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604448 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10604448 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | Q8×#674 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10604367 | C2 | 1 | 2 | yes | maximal |
| 6 | #10604366 | C2 | 1 | 2 | yes | maximal |
| 7 | C2×C4⋊#674 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C4×#3998 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C5×#533 | C4 | 1 | 1 | yes | characteristic |
| 10 | C2×#3998 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#3998 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×#674 | C22 | 1 | 2 | yes | — |
| 13 | #65313 | C4 | 1 | 2 | yes | — |
| 14 | #65313 | C4 | 1 | 2 | yes | — |
| 15 | Q8×#168 | C22 | 1 | 2 | yes | — |
| 16 | Q8×#168 | C22 | 1 | 2 | yes | — |
| 17 | #65288 | C4 | 1 | 2 | yes | — |
| 18 | #65286 | C22 | 1 | 2 | yes | — |
| 19 | #65213 | C4 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×#168 | C22 | 1 | 1 | yes | characteristic |
| 21 | #64859 | C22 | 1 | 4 | yes | — |
| 22 | #64859 | C22 | 1 | 2 | yes | — |
| 23 | C4⋊#674 | C22 | 1 | 4 | yes | — |
| 24 | C4⋊#674 | C22 | 1 | 4 | yes | — |
| 25 | C4⋊#674 | C22 | 1 | 1 | yes | characteristic |
| 26 | C4⋊#674 | C22 | 1 | 1 | yes | characteristic |
| 27 | C4⋊#674 | C22 | 1 | 2 | yes | — |
| 28 | C4×#674 | C22 | 1 | 4 | yes | — |
| 29 | C4×#674 | C22 | 1 | 2 | yes | — |
| 30 | #33476 | — | 5 | 1 | no | maximal, Sylow |
| 31 | C5×C4⋊Q8 | C8 | 1 | 1 | yes | characteristic |
| 32 | C5×C4⋊Q8 | C8 | 1 | 1 | yes | characteristic |
| 33 | C5×#125 | C8 | 1 | 2 | yes | — |
| 34 | C5×C22⋊Q8 | C8 | 1 | 2 | yes | — |
| 35 | C5×C22⋊Q8 | C8 | 1 | 2 | yes | — |
| 36 | C4×C5×Q8 | C2×C4 | 1 | 2 | yes | — |
| 37 | C4×C5×Q8 | C2×C4 | 1 | 2 | yes | — |
| 38 | C5×#117 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×#681 | C2×C4 | 1 | 2 | yes | — |
| 41 | C22×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C22×#168 | C23 | 1 | 1 | yes | characteristic |
| 43 | #3998 | — | 2 | 2 | no | — |
| 44 | #3998 | Q8 | 1 | 4 | yes | complemented |
| 45 | C2×#674 | — | 2 | 4 | no | — |
| 46 | C2×#674 | C23 | 1 | 2 | yes | — |
| 47 | C2×#674 | C23 | 1 | 2 | yes | — |
| 48 | C2×#674 | C23 | 1 | 2 | yes | — |
| 49 | C2×#674 | C23 | 1 | 2 | yes | — |
| 50 | C4⋊#168 | C2×C4 | 1 | 4 | yes | — |
| 51 | C4⋊#168 | C2×C4 | 1 | 4 | yes | — |
| 52 | C4⋊#168 | C2×C4 | 1 | 2 | yes | — |
| 53 | C4⋊#168 | C2×C4 | 1 | 2 | yes | — |
| 54 | #3935 | C2×C4 | 1 | 4 | yes | — |
| 55 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C4×#168 | C23 | 1 | 4 | yes | — |
| 58 | C4×#168 | C23 | 1 | 1 | yes | characteristic |
| 59 | C4×#168 | C23 | 1 | 1 | yes | characteristic |
| 60 | #2960 | — | 5 | 1 | no | — |
| 61 | #2183 | — | 5 | 2 | no | — |
| 62 | #2183 | — | 5 | 4 | no | — |
| 63 | Q8×C16 | — | 5 | 2 | no | — |
| 64 | #2151 | — | 5 | 2 | no | — |
| 65 | C2×C4⋊C16 | — | 5 | 1 | no | — |
| 66 | #2108 | — | 5 | 2 | no | — |
| 67 | C4×M32 | — | 5 | 1 | no | — |
| 68 | C2×C5×Q8 | C2×C8 | 1 | 2 | yes | — |
| 69 | C2×C5×Q8 | C2×C8 | 1 | 2 | yes | — |
| 70 | C22×C4×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 71 | C22×C4×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C5×C4⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 74 | C5×C4⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 75 | C5×C4⋊C4 | C2×C8 | 1 | 4 | yes | — |
| 76 | C5×C4⋊C4 | C2×C8 | 1 | 4 | yes | — |
| 77 | C5×C4⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 78 | C5×C4⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 79 | C5×C22⋊C4 | C2×C8 | 1 | 4 | yes | — |
| 80 | C42×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 81 | C42×C5 | C2×C8 | 1 | 1 | yes | characteristic |
| 82 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C42×C5 | C22×C4 | 1 | 4 | yes | — |
| 85 | #681 | — | 2 | 4 | no | — |
| 86 | C2×#168 | — | 2 | 2 | no | — |
| 87 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×#168 | C22×C4 | 1 | 4 | yes | — |
| 90 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C2×#168 | C2×Q8 | 1 | 2 | yes | — |
| 92 | C2×#168 | #41 | 1 | 2 | yes | — |
| 93 | C2×#168 | C24 | 1 | 1 | yes | characteristic |
| 94 | #674 | — | 2 | 2 | no | — |
| 95 | #674 | — | 2 | 2 | no | — |
| 96 | #674 | — | 2 | 2 | no | — |
| 97 | #674 | C2×Q8 | 1 | 4 | yes | — |
| 98 | #533 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 99 | C2×M32 | — | 5 | 1 | no | — |
| 100 | C2×M32 | — | 5 | 1 | no | — |
| 101 | C22×C16 | — | 5 | 2 | no | — |
| 102 | #446 | — | 5 | 2 | no | — |
| 103 | #446 | — | 5 | 2 | no | — |
| 104 | C8×Q8 | — | 5 | 2 | no | — |
| 105 | C8×Q8 | — | 5 | 2 | no | — |
| 106 | #424 | — | 5 | 2 | no | — |
| 107 | #423 | — | 5 | 1 | no | — |
| 108 | #405 | — | 5 | 2 | no | — |
| 109 | C2×C4×C8 | — | 5 | 1 | no | — |
| 110 | C4⋊C16 | — | 5 | 1 | no | — |
| 111 | C4⋊C16 | — | 5 | 1 | no | — |
| 112 | C4⋊C16 | — | 5 | 2 | no | — |
| 113 | C4⋊C16 | — | 5 | 4 | no | — |
| 114 | C4⋊C16 | — | 5 | 4 | no | — |
| 115 | #346 | — | 5 | 2 | no | — |
| 116 | #346 | — | 5 | 4 | no | — |
| 117 | C4×C16 | — | 5 | 2 | no | — |
| 118 | C4×C16 | — | 5 | 4 | no | — |
| 119 | C23×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 120 | C5×Q8 | — | 2 | 4 | no | — |
| 121 | C5×Q8 | — | 4 | 2 | no | — |
| 122 | C2×C4×C5 | — | 2 | 2 | no | — |
| 123 | C2×C4×C5 | — | 4 | 1 | no | — |
| 124 | C2×C4×C5 | C4×Q8 | 1 | 2 | yes | — |
| 125 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C5 | C22×C8 | 1 | 4 | yes | — |
| 129 | C2×C4×C5 | C22×C8 | 1 | 4 | yes | — |
| 130 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C5 | #131 | 1 | 2 | yes | — |
| 133 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 134 | #168 | — | 2 | 1 | no | — |
| 135 | #168 | — | 2 | 1 | no | — |
| 136 | #168 | — | 2 | 4 | no | — |
| 137 | #168 | C4×Q8 | 1 | 2 | yes | — |
| 138 | #168 | C22×Q8 | 1 | 1 | yes | characteristic |
| 139 | #168 | C2×#41 | 1 | 1 | yes | characteristic |
| 140 | C2×M16 | — | 5 | 2 | no | — |
| 141 | C22×C8 | — | 5 | 1 | no | — |
| 142 | C22×C8 | — | 5 | 1 | no | — |
| 143 | C4⋊Q8 | #170 | 1 | 1 | yes | characteristic |
| 144 | C4⋊Q8 | #170 | 1 | 1 | yes | characteristic |
| 145 | #125 | #170 | 1 | 2 | yes | — |
| 146 | C22⋊Q8 | #170 | 1 | 2 | yes | — |
| 147 | C22⋊Q8 | #170 | 1 | 2 | yes | — |
| 148 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 149 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 150 | #117 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 151 | C2×C42 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 152 | M32 | — | 5 | 4 | no | — |
| 153 | M32 | — | 10 | 2 | no | — |
| 154 | C2×C16 | — | 5 | 2 | no | — |
| 155 | C2×C16 | — | 5 | 2 | no | — |
| 156 | C2×C16 | — | 5 | 2 | no | — |
| 157 | C2×C16 | — | 5 | 2 | no | — |
| 158 | C2×C16 | — | 10 | 4 | no | — |
| 159 | C4⋊C8 | — | 5 | 2 | no | — |
| 160 | C4⋊C8 | — | 5 | 2 | no | — |
| 161 | C4⋊C8 | — | 5 | 4 | no | — |
| 162 | C4⋊C8 | — | 5 | 4 | no | — |
| 163 | #97 | — | 5 | 4 | no | — |
| 164 | C4×C8 | — | 5 | 1 | no | — |
| 165 | C4×C8 | — | 5 | 4 | no | — |
| 166 | C4×C8 | — | 5 | 1 | no | — |
| 167 | C4×C8 | — | 5 | 1 | no | — |
| 168 | C4×C8 | — | 5 | 1 | no | — |
| 169 | C22×C5 | — | 2 | 1 | no | — |
| 170 | C22×C5 | — | 2 | 1 | no | — |
| 171 | C22×C5 | C8×Q8 | 1 | 2 | yes | — |
| 172 | C22×C5 | C23×C8 | 1 | 1 | yes | characteristic |
| 173 | C4×C5 | — | 2 | 1 | no | — |
| 174 | C4×C5 | — | 2 | 4 | no | — |
| 175 | C4×C5 | — | 2 | 1 | no | — |
| 176 | C4×C5 | — | 2 | 1 | no | — |
| 177 | C4×C5 | — | 2 | 4 | no | — |
| 178 | C4×C5 | C8×Q8 | 1 | 2 | yes | — |
| 179 | C4×C5 | #504 | 1 | 2 | yes | — |
| 180 | C4×C5 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 181 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 182 | C2×Q8 | C2×#170 | 1 | 2 | yes | — |
| 183 | C2×Q8 | C2×#170 | 1 | 2 | yes | — |
| 184 | C22×C4 | C2×#170 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | C2×#170 | 1 | 1 | yes | characteristic |
| 186 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 187 | M16 | — | 10 | 4 | no | — |
| 188 | C2×C8 | — | 5 | 2 | no | — |
| 189 | C2×C8 | — | 5 | 2 | no | — |
| 190 | C2×C8 | — | 5 | 1 | no | — |
| 191 | C2×C8 | — | 5 | 1 | no | — |
| 192 | C2×C8 | — | 5 | 1 | no | — |
| 193 | C2×C8 | — | 5 | 1 | no | — |
| 194 | C2×C8 | — | 5 | 4 | no | — |
| 195 | C2×C8 | — | 10 | 2 | no | — |
| 196 | C4⋊C4 | C2×#170 | 1 | 2 | yes | — |
| 197 | C4⋊C4 | C2×#170 | 1 | 2 | yes | — |
| 198 | C4⋊C4 | C2×#170 | 1 | 4 | yes | — |
| 199 | C4⋊C4 | C2×#170 | 1 | 4 | yes | — |
| 200 | C4⋊C4 | C2×#170 | 1 | 2 | yes | — |
| 201 | C4⋊C4 | C2×#170 | 1 | 2 | yes | — |
| 202 | C22⋊C4 | C2×#170 | 1 | 4 | yes | — |
| 203 | C42 | C2×#170 | 1 | 1 | yes | characteristic |
| 204 | C42 | C2×#170 | 1 | 1 | yes | characteristic |
| 205 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 206 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 207 | C42 | C22×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 208 | C16 | — | 5 | 4 | no | — |
| 209 | C16 | — | 10 | 2 | no | — |
| 210 | C16 | — | 10 | 2 | no | — |
| 211 | C16 | — | 10 | 2 | no | — |
| 212 | C2×C5 | — | 2 | 2 | no | — |
| 213 | C2×C5 | C2×C8×Q8 | 1 | 1 | yes | characteristic |
| 214 | C2×C5 | C2×#504 | 1 | 1 | yes | characteristic |
| 215 | C2×C5 | #3407 | 1 | 1 | yes | characteristic |
| 216 | C23 | C22×#170 | 1 | 1 | yes | characteristic |
| 217 | Q8 | — | 2 | 4 | no | — |
| 218 | Q8 | — | 4 | 2 | no | — |
| 219 | C2×C4 | — | 2 | 2 | no | — |
| 220 | C2×C4 | — | 4 | 1 | no | — |
| 221 | C2×C4 | #4133 | 1 | 2 | yes | — |
| 222 | C2×C4 | Q8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 223 | C2×C4 | C22×#170 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | C22×#170 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | C22×#170 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | C22×#170 | 1 | 4 | yes | — |
| 227 | C2×C4 | C22×#170 | 1 | 4 | yes | — |
| 228 | C2×C4 | C22×#170 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | C22×#170 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 231 | C8 | — | 5 | 1 | no | — |
| 232 | C8 | — | 5 | 1 | no | — |
| 233 | C8 | — | 5 | 2 | no | — |
| 234 | C8 | — | 10 | 1 | no | — |
| 235 | C8 | — | 10 | 1 | no | — |
| 236 | C8 | — | 10 | 4 | no | — |
| 237 | C5 | #33476 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 238 | C22 | — | 2 | 1 | no | — |
| 239 | C22 | — | 2 | 1 | no | — |
| 240 | C22 | Q8×#170 | 1 | 2 | yes | — |
| 241 | C22 | C23×#170 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 1 | no | — |
| 243 | C4 | — | 2 | 4 | no | — |
| 244 | C4 | — | 2 | 1 | no | — |
| 245 | C4 | — | 2 | 1 | no | — |
| 246 | C4 | — | 2 | 4 | no | — |
| 247 | C4 | Q8×#170 | 1 | 2 | yes | — |
| 248 | C4 | #65791 | 1 | 2 | yes | — |
| 249 | C4 | C2×#4133 | 1 | 1 | yes | characteristic |
| 250 | C4 | C2×Q8×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 2 | 2 | no | — |
| 252 | C2 | C2×Q8×#170 | 1 | 1 | yes | characteristic |
| 253 | C2 | C2×#65791 | 1 | 1 | yes | characteristic |
| 254 | C2 | #10617530 | 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