← #13017362
Subgroups — #13017362
Order 1280.
410 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×#3994 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10603842 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C8×C4⋊AGL(1,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | D5⋊#335 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | D5×#322 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #10598530 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #10598530 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#3994 | C4 | 1 | 2 | yes | — |
| 9 | C2×#3992 | — | 2 | 1 | no | — |
| 10 | C2×C8×AGL(1,𝔽5) | — | 2 | 1 | no | — |
| 11 | C4×C4⋊AGL(1,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 12 | C4×C4⋊AGL(1,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 13 | D5⋊(C4⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 14 | D5⋊(C4⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 18 | #64893 | C4 | 1 | 2 | yes | — |
| 19 | (C2×AGL(1,𝔽5))⋊C8 | — | 2 | 1 | no | — |
| 20 | (C2×AGL(1,𝔽5))⋊C8 | — | 2 | 1 | no | — |
| 21 | #64884 | C4 | 1 | 2 | yes | complemented |
| 22 | #64882 | C4 | 1 | 2 | yes | complemented |
| 23 | C5×#322 | C4 | 1 | 1 | yes | characteristic, complemented |
| 24 | #64675 | C4 | 1 | 1 | yes | characteristic, complemented |
| 25 | #64674 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | #64674 | C4 | 1 | 1 | yes | characteristic, complemented |
| 27 | #8291 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×C4⋊AGL(1,𝔽5) | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C4⋊AGL(1,𝔽5) | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×C4×AGL(1,𝔽5) | — | 2 | 1 | no | — |
| 31 | C2×C4×AGL(1,𝔽5) | — | 2 | 1 | no | — |
| 32 | C2×D5⋊C8 | — | 2 | 1 | no | — |
| 33 | C2×D5⋊C8 | — | 2 | 1 | no | — |
| 34 | C2×C8×D5 | — | 2 | 1 | no | — |
| 35 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C8×D5 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×C8×D5 | D4 | 1 | 1 | yes | characteristic |
| 39 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 40 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 1 | no | — |
| 41 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 1 | no | — |
| 42 | #4004 | — | 2 | 1 | no | — |
| 43 | #4004 | — | 2 | 1 | no | — |
| 44 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 45 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 46 | #3994 | — | 2 | 4 | no | — |
| 47 | #3992 | — | 4 | 2 | no | — |
| 48 | C8×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 49 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C8×Dic5 | — | 2 | 1 | no | — |
| 53 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C8×Dic5 | Q8 | 1 | 1 | yes | characteristic |
| 56 | C8×Dic5 | Q8 | 1 | 1 | yes | characteristic |
| 57 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C4×#107 | — | 5 | 1 | no | — |
| 61 | #1770 | — | 5 | 1 | no | — |
| 62 | C8×C4⋊C4 | — | 5 | 1 | no | — |
| 63 | C2×#335 | — | 5 | 1 | no | — |
| 64 | C2×#322 | — | 5 | 1 | no | — |
| 65 | #1279 | — | 5 | 1 | no | — |
| 66 | #1279 | — | 5 | 1 | no | — |
| 67 | C22×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 68 | C22×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 69 | C2×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 72 | C2×#170 | — | 2 | 2 | no | — |
| 73 | C2×#170 | — | 2 | 2 | no | — |
| 74 | C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 75 | C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 76 | C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 77 | C4⋊AGL(1,𝔽5) | M16 | 1 | 4 | yes | — |
| 78 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 79 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 80 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 81 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 82 | D5⋊C8 | — | 4 | 1 | no | — |
| 83 | D5⋊C8 | — | 4 | 1 | no | — |
| 84 | D5⋊C8 | — | 4 | 1 | no | — |
| 85 | D5⋊C8 | — | 4 | 1 | no | — |
| 86 | C2×C5×C8 | — | 2 | 1 | no | — |
| 87 | C2×C5×C8 | C42 | 1 | 1 | yes | characteristic |
| 88 | C2×C5×C8 | C42 | 1 | 1 | yes | characteristic |
| 89 | C2×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C4×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 94 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 95 | C2×#168 | — | 2 | 1 | no | — |
| 96 | C2×#168 | C42 | 1 | 1 | yes | characteristic |
| 97 | C2×#168 | C42 | 1 | 1 | yes | characteristic |
| 98 | C2×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 99 | C2×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 100 | C8×D5 | — | 2 | 2 | no | — |
| 101 | C8×D5 | — | 2 | 2 | no | — |
| 102 | C8×D5 | — | 2 | 1 | no | — |
| 103 | C8×D5 | — | 2 | 1 | no | — |
| 104 | C8×D5 | — | 4 | 2 | no | — |
| 105 | C8×D5 | D8 | 1 | 2 | yes | — |
| 106 | C8×D5 | Q16 | 1 | 2 | yes | — |
| 107 | C2×#107 | — | 5 | 2 | no | — |
| 108 | C2×C4⋊C8 | — | 5 | 1 | no | — |
| 109 | C2×C4⋊C8 | — | 5 | 1 | no | — |
| 110 | C2×#97 | — | 10 | 1 | no | — |
| 111 | C2×C4×C8 | — | 5 | 1 | no | — |
| 112 | C2×C4×C8 | — | 5 | 1 | no | — |
| 113 | C2×C4×C8 | — | 5 | 1 | no | — |
| 114 | C2×C4×C8 | — | 10 | 1 | no | — |
| 115 | C4×C4⋊C4 | — | 5 | 1 | no | — |
| 116 | C4×C4⋊C4 | — | 5 | 1 | no | — |
| 117 | #340 | — | 5 | 2 | no | — |
| 118 | #336 | — | 10 | 1 | no | — |
| 119 | #336 | — | 10 | 1 | no | — |
| 120 | #335 | — | 5 | 2 | no | — |
| 121 | #335 | — | 5 | 2 | no | — |
| 122 | #322 | — | 5 | 1 | no | — |
| 123 | #322 | — | 5 | 1 | no | — |
| 124 | #322 | — | 5 | 1 | no | — |
| 125 | #322 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 126 | C22×D5 | C4×D4 | 1 | 1 | yes | characteristic |
| 127 | C2×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 128 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 129 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 130 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 131 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 132 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 133 | C2×C4×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C5 | #117 | 1 | 1 | yes | characteristic |
| 136 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×Dic5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 138 | C2×Dic5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 139 | C4×D5 | — | 2 | 2 | no | — |
| 140 | C4×D5 | C2×M16 | 1 | 2 | yes | — |
| 141 | C4×D5 | #131 | 1 | 2 | yes | — |
| 142 | C4×D5 | C2×D8 | 1 | 1 | yes | characteristic |
| 143 | C4×D5 | C2×Q16 | 1 | 1 | yes | characteristic |
| 144 | C4×D5 | #135 | 1 | 1 | yes | characteristic |
| 145 | C4×D5 | #135 | 1 | 1 | yes | characteristic |
| 146 | #170 | — | 4 | 2 | no | — |
| 147 | #170 | — | 4 | 2 | no | — |
| 148 | C5×C8 | — | 2 | 1 | no | — |
| 149 | C5×C8 | — | 2 | 1 | no | — |
| 150 | C5×C8 | — | 2 | 1 | no | — |
| 151 | C5×C8 | — | 4 | 1 | no | — |
| 152 | C5×C8 | #107 | 1 | 2 | yes | complemented |
| 153 | #168 | — | 2 | 1 | no | — |
| 154 | #168 | — | 2 | 1 | no | — |
| 155 | #168 | — | 2 | 1 | no | — |
| 156 | #168 | — | 4 | 1 | no | — |
| 157 | #168 | #107 | 1 | 2 | yes | complemented |
| 158 | C22×C8 | — | 5 | 1 | no | — |
| 159 | C22×C8 | — | 5 | 1 | no | — |
| 160 | C22×C8 | — | 5 | 1 | no | — |
| 161 | C22×C8 | — | 5 | 1 | no | — |
| 162 | C22×C8 | — | 10 | 1 | no | — |
| 163 | C22×C8 | — | 10 | 1 | no | — |
| 164 | C22×C8 | — | 10 | 1 | no | — |
| 165 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 166 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 167 | C2×C42 | — | 5 | 1 | no | — |
| 168 | C2×C42 | — | 10 | 1 | no | — |
| 169 | C2×C42 | — | 10 | 1 | no | — |
| 170 | #107 | — | 10 | 4 | no | — |
| 171 | C4⋊C8 | — | 5 | 2 | no | — |
| 172 | C4⋊C8 | — | 5 | 2 | no | — |
| 173 | C4⋊C8 | — | 10 | 1 | no | — |
| 174 | C4⋊C8 | — | 10 | 1 | no | — |
| 175 | #97 | — | 20 | 2 | no | — |
| 176 | C4×C8 | — | 5 | 1 | no | — |
| 177 | C4×C8 | — | 5 | 1 | no | — |
| 178 | C4×C8 | — | 5 | 1 | no | — |
| 179 | C4×C8 | — | 5 | 1 | no | — |
| 180 | C4×C8 | — | 5 | 1 | no | — |
| 181 | C4×C8 | — | 5 | 1 | no | — |
| 182 | C4×C8 | — | 5 | 1 | no | — |
| 183 | C4×C8 | — | 10 | 1 | no | — |
| 184 | C4×C8 | — | 20 | 2 | no | — |
| 185 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 188 | #95 | — | 10 | 1 | no | — |
| 189 | #95 | — | 10 | 1 | no | — |
| 190 | C22×C5 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C2×D5 | #435 | 1 | 2 | yes | — |
| 192 | C2×D5 | C4×D8 | 1 | 1 | yes | characteristic |
| 193 | C2×D5 | C4×Q16 | 1 | 1 | yes | characteristic |
| 194 | C2×D5 | #444 | 1 | 2 | yes | — |
| 195 | AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 196 | AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 197 | C4×C5 | — | 2 | 1 | no | — |
| 198 | C4×C5 | C4×M16 | 1 | 1 | yes | characteristic |
| 199 | C4×C5 | #405 | 1 | 1 | yes | characteristic |
| 200 | C4×C5 | C2×#107 | 1 | 1 | yes | characteristic |
| 201 | C4×C5 | #427 | 1 | 1 | yes | characteristic |
| 202 | Dic5 | — | 2 | 1 | no | — |
| 203 | Dic5 | C4×D8 | 1 | 1 | yes | characteristic |
| 204 | Dic5 | C4×Q16 | 1 | 1 | yes | characteristic |
| 205 | Dic5 | #446 | 1 | 2 | yes | — |
| 206 | C22×C4 | — | 5 | 1 | no | — |
| 207 | C22×C4 | — | 5 | 1 | no | — |
| 208 | C22×C4 | — | 5 | 1 | no | — |
| 209 | C22×C4 | — | 10 | 2 | no | — |
| 210 | C22×C4 | — | 10 | 2 | no | — |
| 211 | C2×C8 | — | 2 | 1 | no | — |
| 212 | C2×C8 | — | 5 | 1 | no | — |
| 213 | C2×C8 | — | 5 | 2 | no | — |
| 214 | C2×C8 | — | 5 | 2 | no | — |
| 215 | C2×C8 | — | 5 | 1 | no | — |
| 216 | C2×C8 | — | 5 | 1 | no | — |
| 217 | C2×C8 | — | 5 | 1 | no | — |
| 218 | C2×C8 | — | 10 | 2 | no | — |
| 219 | C2×C8 | — | 10 | 1 | no | — |
| 220 | C2×C8 | — | 10 | 1 | no | — |
| 221 | C2×C8 | — | 10 | 2 | no | — |
| 222 | C2×C8 | — | 10 | 2 | no | — |
| 223 | C2×C8 | — | 10 | 2 | no | — |
| 224 | C2×C8 | — | 10 | 1 | no | — |
| 225 | C2×C8 | — | 20 | 2 | no | — |
| 226 | C2×C8 | — | 20 | 1 | no | — |
| 227 | C2×C8 | — | 20 | 1 | no | — |
| 228 | C2×C8 | — | 20 | 1 | no | — |
| 229 | C2×C8 | — | 20 | 1 | no | — |
| 230 | C2×C8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 231 | C2×C8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 234 | C4⋊C4 | — | 5 | 4 | no | — |
| 235 | C4⋊C4 | — | 10 | 2 | no | — |
| 236 | C4⋊C4 | — | 10 | 2 | no | — |
| 237 | C4⋊C4 | — | 10 | 2 | no | — |
| 238 | C42 | — | 5 | 1 | no | — |
| 239 | C42 | — | 5 | 1 | no | — |
| 240 | C42 | — | 5 | 1 | no | — |
| 241 | C42 | — | 20 | 1 | no | — |
| 242 | C42 | — | 20 | 1 | no | — |
| 243 | C42 | — | 20 | 1 | no | — |
| 244 | C42 | — | 20 | 1 | no | — |
| 245 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 246 | C2×C5 | #1771 | 1 | 1 | yes | characteristic |
| 247 | C2×C5 | #1773 | 1 | 1 | yes | characteristic |
| 248 | C2×C5 | C4×#107 | 1 | 1 | yes | characteristic |
| 249 | D5 | #1581 | 1 | 2 | yes | — |
| 250 | D5 | #1584 | 1 | 2 | yes | — |
| 251 | C23 | — | 5 | 1 | no | — |
| 252 | C2×C4 | — | 5 | 1 | no | — |
| 253 | C2×C4 | — | 5 | 1 | no | — |
| 254 | C2×C4 | — | 5 | 1 | no | — |
| 255 | C2×C4 | — | 5 | 1 | no | — |
| 256 | C2×C4 | — | 5 | 1 | no | — |
| 257 | C2×C4 | — | 5 | 1 | no | — |
| 258 | C2×C4 | — | 5 | 2 | no | — |
| 259 | C2×C4 | — | 5 | 2 | no | — |
| 260 | C2×C4 | — | 5 | 1 | no | — |
| 261 | C2×C4 | — | 10 | 2 | no | — |
| 262 | C2×C4 | — | 10 | 4 | no | — |
| 263 | C2×C4 | — | 20 | 2 | no | — |
| 264 | C2×C4 | — | 20 | 2 | no | — |
| 265 | C2×C4 | — | 20 | 2 | no | — |
| 266 | C2×C4 | — | 20 | 2 | no | — |
| 267 | C2×C4 | — | 20 | 2 | no | — |
| 268 | C2×C4 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | C2×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #4130 | 1 | 1 | yes | characteristic |
| 271 | C8 | — | 2 | 1 | no | — |
| 272 | C8 | — | 2 | 1 | no | — |
| 273 | C8 | — | 2 | 1 | no | — |
| 274 | C8 | — | 4 | 1 | no | — |
| 275 | C8 | — | 5 | 2 | no | — |
| 276 | C8 | — | 10 | 1 | no | — |
| 277 | C8 | — | 10 | 1 | no | — |
| 278 | C8 | — | 10 | 1 | no | — |
| 279 | C8 | — | 20 | 1 | no | — |
| 280 | C8 | — | 20 | 2 | no | — |
| 281 | C8 | — | 20 | 2 | no | — |
| 282 | C8 | #3994 | 1 | 2 | yes | complemented |
| 283 | C5 | #8291 | 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 | C4×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 2 | 1 | no | — |
| 290 | C4 | — | 5 | 1 | no | — |
| 291 | C4 | — | 5 | 1 | no | — |
| 292 | C4 | — | 5 | 2 | no | — |
| 293 | C4 | — | 10 | 1 | no | — |
| 294 | C4 | — | 20 | 4 | no | — |
| 295 | C4 | — | 40 | 2 | no | — |
| 296 | C4 | C2×#3994 | 1 | 1 | yes | characteristic |
| 297 | C4 | #65721 | 1 | 1 | yes | characteristic |
| 298 | C4 | M16×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 299 | C4 | #65728 | 1 | 1 | yes | characteristic |
| 300 | C2 | — | 5 | 2 | no | — |
| 301 | C2 | — | 5 | 2 | no | — |
| 302 | C2 | C4×#3994 | 1 | 1 | yes | characteristic |
| 303 | C2 | #10603880 | 1 | 1 | yes | characteristic |
| 304 | C2 | #10603885 | 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