← #13036903
Subgroups — #13036903
Order 1280.
854 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 | AGL(1,𝔽5)×C4⋊Q8 | C2 | 1 | 2 | yes | maximal |
| 2 | D5×#500 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604674 | C2 | 1 | 2 | yes | maximal |
| 4 | #10604076 | C2 | 1 | 4 | yes | maximal |
| 5 | #10604072 | C2 | 1 | 4 | yes | maximal |
| 6 | #10603847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C4×#3992 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×Q8×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 9 | D5×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 10 | (C2×AGL(1,𝔽5))⋊Q8 | — | 2 | 2 | no | — |
| 11 | C2×#3993 | C22 | 1 | 4 | yes | — |
| 12 | C2×#3992 | C22 | 1 | 2 | yes | — |
| 13 | C2×#3992 | C22 | 1 | 2 | yes | — |
| 14 | #65712 | — | 2 | 2 | no | — |
| 15 | #65712 | C22 | 1 | 4 | yes | — |
| 16 | C4⋊C4×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 17 | C4⋊C4×AGL(1,𝔽5) | C22 | 1 | 4 | yes | — |
| 18 | #65688 | C22 | 1 | 2 | yes | — |
| 19 | C42×AGL(1,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | C5×#500 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #65367 | C4 | 1 | 2 | yes | complemented |
| 23 | D5×#107 | C22 | 1 | 4 | yes | — |
| 24 | #65163 | C4 | 1 | 4 | yes | complemented |
| 25 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 26 | #64971 | C4 | 1 | 1 | yes | characteristic, complemented |
| 27 | #64874 | C22 | 1 | 8 | yes | — |
| 28 | #21726 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C2×Q8×D5 | — | 2 | 2 | no | — |
| 30 | Q8×AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 31 | C2×C4⋊AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 32 | C2×C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 33 | C2×C4⋊AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 34 | C2×C4⋊AGL(1,𝔽5) | C23 | 1 | 4 | yes | — |
| 35 | C2×C4×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 36 | C2×C4×AGL(1,𝔽5) | D4 | 1 | 2 | yes | — |
| 37 | C2×C4×AGL(1,𝔽5) | D4 | 1 | 2 | yes | — |
| 38 | C2×C4×AGL(1,𝔽5) | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C4×AGL(1,𝔽5) | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×D5⋊C8 | C23 | 1 | 2 | yes | — |
| 41 | C5×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 42 | #4090 | — | 2 | 2 | no | — |
| 43 | C2×C8×D5 | C23 | 1 | 2 | yes | — |
| 44 | D5×C4⋊C4 | — | 2 | 2 | no | — |
| 45 | D5×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 46 | #4034 | — | 2 | 2 | no | — |
| 47 | #4034 | C2×C4 | 1 | 4 | yes | — |
| 48 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 49 | #4015 | C2×C4 | 1 | 2 | yes | — |
| 50 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 4 | no | — |
| 51 | (C2×AGL(1,𝔽5))⋊C4 | — | 2 | 4 | no | — |
| 52 | C4×#170 | — | 2 | 1 | no | — |
| 53 | C4×#170 | — | 2 | 1 | no | — |
| 54 | #3993 | — | 2 | 8 | no | — |
| 55 | #3992 | — | 2 | 4 | no | — |
| 56 | #3992 | Q8 | 1 | 8 | yes | complemented |
| 57 | C5×#107 | C2×C4 | 1 | 4 | yes | — |
| 58 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | #3950 | C2×C4 | 1 | 4 | yes | — |
| 60 | C8×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 61 | #3939 | C2×C4 | 1 | 8 | yes | — |
| 62 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C2×#500 | — | 5 | 1 | no | — |
| 64 | C4×C4⋊Q8 | — | 5 | 2 | no | — |
| 65 | #1990 | — | 5 | 2 | no | — |
| 66 | #1943 | — | 5 | 4 | no | — |
| 67 | #1844 | — | 5 | 1 | no | — |
| 68 | #1776 | — | 5 | 4 | no | — |
| 69 | C4×#97 | — | 5 | 1 | no | — |
| 70 | C22×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 71 | C22×AGL(1,𝔽5) | C2×D4 | 1 | 2 | yes | — |
| 72 | C22×AGL(1,𝔽5) | C2×D4 | 1 | 2 | yes | — |
| 73 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 74 | Q8×D5 | — | 4 | 4 | no | — |
| 75 | Q8×D5 | — | 4 | 4 | no | — |
| 76 | C2×C4×D5 | — | 2 | 4 | no | — |
| 77 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×D5 | C24 | 1 | 1 | yes | characteristic |
| 80 | C2×Dic10 | — | 2 | 2 | no | — |
| 81 | C2×Dic10 | — | 2 | 4 | no | — |
| 82 | C2×#170 | — | 2 | 2 | no | — |
| 83 | C4⋊AGL(1,𝔽5) | — | 2 | 8 | no | — |
| 84 | C4⋊AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 85 | C4⋊AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 86 | C4⋊AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 87 | C4×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 88 | C4×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 89 | C4×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 90 | C4×AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 91 | C4×AGL(1,𝔽5) | C2×Q8 | 1 | 4 | yes | — |
| 92 | D5⋊C8 | — | 2 | 2 | no | — |
| 93 | D5⋊C8 | C2×Q8 | 1 | 4 | yes | — |
| 94 | C2×C5×C8 | C22×C4 | 1 | 2 | yes | — |
| 95 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 96 | C5×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 97 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 99 | C4⋊Dic5 | C22×C4 | 1 | 4 | yes | — |
| 100 | #683 | — | 2 | 4 | no | — |
| 101 | #683 | — | 2 | 4 | no | — |
| 102 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 103 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 104 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 105 | C2×#168 | C22×C4 | 1 | 2 | yes | — |
| 106 | C8×D5 | C2×Q8 | 1 | 4 | yes | — |
| 107 | C8×D5 | #41 | 1 | 4 | yes | — |
| 108 | C2×C4⋊Q8 | — | 5 | 2 | no | — |
| 109 | C2×C4×Q8 | — | 10 | 2 | no | — |
| 110 | #500 | — | 5 | 2 | no | — |
| 111 | #500 | — | 5 | 4 | no | — |
| 112 | #500 | — | 5 | 1 | no | — |
| 113 | #500 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 114 | C2×#107 | — | 5 | 4 | no | — |
| 115 | C2×#106 | — | 5 | 4 | no | — |
| 116 | C2×#97 | — | 5 | 2 | no | — |
| 117 | C2×#97 | — | 5 | 2 | no | — |
| 118 | C2×C4×C8 | — | 5 | 1 | no | — |
| 119 | C2×C4×C8 | — | 5 | 1 | no | — |
| 120 | #391 | — | 10 | 2 | no | — |
| 121 | #389 | — | 5 | 4 | no | — |
| 122 | #389 | — | 10 | 2 | no | — |
| 123 | #384 | — | 5 | 2 | no | — |
| 124 | C4×C4⋊C4 | — | 5 | 4 | no | — |
| 125 | C4×C4⋊C4 | — | 10 | 2 | no | — |
| 126 | C43 | — | 5 | 1 | no | — |
| 127 | #340 | — | 5 | 8 | no | — |
| 128 | C22×D5 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 130 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 131 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 132 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 133 | C2×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 134 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 135 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 136 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 137 | C2×AGL(1,𝔽5) | C4⋊Q8 | 1 | 4 | yes | — |
| 138 | C5×Q8 | — | 4 | 4 | no | — |
| 139 | C2×C4×C5 | — | 2 | 4 | no | — |
| 140 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 142 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic5 | — | 2 | 4 | no | — |
| 144 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 145 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×Dic5 | C2×#41 | 1 | 1 | yes | characteristic |
| 147 | C4×D5 | — | 4 | 4 | no | — |
| 148 | C4×D5 | — | 4 | 4 | no | — |
| 149 | C4×D5 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 150 | C4×D5 | #137 | 1 | 4 | yes | — |
| 151 | C4×D5 | C22×Q8 | 1 | 2 | yes | — |
| 152 | C4×D5 | C2×#41 | 1 | 2 | yes | — |
| 153 | Dic10 | — | 4 | 4 | no | — |
| 154 | Dic10 | — | 4 | 8 | no | — |
| 155 | #170 | — | 2 | 4 | no | — |
| 156 | C5×C8 | C4×Q8 | 1 | 4 | yes | complemented |
| 157 | #168 | C4×Q8 | 1 | 4 | yes | complemented |
| 158 | C22×Q8 | — | 10 | 2 | no | — |
| 159 | C22×C8 | — | 5 | 2 | no | — |
| 160 | C22×C8 | — | 5 | 2 | no | — |
| 161 | C4⋊Q8 | — | 5 | 4 | no | — |
| 162 | C4⋊Q8 | — | 5 | 2 | no | — |
| 163 | C4⋊Q8 | — | 10 | 2 | no | — |
| 164 | C4⋊Q8 | — | 10 | 2 | no | — |
| 165 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 166 | C4×Q8 | — | 20 | 8 | no | — |
| 167 | C2×C4⋊C4 | — | 5 | 4 | no | — |
| 168 | C2×C4⋊C4 | — | 5 | 4 | no | — |
| 169 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 170 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 171 | C2×C4⋊C4 | — | 10 | 4 | no | — |
| 172 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 173 | C2×C42 | — | 5 | 1 | no | — |
| 174 | C2×C42 | — | 5 | 1 | no | — |
| 175 | C2×C42 | — | 5 | 2 | no | — |
| 176 | C2×C42 | — | 5 | 2 | no | — |
| 177 | C2×C42 | — | 5 | 1 | no | — |
| 178 | C2×C42 | — | 10 | 4 | no | — |
| 179 | #107 | — | 5 | 4 | no | — |
| 180 | #107 | — | 5 | 8 | no | — |
| 181 | #107 | C2×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 182 | #106 | — | 10 | 8 | no | — |
| 183 | #97 | — | 5 | 8 | no | — |
| 184 | #97 | — | 10 | 4 | no | — |
| 185 | C4×C8 | — | 5 | 2 | no | — |
| 186 | C4×C8 | — | 5 | 1 | no | — |
| 187 | C4×C8 | — | 10 | 1 | no | — |
| 188 | C4×C8 | — | 10 | 1 | no | — |
| 189 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 190 | #95 | — | 10 | 4 | no | — |
| 191 | #95 | — | 10 | 4 | no | — |
| 192 | C22×C5 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 193 | C2×D5 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 194 | C2×D5 | #532 | 1 | 1 | yes | characteristic |
| 195 | C2×D5 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 196 | C2×D5 | C2×#137 | 1 | 2 | yes | — |
| 197 | AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 198 | AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 199 | AGL(1,𝔽5) | — | 8 | 4 | no | — |
| 200 | C4×C5 | — | 4 | 4 | no | — |
| 201 | C4×C5 | #441 | 1 | 2 | yes | — |
| 202 | C4×C5 | #442 | 1 | 2 | yes | — |
| 203 | C4×C5 | C2×C4×Q8 | 1 | 2 | yes | — |
| 204 | Dic5 | — | 4 | 4 | no | — |
| 205 | Dic5 | #441 | 1 | 2 | yes | — |
| 206 | Dic5 | #442 | 1 | 2 | yes | — |
| 207 | Dic5 | #533 | 1 | 2 | yes | — |
| 208 | C2×Q8 | — | 2 | 2 | no | — |
| 209 | C2×Q8 | — | 10 | 4 | no | — |
| 210 | C2×Q8 | — | 10 | 2 | no | — |
| 211 | C2×Q8 | — | 20 | 4 | no | — |
| 212 | C2×Q8 | — | 20 | 4 | no | — |
| 213 | C22×C4 | — | 5 | 1 | no | — |
| 214 | C22×C4 | — | 5 | 1 | no | — |
| 215 | C22×C4 | — | 5 | 1 | no | — |
| 216 | C22×C4 | — | 5 | 2 | no | — |
| 217 | C22×C4 | — | 5 | 2 | no | — |
| 218 | C22×C4 | — | 10 | 4 | no | — |
| 219 | C22×C4 | — | 10 | 4 | no | — |
| 220 | C2×C8 | — | 5 | 2 | no | — |
| 221 | C2×C8 | — | 5 | 4 | no | — |
| 222 | C2×C8 | — | 5 | 4 | no | — |
| 223 | C2×C8 | — | 5 | 4 | no | — |
| 224 | C2×C8 | — | 10 | 2 | no | — |
| 225 | C2×C8 | — | 10 | 2 | no | — |
| 226 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 227 | C4⋊C4 | — | 2 | 2 | no | — |
| 228 | C4⋊C4 | — | 5 | 4 | no | — |
| 229 | C4⋊C4 | — | 10 | 4 | no | — |
| 230 | C4⋊C4 | — | 10 | 4 | no | — |
| 231 | C4⋊C4 | — | 10 | 8 | no | — |
| 232 | C4⋊C4 | — | 10 | 2 | no | — |
| 233 | C4⋊C4 | — | 20 | 4 | no | — |
| 234 | C4⋊C4 | — | 20 | 4 | no | — |
| 235 | C4⋊C4 | — | 20 | 8 | no | — |
| 236 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 237 | C42 | — | 5 | 4 | no | — |
| 238 | C42 | — | 5 | 1 | no | — |
| 239 | C42 | — | 5 | 1 | no | — |
| 240 | C42 | — | 5 | 1 | no | — |
| 241 | C42 | — | 10 | 2 | no | — |
| 242 | C42 | — | 10 | 4 | no | — |
| 243 | C42 | — | 10 | 4 | no | — |
| 244 | C42 | — | 20 | 8 | no | — |
| 245 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 246 | C2×C5 | C4×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 247 | C2×C5 | C2×#441 | 1 | 1 | yes | characteristic |
| 248 | C2×C5 | C2×#442 | 1 | 1 | yes | characteristic |
| 249 | D5 | #3165 | 1 | 2 | yes | — |
| 250 | D5 | #3182 | 1 | 2 | yes | — |
| 251 | C23 | — | 5 | 1 | no | — |
| 252 | Q8 | — | 4 | 4 | no | — |
| 253 | Q8 | — | 20 | 8 | no | — |
| 254 | Q8 | — | 20 | 4 | no | — |
| 255 | C2×C4 | — | 2 | 4 | no | — |
| 256 | C2×C4 | — | 5 | 1 | no | — |
| 257 | C2×C4 | — | 5 | 1 | no | — |
| 258 | C2×C4 | — | 5 | 1 | no | — |
| 259 | C2×C4 | — | 5 | 4 | no | — |
| 260 | C2×C4 | — | 5 | 4 | no | — |
| 261 | C2×C4 | — | 5 | 2 | no | — |
| 262 | C2×C4 | — | 5 | 2 | no | — |
| 263 | C2×C4 | — | 5 | 4 | no | — |
| 264 | C2×C4 | — | 10 | 2 | no | — |
| 265 | C2×C4 | — | 10 | 2 | no | — |
| 266 | C2×C4 | — | 10 | 2 | no | — |
| 267 | C2×C4 | — | 10 | 2 | no | — |
| 268 | C2×C4 | — | 10 | 2 | no | — |
| 269 | C2×C4 | — | 10 | 4 | no | — |
| 270 | C2×C4 | — | 20 | 4 | no | — |
| 271 | C2×C4 | — | 20 | 4 | no | — |
| 272 | C2×C4 | — | 20 | 4 | no | — |
| 273 | C2×C4 | — | 20 | 4 | no | — |
| 274 | C2×C4 | — | 20 | 4 | no | — |
| 275 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 5 | 4 | no | — |
| 279 | C8 | — | 10 | 4 | no | — |
| 280 | C8 | Q8×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 281 | C5 | #21726 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 282 | C22 | — | 5 | 1 | no | — |
| 283 | C22 | — | 5 | 1 | no | — |
| 284 | C22 | — | 5 | 2 | no | — |
| 285 | C22 | — | 5 | 2 | no | — |
| 286 | C22 | C2×D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 287 | C4 | — | 4 | 4 | no | — |
| 288 | C4 | — | 5 | 2 | no | — |
| 289 | C4 | — | 5 | 2 | no | — |
| 290 | C4 | — | 5 | 2 | no | — |
| 291 | C4 | — | 10 | 4 | no | — |
| 292 | C4 | — | 20 | 2 | no | — |
| 293 | C4 | — | 20 | 4 | no | — |
| 294 | C4 | — | 40 | 4 | no | — |
| 295 | C4 | #65731 | 1 | 2 | yes | — |
| 296 | C4 | #65739 | 1 | 2 | yes | — |
| 297 | C4 | C2×Q8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 298 | C2 | — | 5 | 2 | no | — |
| 299 | C2 | — | 5 | 2 | no | — |
| 300 | C2 | AGL(1,𝔽5)×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 301 | C2 | C2×#65731 | 1 | 1 | yes | characteristic |
| 302 | C2 | C2×#65739 | 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