← C5×#17722 #11940442
Subgroups — C5×#17722 #11940442
Order 1280.
964 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#3503 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#2988 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×#2906 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C5×#1550 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C5×#1548 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C5×#1542 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×C5×#333 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×C5×#131 | — | 2 | 1 | no | — |
| 9 | C5×Q82 | — | 2 | 1 | no | — |
| 10 | C5×#557 | C4 | 1 | 2 | yes | — |
| 11 | C5×#554 | — | 2 | 3 | no | — |
| 12 | C5×D4×Q8 | C4 | 1 | 6 | yes | — |
| 13 | C5×#544 | — | 2 | 3 | no | — |
| 14 | C5×#533 | C22 | 1 | 3 | yes | — |
| 15 | C2×C5×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 16 | C5×#520 | — | 2 | 1 | no | — |
| 17 | C4×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 18 | C5×C8×Q8 | — | 2 | 1 | no | — |
| 19 | C5×C8×D4 | — | 2 | 3 | no | — |
| 20 | C5×#431 | C22 | 1 | 3 | yes | — |
| 21 | C5×#424 | — | 2 | 3 | no | — |
| 22 | C5×#423 | C22 | 1 | 3 | yes | — |
| 23 | C2×C5×C4⋊C8 | C22 | 1 | 3 | yes | — |
| 24 | C2×C5×C4⋊C8 | C22 | 1 | 3 | yes | — |
| 25 | C5×#333 | C22 | 1 | 3 | yes | — |
| 26 | C5×#333 | C22 | 1 | 6 | yes | — |
| 27 | C5×#333 | C22 | 1 | 3 | yes | — |
| 28 | C5×#332 | C22 | 1 | 1 | yes | characteristic |
| 29 | C5×#332 | C22 | 1 | 3 | yes | — |
| 30 | #17722 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 31 | C2×C5×#41 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C5×Q8 | C2×C4 | 1 | 6 | yes | — |
| 33 | C5×#131 | — | 4 | 4 | no | — |
| 34 | C2×C5×M16 | — | 2 | 3 | no | — |
| 35 | C22×C5×C8 | — | 2 | 3 | no | — |
| 36 | C22×C5×C8 | — | 2 | 3 | no | — |
| 37 | C5×C4⋊Q8 | — | 2 | 6 | no | — |
| 38 | C5×C4⋊Q8 | — | 2 | 3 | no | — |
| 39 | C5×C4⋊Q8 | — | 2 | 3 | no | — |
| 40 | C5×C4⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 41 | C5×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 42 | C5×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C5×#125 | — | 2 | 6 | no | — |
| 44 | C5×#125 | — | 2 | 3 | no | — |
| 45 | C5×C22⋊Q8 | — | 2 | 3 | no | — |
| 46 | C5×C22⋊Q8 | — | 2 | 12 | no | — |
| 47 | C5×C22⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 48 | C4×C5×Q8 | — | 2 | 2 | no | — |
| 49 | C4×C5×Q8 | — | 2 | 3 | no | — |
| 50 | C4×C5×Q8 | C2×C4 | 1 | 6 | yes | — |
| 51 | C4×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C4×C5×Q8 | D4 | 1 | 1 | yes | characteristic |
| 53 | C4×C5×D4 | — | 2 | 6 | no | — |
| 54 | C4×C5×D4 | C2×C4 | 1 | 3 | yes | — |
| 55 | C4×C5×D4 | D4 | 1 | 3 | yes | — |
| 56 | C5×#117 | — | 2 | 3 | no | — |
| 57 | C5×#117 | D4 | 1 | 3 | yes | — |
| 58 | C2×C5×C4⋊C4 | — | 2 | 3 | no | — |
| 59 | C2×C5×C4⋊C4 | — | 2 | 3 | no | — |
| 60 | C2×C42×C5 | C23 | 1 | 3 | yes | — |
| 61 | C5×C4⋊C8 | — | 2 | 3 | no | — |
| 62 | C5×C4⋊C8 | — | 2 | 3 | no | — |
| 63 | C5×C4⋊C8 | — | 2 | 3 | no | — |
| 64 | C5×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C5×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 66 | C5×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 67 | C5×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | C5×C22⋊C8 | — | 2 | 3 | no | — |
| 69 | C4×C5×C8 | — | 2 | 3 | no | — |
| 70 | #3503 | C2×C5 | 1 | 1 | yes | characteristic |
| 71 | #2988 | C2×C5 | 1 | 1 | yes | characteristic |
| 72 | #2906 | C2×C5 | 1 | 1 | yes | characteristic |
| 73 | #1550 | C2×C5 | 1 | 3 | yes | complemented |
| 74 | #1548 | C2×C5 | 1 | 3 | yes | complemented |
| 75 | #1542 | C2×C5 | 1 | 3 | yes | complemented |
| 76 | C2×#333 | C2×C5 | 1 | 3 | yes | complemented |
| 77 | C5×#41 | SD16 | 1 | 4 | yes | — |
| 78 | C5×#41 | Q16 | 1 | 4 | yes | — |
| 79 | C2×C5×Q8 | — | 2 | 6 | no | — |
| 80 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 81 | C2×C5×Q8 | — | 8 | 6 | no | — |
| 82 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C5×Q8 | C22×C4 | 1 | 6 | yes | — |
| 84 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 85 | C22×C4×C5 | — | 2 | 6 | no | — |
| 86 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 87 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 88 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 89 | C5×M16 | — | 4 | 6 | no | — |
| 90 | C2×C5×C8 | — | 2 | 1 | no | — |
| 91 | C2×C5×C8 | — | 2 | 3 | no | — |
| 92 | C2×C5×C8 | — | 2 | 3 | no | — |
| 93 | C2×C5×C8 | — | 2 | 1 | no | — |
| 94 | C2×C5×C8 | — | 4 | 6 | no | — |
| 95 | C2×C5×C8 | — | 8 | 3 | no | — |
| 96 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 97 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 98 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 99 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 100 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 101 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 102 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 103 | C5×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 104 | C5×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 105 | C5×C4⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 106 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 107 | C5×C22⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 108 | C42×C5 | — | 2 | 6 | no | — |
| 109 | C42×C5 | C22×C4 | 1 | 3 | yes | — |
| 110 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 111 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 112 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 113 | C2×#131 | — | 2 | 1 | no | — |
| 114 | Q82 | — | 2 | 1 | no | — |
| 115 | #557 | C4×C5 | 1 | 2 | yes | — |
| 116 | #554 | — | 2 | 3 | no | — |
| 117 | D4×Q8 | C4×C5 | 1 | 6 | yes | — |
| 118 | #544 | — | 2 | 3 | no | — |
| 119 | #533 | C22×C5 | 1 | 3 | yes | — |
| 120 | C2×C4⋊Q8 | C22×C5 | 1 | 3 | yes | — |
| 121 | #520 | — | 2 | 1 | no | — |
| 122 | C4×#41 | C22×C5 | 1 | 1 | yes | characteristic |
| 123 | C8×Q8 | — | 2 | 1 | no | — |
| 124 | C8×D4 | — | 2 | 3 | no | — |
| 125 | #431 | C22×C5 | 1 | 3 | yes | — |
| 126 | #424 | — | 2 | 3 | no | — |
| 127 | #423 | C22×C5 | 1 | 3 | yes | — |
| 128 | C2×C4⋊C8 | C22×C5 | 1 | 3 | yes | — |
| 129 | C2×C4⋊C8 | C22×C5 | 1 | 3 | yes | — |
| 130 | #333 | C22×C5 | 1 | 3 | yes | — |
| 131 | #333 | C22×C5 | 1 | 6 | yes | — |
| 132 | #333 | C22×C5 | 1 | 3 | yes | — |
| 133 | #332 | C22×C5 | 1 | 1 | yes | characteristic |
| 134 | #332 | C22×C5 | 1 | 3 | yes | — |
| 135 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 136 | C5×Q8 | — | 4 | 6 | no | — |
| 137 | C5×Q8 | — | 4 | 1 | no | — |
| 138 | C5×Q8 | — | 4 | 1 | no | — |
| 139 | C5×Q8 | — | 8 | 6 | no | — |
| 140 | C5×Q8 | #103 | 1 | 4 | yes | — |
| 141 | C5×D4 | #103 | 1 | 12 | yes | — |
| 142 | C2×C4×C5 | — | 2 | 3 | no | — |
| 143 | C2×C4×C5 | — | 2 | 3 | no | — |
| 144 | C2×C4×C5 | — | 2 | 2 | no | — |
| 145 | C2×C4×C5 | — | 2 | 6 | no | — |
| 146 | C2×C4×C5 | — | 4 | 3 | no | — |
| 147 | C2×C4×C5 | — | 8 | 6 | no | — |
| 148 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 149 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 150 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 151 | C2×C4×C5 | C2×SD16 | 1 | 6 | yes | — |
| 152 | C2×C4×C5 | C2×Q16 | 1 | 6 | yes | — |
| 153 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 156 | C5×C8 | — | 4 | 1 | no | — |
| 157 | C5×C8 | — | 4 | 3 | no | — |
| 158 | C5×C8 | — | 4 | 3 | no | — |
| 159 | C5×C8 | — | 4 | 1 | no | — |
| 160 | C2×#41 | C5×D4 | 1 | 1 | yes | characteristic |
| 161 | C22×Q8 | C2×C4×C5 | 1 | 6 | yes | — |
| 162 | #131 | — | 4 | 4 | no | — |
| 163 | C2×M16 | — | 2 | 3 | no | — |
| 164 | C22×C8 | — | 2 | 3 | no | — |
| 165 | C22×C8 | — | 2 | 3 | no | — |
| 166 | C4⋊Q8 | — | 2 | 6 | no | — |
| 167 | C4⋊Q8 | — | 2 | 3 | no | — |
| 168 | C4⋊Q8 | — | 2 | 3 | no | — |
| 169 | C4⋊Q8 | C2×C4×C5 | 1 | 6 | yes | — |
| 170 | C4⋊Q8 | C23×C5 | 1 | 3 | yes | — |
| 171 | C4⋊Q8 | C23×C5 | 1 | 1 | yes | characteristic |
| 172 | #125 | — | 2 | 6 | no | — |
| 173 | #125 | — | 2 | 3 | no | — |
| 174 | C22⋊Q8 | — | 2 | 3 | no | — |
| 175 | C22⋊Q8 | — | 2 | 12 | no | — |
| 176 | C22⋊Q8 | C2×C4×C5 | 1 | 6 | yes | — |
| 177 | C4×Q8 | — | 2 | 2 | no | — |
| 178 | C4×Q8 | — | 2 | 3 | no | — |
| 179 | C4×Q8 | C2×C4×C5 | 1 | 6 | yes | — |
| 180 | C4×Q8 | C2×C4×C5 | 1 | 1 | yes | characteristic |
| 181 | C4×Q8 | C5×D4 | 1 | 1 | yes | characteristic |
| 182 | C4×D4 | — | 2 | 6 | no | — |
| 183 | C4×D4 | C2×C4×C5 | 1 | 3 | yes | — |
| 184 | C4×D4 | C5×D4 | 1 | 3 | yes | — |
| 185 | #117 | — | 2 | 3 | no | — |
| 186 | #117 | C5×D4 | 1 | 3 | yes | — |
| 187 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 188 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 189 | C2×C42 | C23×C5 | 1 | 3 | yes | — |
| 190 | C4⋊C8 | — | 2 | 3 | no | — |
| 191 | C4⋊C8 | — | 2 | 3 | no | — |
| 192 | C4⋊C8 | — | 2 | 3 | no | — |
| 193 | C4⋊C8 | C23×C5 | 1 | 1 | yes | characteristic |
| 194 | C4⋊C8 | C23×C5 | 1 | 3 | yes | — |
| 195 | C4⋊C8 | C23×C5 | 1 | 3 | yes | — |
| 196 | C4⋊C8 | C23×C5 | 1 | 1 | yes | characteristic |
| 197 | C22⋊C8 | — | 2 | 3 | no | — |
| 198 | C4×C8 | — | 2 | 3 | no | — |
| 199 | C22×C5 | — | 2 | 3 | no | — |
| 200 | C22×C5 | — | 2 | 3 | no | — |
| 201 | C22×C5 | C2×#103 | 1 | 6 | yes | — |
| 202 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 203 | C4×C5 | — | 2 | 1 | no | — |
| 204 | C4×C5 | — | 2 | 3 | no | — |
| 205 | C4×C5 | — | 2 | 3 | no | — |
| 206 | C4×C5 | — | 4 | 6 | no | — |
| 207 | C4×C5 | — | 4 | 2 | no | — |
| 208 | C4×C5 | C2×#103 | 1 | 6 | yes | — |
| 209 | C4×C5 | C22×SD16 | 1 | 1 | yes | characteristic |
| 210 | C4×C5 | C22×Q16 | 1 | 1 | yes | characteristic |
| 211 | C4×C5 | #577 | 1 | 1 | yes | characteristic |
| 212 | C4×C5 | #578 | 1 | 1 | yes | characteristic |
| 213 | #41 | C5×SD16 | 1 | 4 | yes | — |
| 214 | #41 | C5×Q16 | 1 | 4 | yes | — |
| 215 | C2×Q8 | — | 2 | 6 | no | — |
| 216 | C2×Q8 | — | 2 | 1 | no | — |
| 217 | C2×Q8 | — | 8 | 6 | no | — |
| 218 | C2×Q8 | C5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 219 | C2×Q8 | C22×C4×C5 | 1 | 6 | yes | — |
| 220 | C2×D4 | C5×C22⋊C4 | 1 | 3 | yes | — |
| 221 | C22×C4 | — | 2 | 6 | no | — |
| 222 | C22×C4 | C22×C4×C5 | 1 | 3 | yes | — |
| 223 | C22×C4 | C2×C5×D4 | 1 | 3 | yes | — |
| 224 | C22×C4 | C2×C5×D4 | 1 | 3 | yes | — |
| 225 | M16 | — | 4 | 6 | no | — |
| 226 | C2×C8 | — | 2 | 1 | no | — |
| 227 | C2×C8 | — | 2 | 3 | no | — |
| 228 | C2×C8 | — | 2 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 3 | no | — |
| 230 | C2×C8 | — | 4 | 6 | no | — |
| 231 | C2×C8 | — | 8 | 3 | no | — |
| 232 | C4⋊C4 | — | 2 | 6 | no | — |
| 233 | C4⋊C4 | — | 2 | 6 | no | — |
| 234 | C4⋊C4 | — | 2 | 6 | no | — |
| 235 | C4⋊C4 | — | 2 | 6 | no | — |
| 236 | C4⋊C4 | — | 2 | 3 | no | — |
| 237 | C4⋊C4 | — | 2 | 3 | no | — |
| 238 | C4⋊C4 | — | 2 | 1 | no | — |
| 239 | C4⋊C4 | C5×C22⋊C4 | 1 | 3 | yes | — |
| 240 | C4⋊C4 | C5×C22⋊C4 | 1 | 3 | yes | — |
| 241 | C4⋊C4 | C22×C4×C5 | 1 | 2 | yes | — |
| 242 | C22⋊C4 | — | 2 | 6 | no | — |
| 243 | C22⋊C4 | C5×C22⋊C4 | 1 | 6 | yes | — |
| 244 | C42 | — | 2 | 6 | no | — |
| 245 | C42 | C22×C4×C5 | 1 | 3 | yes | — |
| 246 | C42 | C2×C5×D4 | 1 | 3 | yes | — |
| 247 | C42 | C2×C5×D4 | 1 | 3 | yes | — |
| 248 | C42 | C24×C5 | 1 | 1 | yes | characteristic |
| 249 | C2×C5 | — | 2 | 6 | no | — |
| 250 | C2×C5 | #2888 | 1 | 1 | yes | characteristic |
| 251 | C2×C5 | C22×#103 | 1 | 1 | yes | characteristic |
| 252 | C2×C5 | #2898 | 1 | 1 | yes | characteristic |
| 253 | C23 | C2×C5×C22⋊C4 | 1 | 3 | yes | — |
| 254 | Q8 | — | 4 | 6 | no | — |
| 255 | Q8 | — | 4 | 1 | no | — |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | Q8 | — | 8 | 6 | no | — |
| 258 | Q8 | C5×#103 | 1 | 4 | yes | — |
| 259 | D4 | C5×#103 | 1 | 12 | yes | — |
| 260 | C2×C4 | — | 2 | 3 | no | — |
| 261 | C2×C4 | — | 2 | 3 | no | — |
| 262 | C2×C4 | — | 2 | 2 | no | — |
| 263 | C2×C4 | — | 2 | 6 | no | — |
| 264 | C2×C4 | — | 4 | 3 | no | — |
| 265 | C2×C4 | — | 8 | 6 | no | — |
| 266 | C2×C4 | C2×C5×C22⋊C4 | 1 | 3 | yes | — |
| 267 | C2×C4 | C2×C5×C22⋊C4 | 1 | 3 | yes | — |
| 268 | C2×C4 | C2×C5×C22⋊C4 | 1 | 3 | yes | — |
| 269 | C2×C4 | C2×C5×SD16 | 1 | 6 | yes | — |
| 270 | C2×C4 | C2×C5×Q16 | 1 | 6 | yes | — |
| 271 | C2×C4 | C23×C4×C5 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | C22×C5×D4 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C22×C5×D4 | 1 | 1 | yes | characteristic |
| 274 | C8 | — | 4 | 1 | no | — |
| 275 | C8 | — | 4 | 3 | no | — |
| 276 | C8 | — | 4 | 3 | no | — |
| 277 | C8 | — | 4 | 1 | no | — |
| 278 | C5 | #17722 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 279 | C22 | — | 2 | 3 | no | — |
| 280 | C22 | — | 2 | 3 | no | — |
| 281 | C22 | C2×C5×#103 | 1 | 6 | yes | — |
| 282 | C22 | C22×C5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 2 | 3 | no | — |
| 285 | C4 | — | 2 | 3 | no | — |
| 286 | C4 | — | 4 | 6 | no | — |
| 287 | C4 | — | 4 | 2 | no | — |
| 288 | C4 | C2×C5×#103 | 1 | 6 | yes | — |
| 289 | C4 | C22×C5×SD16 | 1 | 1 | yes | characteristic |
| 290 | C4 | C22×C5×Q16 | 1 | 1 | yes | characteristic |
| 291 | C4 | C5×#577 | 1 | 1 | yes | characteristic |
| 292 | C4 | C5×#578 | 1 | 1 | yes | characteristic |
| 293 | C2 | — | 2 | 6 | no | — |
| 294 | C2 | C5×#2888 | 1 | 1 | yes | characteristic |
| 295 | C2 | C22×C5×#103 | 1 | 1 | yes | characteristic |
| 296 | C2 | C5×#2898 | 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