← C23×#17688 #19165314
Subgroups — C23×#17688 #19165314
Order 5888.
1148 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×#3500 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×#2971 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C23×#1536 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C23×#1527 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C23×#1525 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×C23×2−1+4 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C23×H(5,𝔽2) | C4 | 1 | 1 | yes | characteristic |
| 8 | C23×#548 | — | 2 | 2 | no | — |
| 9 | C23×#546 | — | 2 | 6 | no | — |
| 10 | C23×#536 | C4 | 1 | 3 | yes | — |
| 11 | C23×#535 | C4 | 1 | 3 | yes | — |
| 12 | C23×#532 | C22 | 1 | 3 | yes | — |
| 13 | C2×C23×#124 | C22 | 1 | 3 | yes | — |
| 14 | C4×#41×C23 | C22 | 1 | 1 | yes | characteristic |
| 15 | C8×Q8×C23 | — | 2 | 2 | no | — |
| 16 | C23×#436 | — | 2 | 6 | no | — |
| 17 | C23×#432 | C22 | 1 | 6 | yes | complemented |
| 18 | C4×M16×C23 | C22 | 1 | 6 | yes | complemented |
| 19 | C23×#329 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | C23×#329 | C22 | 1 | 6 | yes | complemented |
| 21 | C23×#329 | C22 | 1 | 3 | yes | complemented |
| 22 | C23×#329 | C22 | 1 | 6 | yes | complemented |
| 23 | C23×2−1+4 | — | 4 | 4 | no | — |
| 24 | C23×H(5,𝔽2) | — | 4 | 4 | no | — |
| 25 | C2×#41×C23 | — | 2 | 1 | no | — |
| 26 | C2×#41×C23 | — | 2 | 3 | no | — |
| 27 | C2×#41×C23 | C2×C4 | 1 | 3 | yes | — |
| 28 | C2×#41×C23 | C2×C4 | 1 | 3 | yes | — |
| 29 | C2×#41×C23 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C22×Q8×C23 | — | 2 | 1 | no | — |
| 31 | C22×Q8×C23 | C2×C4 | 1 | 3 | yes | — |
| 32 | C22×D4×C23 | — | 2 | 3 | no | — |
| 33 | C22×D4×C23 | C2×C4 | 1 | 3 | yes | — |
| 34 | C2×M16×C23 | — | 2 | 6 | no | — |
| 35 | C23×C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 36 | C23×#127 | C2×C4 | 1 | 3 | yes | — |
| 37 | C23×#124 | — | 2 | 6 | no | — |
| 38 | C23×#124 | C2×C4 | 1 | 6 | yes | — |
| 39 | C23×#124 | C23 | 1 | 3 | yes | — |
| 40 | C23×#124 | C23 | 1 | 1 | yes | characteristic |
| 41 | C23×#123 | — | 2 | 6 | no | — |
| 42 | C23×C22⋊Q8 | — | 2 | 6 | no | — |
| 43 | C23×#121 | — | 2 | 6 | no | — |
| 44 | C23×#121 | — | 2 | 6 | no | — |
| 45 | C23×#120 | — | 2 | 6 | no | — |
| 46 | C4×Q8×C23 | D4 | 1 | 2 | yes | — |
| 47 | C4×D4×C23 | — | 2 | 6 | no | — |
| 48 | C4×D4×C23 | D4 | 1 | 6 | yes | — |
| 49 | C23×#117 | C2×C4 | 1 | 3 | yes | — |
| 50 | C2×C22⋊C4×C23 | — | 2 | 6 | no | — |
| 51 | C2×C42×C23 | C23 | 1 | 3 | yes | — |
| 52 | C23×C4⋊C8 | — | 2 | 6 | no | — |
| 53 | C23×C22⋊C8 | — | 2 | 6 | no | — |
| 54 | C23×#97 | C23 | 1 | 2 | yes | — |
| 55 | C23×#97 | C23 | 1 | 6 | yes | — |
| 56 | C4×C8×C23 | — | 2 | 6 | no | — |
| 57 | C24×C23 | — | 2 | 3 | no | — |
| 58 | #41×C23 | — | 2 | 4 | no | — |
| 59 | #41×C23 | — | 4 | 6 | no | — |
| 60 | #41×C23 | — | 4 | 6 | no | — |
| 61 | #41×C23 | — | 4 | 12 | no | — |
| 62 | #41×C23 | — | 4 | 4 | no | — |
| 63 | C2×Q8×C23 | — | 2 | 3 | no | — |
| 64 | C2×Q8×C23 | — | 4 | 6 | no | — |
| 65 | C2×Q8×C23 | — | 4 | 4 | no | — |
| 66 | C2×Q8×C23 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C2×Q8×C23 | C22×C4 | 1 | 3 | yes | — |
| 68 | C2×Q8×C23 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C2×D4×C23 | — | 2 | 3 | no | — |
| 70 | C2×D4×C23 | — | 2 | 3 | no | — |
| 71 | C2×D4×C23 | — | 2 | 3 | no | — |
| 72 | C2×D4×C23 | — | 2 | 6 | no | — |
| 73 | C2×D4×C23 | — | 4 | 6 | no | — |
| 74 | C2×D4×C23 | — | 4 | 12 | no | — |
| 75 | C2×D4×C23 | C22⋊C4 | 1 | 3 | yes | — |
| 76 | C2×D4×C23 | C22×C4 | 1 | 3 | yes | — |
| 77 | C2×D4×C23 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C23 | — | 2 | 3 | no | — |
| 79 | C22×C4×C23 | — | 2 | 3 | no | — |
| 80 | C22×C4×C23 | — | 2 | 3 | no | — |
| 81 | C22×C4×C23 | C22×C4 | 1 | 3 | yes | — |
| 82 | C22×C4×C23 | C2×D4 | 1 | 6 | yes | — |
| 83 | M16×C23 | — | 4 | 12 | no | — |
| 84 | C2×C8×C23 | — | 2 | 2 | no | — |
| 85 | C2×C8×C23 | — | 2 | 6 | no | — |
| 86 | C4⋊C4×C23 | — | 2 | 6 | no | — |
| 87 | C4⋊C4×C23 | C22⋊C4 | 1 | 6 | yes | — |
| 88 | C22⋊C4×C23 | — | 2 | 12 | no | — |
| 89 | C22⋊C4×C23 | — | 2 | 6 | no | — |
| 90 | C22⋊C4×C23 | — | 2 | 2 | no | — |
| 91 | C22⋊C4×C23 | C22⋊C4 | 1 | 6 | yes | — |
| 92 | C42×C23 | C22×C4 | 1 | 3 | yes | — |
| 93 | C42×C23 | C2×D4 | 1 | 6 | yes | — |
| 94 | C42×C23 | C24 | 1 | 1 | yes | characteristic |
| 95 | #17688 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 96 | C23×C23 | — | 2 | 3 | no | — |
| 97 | C23×C23 | — | 2 | 1 | no | — |
| 98 | C23×C23 | — | 4 | 6 | no | — |
| 99 | C23×C23 | — | 8 | 6 | no | — |
| 100 | C23×C23 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 101 | Q8×C23 | — | 4 | 3 | no | — |
| 102 | Q8×C23 | — | 4 | 1 | no | — |
| 103 | Q8×C23 | — | 4 | 6 | no | — |
| 104 | Q8×C23 | #100 | 1 | 4 | yes | — |
| 105 | D4×C23 | — | 2 | 6 | no | — |
| 106 | D4×C23 | — | 4 | 3 | no | — |
| 107 | D4×C23 | — | 4 | 1 | no | — |
| 108 | D4×C23 | — | 4 | 6 | no | — |
| 109 | D4×C23 | — | 4 | 6 | no | — |
| 110 | D4×C23 | — | 4 | 6 | no | — |
| 111 | D4×C23 | — | 8 | 6 | no | — |
| 112 | C2×C4×C23 | — | 2 | 6 | no | — |
| 113 | C2×C4×C23 | — | 2 | 3 | no | — |
| 114 | C2×C4×C23 | — | 2 | 1 | no | — |
| 115 | C2×C4×C23 | — | 4 | 6 | no | — |
| 116 | C2×C4×C23 | — | 4 | 6 | no | — |
| 117 | C2×C4×C23 | — | 4 | 6 | no | — |
| 118 | C2×C4×C23 | — | 4 | 6 | no | — |
| 119 | C2×C4×C23 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 120 | C2×C4×C23 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 121 | C2×C4×C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C23 | C22×D4 | 1 | 2 | yes | — |
| 123 | C8×C23 | — | 2 | 4 | no | — |
| 124 | C8×C23 | — | 4 | 6 | no | — |
| 125 | #3500 | C2×C23 | 1 | 1 | yes | characteristic |
| 126 | #2971 | C2×C23 | 1 | 2 | yes | complemented |
| 127 | #1536 | C2×C23 | 1 | 3 | yes | complemented |
| 128 | #1527 | C2×C23 | 1 | 6 | yes | complemented |
| 129 | #1525 | C2×C23 | 1 | 3 | yes | complemented |
| 130 | C22×C23 | — | 2 | 3 | no | — |
| 131 | C22×C23 | — | 2 | 6 | no | — |
| 132 | C22×C23 | — | 4 | 1 | no | — |
| 133 | C22×C23 | — | 4 | 3 | no | — |
| 134 | C22×C23 | — | 4 | 2 | no | — |
| 135 | C22×C23 | — | 4 | 6 | no | — |
| 136 | C22×C23 | — | 16 | 6 | no | — |
| 137 | C22×C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 138 | C4×C23 | — | 2 | 1 | no | — |
| 139 | C4×C23 | — | 2 | 2 | no | — |
| 140 | C4×C23 | — | 2 | 6 | no | — |
| 141 | C4×C23 | — | 4 | 1 | no | — |
| 142 | C4×C23 | — | 4 | 3 | no | — |
| 143 | C4×C23 | C2×#100 | 1 | 6 | yes | — |
| 144 | C2×2−1+4 | C4×C23 | 1 | 1 | yes | characteristic |
| 145 | C2×H(5,𝔽2) | C4×C23 | 1 | 1 | yes | characteristic |
| 146 | #548 | — | 2 | 2 | no | — |
| 147 | #546 | — | 2 | 6 | no | — |
| 148 | #536 | C4×C23 | 1 | 3 | yes | — |
| 149 | #535 | C4×C23 | 1 | 3 | yes | — |
| 150 | #532 | C22×C23 | 1 | 3 | yes | — |
| 151 | C2×#124 | C22×C23 | 1 | 3 | yes | — |
| 152 | C4×#41 | C22×C23 | 1 | 1 | yes | characteristic |
| 153 | C8×Q8 | — | 2 | 2 | no | — |
| 154 | #436 | — | 2 | 6 | no | — |
| 155 | #432 | C22×C23 | 1 | 6 | yes | complemented |
| 156 | C4×M16 | C22×C23 | 1 | 6 | yes | complemented |
| 157 | #329 | C22×C23 | 1 | 3 | yes | complemented |
| 158 | #329 | C22×C23 | 1 | 6 | yes | complemented |
| 159 | #329 | C22×C23 | 1 | 1 | yes | characteristic, complemented |
| 160 | #329 | C22×C23 | 1 | 6 | yes | complemented |
| 161 | C2×C23 | — | 4 | 3 | no | — |
| 162 | C2×C23 | — | 8 | 1 | no | — |
| 163 | C2×C23 | — | 8 | 3 | no | — |
| 164 | C2×C23 | C22×#100 | 1 | 1 | yes | characteristic |
| 165 | C2×C23 | #2901 | 1 | 2 | yes | — |
| 166 | 2−1+4 | — | 4 | 4 | no | — |
| 167 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 168 | C2×#41 | — | 2 | 1 | no | — |
| 169 | C2×#41 | — | 2 | 3 | no | — |
| 170 | C2×#41 | C2×C4×C23 | 1 | 3 | yes | — |
| 171 | C2×#41 | C2×C4×C23 | 1 | 3 | yes | — |
| 172 | C2×#41 | C2×C4×C23 | 1 | 1 | yes | characteristic |
| 173 | C22×Q8 | — | 2 | 1 | no | — |
| 174 | C22×Q8 | C2×C4×C23 | 1 | 3 | yes | — |
| 175 | C22×D4 | — | 2 | 3 | no | — |
| 176 | C22×D4 | C2×C4×C23 | 1 | 3 | yes | — |
| 177 | C2×M16 | — | 2 | 6 | no | — |
| 178 | C4⋊Q8 | C2×C4×C23 | 1 | 3 | yes | — |
| 179 | #127 | C2×C4×C23 | 1 | 3 | yes | — |
| 180 | #124 | — | 2 | 6 | no | — |
| 181 | #124 | C2×C4×C23 | 1 | 6 | yes | — |
| 182 | #124 | C23×C23 | 1 | 3 | yes | — |
| 183 | #124 | C23×C23 | 1 | 1 | yes | characteristic |
| 184 | #123 | — | 2 | 6 | no | — |
| 185 | C22⋊Q8 | — | 2 | 6 | no | — |
| 186 | #121 | — | 2 | 6 | no | — |
| 187 | #121 | — | 2 | 6 | no | — |
| 188 | #120 | — | 2 | 6 | no | — |
| 189 | C4×Q8 | D4×C23 | 1 | 2 | yes | — |
| 190 | C4×D4 | — | 2 | 6 | no | — |
| 191 | C4×D4 | D4×C23 | 1 | 6 | yes | — |
| 192 | #117 | C2×C4×C23 | 1 | 3 | yes | — |
| 193 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 194 | C2×C42 | C23×C23 | 1 | 3 | yes | — |
| 195 | C4⋊C8 | — | 2 | 6 | no | — |
| 196 | C22⋊C8 | — | 2 | 6 | no | — |
| 197 | #97 | C23×C23 | 1 | 2 | yes | — |
| 198 | #97 | C23×C23 | 1 | 6 | yes | — |
| 199 | C4×C8 | — | 2 | 6 | no | — |
| 200 | C23 | #17688 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 201 | C24 | — | 2 | 3 | no | — |
| 202 | #41 | — | 2 | 4 | no | — |
| 203 | #41 | — | 4 | 6 | no | — |
| 204 | #41 | — | 4 | 6 | no | — |
| 205 | #41 | — | 4 | 12 | no | — |
| 206 | #41 | — | 4 | 4 | no | — |
| 207 | C2×Q8 | — | 2 | 3 | no | — |
| 208 | C2×Q8 | — | 4 | 6 | no | — |
| 209 | C2×Q8 | — | 4 | 4 | no | — |
| 210 | C2×Q8 | C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 211 | C2×Q8 | C22×C4×C23 | 1 | 3 | yes | — |
| 212 | C2×Q8 | C22×C4×C23 | 1 | 1 | yes | characteristic |
| 213 | C2×D4 | — | 2 | 3 | no | — |
| 214 | C2×D4 | — | 2 | 3 | no | — |
| 215 | C2×D4 | — | 2 | 3 | no | — |
| 216 | C2×D4 | — | 2 | 6 | no | — |
| 217 | C2×D4 | — | 4 | 6 | no | — |
| 218 | C2×D4 | — | 4 | 12 | no | — |
| 219 | C2×D4 | C22⋊C4×C23 | 1 | 3 | yes | — |
| 220 | C2×D4 | C22×C4×C23 | 1 | 3 | yes | — |
| 221 | C2×D4 | C22×C4×C23 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | — | 2 | 3 | no | — |
| 223 | C22×C4 | — | 2 | 3 | no | — |
| 224 | C22×C4 | — | 2 | 3 | no | — |
| 225 | C22×C4 | C22×C4×C23 | 1 | 3 | yes | — |
| 226 | C22×C4 | C2×D4×C23 | 1 | 6 | yes | — |
| 227 | M16 | — | 4 | 12 | no | — |
| 228 | C2×C8 | — | 2 | 2 | no | — |
| 229 | C2×C8 | — | 2 | 6 | no | — |
| 230 | C4⋊C4 | — | 2 | 6 | no | — |
| 231 | C4⋊C4 | C22⋊C4×C23 | 1 | 6 | yes | — |
| 232 | C22⋊C4 | — | 2 | 12 | no | — |
| 233 | C22⋊C4 | — | 2 | 6 | no | — |
| 234 | C22⋊C4 | — | 2 | 2 | no | — |
| 235 | C22⋊C4 | C22⋊C4×C23 | 1 | 6 | yes | — |
| 236 | C42 | C22×C4×C23 | 1 | 3 | yes | — |
| 237 | C42 | C2×D4×C23 | 1 | 6 | yes | — |
| 238 | C42 | C24×C23 | 1 | 1 | yes | characteristic |
| 239 | C23 | — | 2 | 3 | no | — |
| 240 | C23 | — | 2 | 1 | no | — |
| 241 | C23 | — | 4 | 6 | no | — |
| 242 | C23 | — | 8 | 6 | no | — |
| 243 | C23 | C2×C22⋊C4×C23 | 1 | 3 | yes | — |
| 244 | Q8 | — | 4 | 3 | no | — |
| 245 | Q8 | — | 4 | 1 | no | — |
| 246 | Q8 | — | 4 | 6 | no | — |
| 247 | Q8 | C23×#100 | 1 | 4 | yes | — |
| 248 | D4 | — | 2 | 6 | no | — |
| 249 | D4 | — | 4 | 3 | no | — |
| 250 | D4 | — | 4 | 1 | no | — |
| 251 | D4 | — | 4 | 6 | no | — |
| 252 | D4 | — | 4 | 6 | no | — |
| 253 | D4 | — | 4 | 6 | no | — |
| 254 | D4 | — | 8 | 6 | no | — |
| 255 | C2×C4 | — | 2 | 6 | no | — |
| 256 | C2×C4 | — | 2 | 3 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 6 | no | — |
| 259 | C2×C4 | — | 4 | 6 | no | — |
| 260 | C2×C4 | — | 4 | 6 | no | — |
| 261 | C2×C4 | — | 4 | 6 | no | — |
| 262 | C2×C4 | C2×C22⋊C4×C23 | 1 | 6 | yes | — |
| 263 | C2×C4 | C2×C22⋊C4×C23 | 1 | 3 | yes | — |
| 264 | C2×C4 | C23×C4×C23 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | C22×D4×C23 | 1 | 2 | yes | — |
| 266 | C8 | — | 2 | 4 | no | — |
| 267 | C8 | — | 4 | 6 | no | — |
| 268 | C22 | — | 2 | 3 | no | — |
| 269 | C22 | — | 2 | 6 | no | — |
| 270 | C22 | — | 4 | 1 | no | — |
| 271 | C22 | — | 4 | 3 | no | — |
| 272 | C22 | — | 4 | 2 | no | — |
| 273 | C22 | — | 4 | 6 | no | — |
| 274 | C22 | — | 16 | 6 | no | — |
| 275 | C22 | C22×C22⋊C4×C23 | 1 | 1 | yes | characteristic |
| 276 | C4 | — | 2 | 1 | no | — |
| 277 | C4 | — | 2 | 2 | no | — |
| 278 | C4 | — | 2 | 6 | no | — |
| 279 | C4 | — | 4 | 1 | no | — |
| 280 | C4 | — | 4 | 3 | no | — |
| 281 | C4 | C2×C23×#100 | 1 | 6 | yes | — |
| 282 | C2 | — | 4 | 3 | no | — |
| 283 | C2 | — | 8 | 3 | no | — |
| 284 | C2 | — | 8 | 1 | no | — |
| 285 | C2 | C22×C23×#100 | 1 | 1 | yes | characteristic |
| 286 | C2 | C23×#2901 | 1 | 2 | yes | — |
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