← C23×#33954 #19181580
Subgroups — C23×#33954 #19181580
Order 5888.
450 conjugacy classes of subgroups.
The subgroup lattice has 310 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×#3383 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×Q16×C23 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×#2255 | C2 | 1 | 2 | yes | maximal |
| 4 | C23×#2254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C23×#2229 | C2 | 1 | 2 | yes | maximal |
| 6 | C23×#2227 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C23×#2225 | C2 | 1 | 2 | yes | maximal |
| 8 | C23×#2225 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C23×#2183 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C23×#2176 | C2 | 1 | 2 | yes | maximal |
| 11 | C4×C23×Q32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | Q82×C23 | — | 2 | 1 | no | — |
| 13 | C23×#557 | — | 2 | 1 | no | — |
| 14 | C2×C23×Q32 | C22 | 1 | 1 | yes | characteristic |
| 15 | C23×#500 | C22 | 1 | 1 | yes | characteristic |
| 16 | C23×#500 | C22 | 1 | 2 | yes | — |
| 17 | C23×#500 | C22 | 1 | 1 | yes | characteristic |
| 18 | C23×#499 | C22 | 1 | 2 | yes | — |
| 19 | C23×#479 | — | 2 | 2 | no | — |
| 20 | C23×#477 | — | 2 | 1 | no | — |
| 21 | C23×#477 | — | 2 | 2 | no | — |
| 22 | C23×#475 | — | 2 | 1 | no | — |
| 23 | C8×Q8×C23 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×Q16×C23 | C22 | 1 | 1 | yes | characteristic |
| 25 | C4×Q16×C23 | C22 | 1 | 1 | yes | characteristic |
| 26 | C4×Q16×C23 | C22 | 1 | 2 | yes | — |
| 27 | C4×Q16×C23 | C22 | 1 | 2 | yes | — |
| 28 | C23×#367 | C22 | 1 | 2 | yes | — |
| 29 | C23×#367 | C22 | 1 | 2 | yes | — |
| 30 | C23×#367 | C22 | 1 | 2 | yes | — |
| 31 | C23×#366 | C22 | 1 | 1 | yes | characteristic |
| 32 | C23×#366 | C22 | 1 | 2 | yes | — |
| 33 | C23×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 34 | C23×C4⋊C16 | C22 | 1 | 2 | yes | — |
| 35 | C23×#358 | C22 | 1 | 1 | yes | characteristic |
| 36 | C23×#358 | C22 | 1 | 1 | yes | characteristic |
| 37 | C23×#358 | C22 | 1 | 2 | yes | — |
| 38 | C23×#358 | C22 | 1 | 2 | yes | — |
| 39 | C23×#346 | C22 | 1 | 2 | yes | — |
| 40 | C4×C16×C23 | C22 | 1 | 1 | yes | characteristic |
| 41 | C2×Q16×C23 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×Q16×C23 | C23 | 1 | 1 | yes | characteristic |
| 43 | C23×C4⋊Q8 | — | 2 | 1 | no | — |
| 44 | C23×C4⋊Q8 | — | 2 | 2 | no | — |
| 45 | C23×C4⋊Q8 | — | 2 | 1 | no | — |
| 46 | C23×C4⋊Q8 | — | 4 | 1 | no | — |
| 47 | C23×C4⋊Q8 | — | 4 | 1 | no | — |
| 48 | C23×C4⋊Q8 | — | 4 | 2 | no | — |
| 49 | C23×#125 | — | 2 | 2 | no | — |
| 50 | C23×#125 | — | 4 | 2 | no | — |
| 51 | C4×Q8×C23 | — | 2 | 1 | no | — |
| 52 | C4×Q8×C23 | — | 2 | 2 | no | — |
| 53 | C4×Q8×C23 | — | 2 | 2 | no | — |
| 54 | C4×Q8×C23 | — | 2 | 1 | no | — |
| 55 | C4×Q8×C23 | — | 4 | 1 | no | — |
| 56 | C4×Q8×C23 | — | 4 | 1 | no | — |
| 57 | C4×Q8×C23 | D4 | 1 | 1 | yes | characteristic |
| 58 | C23×Q32 | Q8 | 1 | 4 | yes | complemented |
| 59 | C2×C16×C23 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C16×C23 | C23 | 1 | 2 | yes | — |
| 61 | C2×C16×C23 | C23 | 1 | 1 | yes | characteristic |
| 62 | C23×#107 | — | 2 | 2 | no | — |
| 63 | C23×#107 | — | 2 | 1 | no | — |
| 64 | C23×#107 | — | 2 | 1 | no | — |
| 65 | C23×#107 | C23 | 1 | 1 | yes | characteristic |
| 66 | C23×#107 | C23 | 1 | 1 | yes | characteristic |
| 67 | C23×#107 | C23 | 1 | 2 | yes | — |
| 68 | C23×#107 | C23 | 1 | 2 | yes | — |
| 69 | C23×#106 | — | 2 | 2 | no | — |
| 70 | C23×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 71 | C23×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 72 | C23×#103 | — | 2 | 2 | no | — |
| 73 | C23×#103 | — | 2 | 2 | no | — |
| 74 | C23×#103 | — | 2 | 1 | no | — |
| 75 | C23×#103 | — | 2 | 1 | no | — |
| 76 | C4×C8×C23 | C23 | 1 | 1 | yes | characteristic |
| 77 | C4×C8×C23 | C23 | 1 | 2 | yes | — |
| 78 | C2×Q8×C23 | — | 2 | 1 | no | — |
| 79 | C2×Q8×C23 | — | 2 | 1 | no | — |
| 80 | C2×Q8×C23 | — | 4 | 1 | no | — |
| 81 | C2×Q8×C23 | — | 4 | 2 | no | — |
| 82 | C2×Q8×C23 | — | 4 | 1 | no | — |
| 83 | C2×Q8×C23 | D8 | 1 | 1 | yes | characteristic |
| 84 | Q16×C23 | — | 2 | 1 | no | — |
| 85 | Q16×C23 | — | 2 | 1 | no | — |
| 86 | Q16×C23 | C2×Q8 | 1 | 2 | yes | — |
| 87 | Q16×C23 | C2×Q8 | 1 | 2 | yes | — |
| 88 | C2×C8×C23 | C2×D4 | 1 | 2 | yes | — |
| 89 | C2×C8×C23 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C23 | C24 | 1 | 1 | yes | characteristic |
| 91 | C4⋊C4×C23 | — | 2 | 1 | no | — |
| 92 | C4⋊C4×C23 | — | 2 | 2 | no | — |
| 93 | C4⋊C4×C23 | — | 2 | 2 | no | — |
| 94 | C4⋊C4×C23 | — | 2 | 1 | no | — |
| 95 | C4⋊C4×C23 | — | 4 | 1 | no | — |
| 96 | C4⋊C4×C23 | — | 4 | 2 | no | — |
| 97 | C4⋊C4×C23 | — | 4 | 2 | no | — |
| 98 | C4⋊C4×C23 | — | 4 | 1 | no | — |
| 99 | C4⋊C4×C23 | — | 4 | 2 | no | — |
| 100 | C4⋊C4×C23 | — | 4 | 2 | no | — |
| 101 | C4⋊C4×C23 | — | 4 | 1 | no | — |
| 102 | C4⋊C4×C23 | — | 4 | 2 | no | — |
| 103 | C4⋊C4×C23 | — | 4 | 1 | no | — |
| 104 | C4⋊C4×C23 | D8 | 1 | 1 | yes | characteristic |
| 105 | C4⋊C4×C23 | D8 | 1 | 2 | yes | — |
| 106 | C42×C23 | — | 4 | 2 | no | — |
| 107 | C42×C23 | — | 4 | 1 | no | — |
| 108 | C42×C23 | — | 4 | 2 | no | — |
| 109 | C42×C23 | — | 4 | 1 | no | — |
| 110 | C42×C23 | C2×D4 | 1 | 2 | yes | — |
| 111 | C42×C23 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C16×C23 | — | 2 | 1 | no | — |
| 113 | C16×C23 | — | 2 | 2 | no | — |
| 114 | C16×C23 | C2×Q8 | 1 | 2 | yes | — |
| 115 | #33954 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 116 | Q8×C23 | — | 2 | 2 | no | — |
| 117 | Q8×C23 | — | 2 | 2 | no | — |
| 118 | Q8×C23 | — | 2 | 2 | no | — |
| 119 | Q8×C23 | — | 4 | 1 | no | — |
| 120 | Q8×C23 | — | 4 | 1 | no | — |
| 121 | Q8×C23 | — | 8 | 2 | no | — |
| 122 | Q8×C23 | — | 16 | 2 | no | — |
| 123 | Q8×C23 | — | 16 | 1 | no | — |
| 124 | C2×C4×C23 | — | 4 | 1 | no | — |
| 125 | C2×C4×C23 | — | 4 | 1 | no | — |
| 126 | C2×C4×C23 | — | 4 | 1 | no | — |
| 127 | C2×C4×C23 | — | 4 | 2 | no | — |
| 128 | C2×C4×C23 | — | 4 | 2 | no | — |
| 129 | C2×C4×C23 | — | 4 | 1 | no | — |
| 130 | C2×C4×C23 | C2×D8 | 1 | 2 | yes | — |
| 131 | C2×C4×C23 | C2×D8 | 1 | 2 | yes | — |
| 132 | C2×C4×C23 | C2×D8 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C23 | C2×D8 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C8×C23 | — | 2 | 2 | no | — |
| 136 | C8×C23 | — | 2 | 1 | no | — |
| 137 | C8×C23 | C22×Q8 | 1 | 1 | yes | characteristic |
| 138 | C8×C23 | 2−1+4 | 1 | 1 | yes | characteristic |
| 139 | #3383 | C2×C23 | 1 | 1 | yes | characteristic |
| 140 | Q8×Q16 | C2×C23 | 1 | 1 | yes | characteristic |
| 141 | #2255 | C2×C23 | 1 | 2 | yes | — |
| 142 | #2254 | C2×C23 | 1 | 1 | yes | characteristic |
| 143 | #2229 | C2×C23 | 1 | 2 | yes | — |
| 144 | #2227 | C2×C23 | 1 | 1 | yes | characteristic |
| 145 | #2225 | C2×C23 | 1 | 2 | yes | — |
| 146 | #2225 | C2×C23 | 1 | 1 | yes | characteristic |
| 147 | #2183 | C2×C23 | 1 | 1 | yes | characteristic |
| 148 | #2176 | C2×C23 | 1 | 2 | yes | — |
| 149 | C4×Q32 | C2×C23 | 1 | 1 | yes | characteristic |
| 150 | C22×C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 151 | C4×C23 | — | 2 | 2 | no | — |
| 152 | C4×C23 | — | 2 | 2 | no | — |
| 153 | C4×C23 | — | 2 | 1 | no | — |
| 154 | C4×C23 | — | 4 | 2 | no | — |
| 155 | C4×C23 | — | 4 | 2 | no | — |
| 156 | C4×C23 | — | 8 | 2 | no | — |
| 157 | C4×C23 | — | 8 | 1 | no | — |
| 158 | C4×C23 | — | 8 | 2 | no | — |
| 159 | C4×C23 | — | 8 | 1 | no | — |
| 160 | C4×C23 | #510 | 1 | 2 | yes | — |
| 161 | C4×C23 | D4×Q8 | 1 | 1 | yes | characteristic |
| 162 | C4×C23 | #578 | 1 | 1 | yes | characteristic |
| 163 | Q82 | — | 2 | 1 | no | — |
| 164 | #557 | — | 2 | 1 | no | — |
| 165 | C2×Q32 | C22×C23 | 1 | 1 | yes | characteristic |
| 166 | #500 | C22×C23 | 1 | 2 | yes | — |
| 167 | #500 | C22×C23 | 1 | 1 | yes | characteristic |
| 168 | #500 | C22×C23 | 1 | 1 | yes | characteristic |
| 169 | #499 | C22×C23 | 1 | 2 | yes | — |
| 170 | #479 | — | 2 | 2 | no | — |
| 171 | #477 | — | 2 | 1 | no | — |
| 172 | #477 | — | 2 | 2 | no | — |
| 173 | #475 | — | 2 | 1 | no | — |
| 174 | C8×Q8 | C22×C23 | 1 | 1 | yes | characteristic |
| 175 | C4×Q16 | C22×C23 | 1 | 1 | yes | characteristic |
| 176 | C4×Q16 | C22×C23 | 1 | 2 | yes | — |
| 177 | C4×Q16 | C22×C23 | 1 | 1 | yes | characteristic |
| 178 | C4×Q16 | C22×C23 | 1 | 2 | yes | — |
| 179 | #367 | C22×C23 | 1 | 2 | yes | — |
| 180 | #367 | C22×C23 | 1 | 2 | yes | — |
| 181 | #367 | C22×C23 | 1 | 2 | yes | — |
| 182 | #366 | C22×C23 | 1 | 1 | yes | characteristic |
| 183 | #366 | C22×C23 | 1 | 2 | yes | — |
| 184 | C4⋊C16 | C22×C23 | 1 | 1 | yes | characteristic |
| 185 | C4⋊C16 | C22×C23 | 1 | 2 | yes | — |
| 186 | #358 | C22×C23 | 1 | 1 | yes | characteristic |
| 187 | #358 | C22×C23 | 1 | 1 | yes | characteristic |
| 188 | #358 | C22×C23 | 1 | 2 | yes | — |
| 189 | #358 | C22×C23 | 1 | 2 | yes | — |
| 190 | #346 | C22×C23 | 1 | 2 | yes | — |
| 191 | C4×C16 | C22×C23 | 1 | 1 | yes | characteristic |
| 192 | C2×C23 | Q8×D8 | 1 | 1 | yes | characteristic |
| 193 | C2×C23 | C2×#510 | 1 | 1 | yes | characteristic |
| 194 | C2×C23 | #3416 | 1 | 1 | yes | characteristic |
| 195 | C2×Q16 | C23×C23 | 1 | 1 | yes | characteristic |
| 196 | C2×Q16 | C23×C23 | 1 | 1 | yes | characteristic |
| 197 | C4⋊Q8 | — | 2 | 1 | no | — |
| 198 | C4⋊Q8 | — | 2 | 2 | no | — |
| 199 | C4⋊Q8 | — | 2 | 1 | no | — |
| 200 | C4⋊Q8 | — | 4 | 1 | no | — |
| 201 | C4⋊Q8 | — | 4 | 1 | no | — |
| 202 | C4⋊Q8 | — | 4 | 2 | no | — |
| 203 | #125 | — | 2 | 2 | no | — |
| 204 | #125 | — | 4 | 2 | no | — |
| 205 | C4×Q8 | — | 2 | 1 | no | — |
| 206 | C4×Q8 | — | 2 | 2 | no | — |
| 207 | C4×Q8 | — | 2 | 2 | no | — |
| 208 | C4×Q8 | — | 2 | 1 | no | — |
| 209 | C4×Q8 | — | 4 | 1 | no | — |
| 210 | C4×Q8 | — | 4 | 1 | no | — |
| 211 | C4×Q8 | D4×C23 | 1 | 1 | yes | characteristic |
| 212 | Q32 | Q8×C23 | 1 | 4 | yes | complemented |
| 213 | C2×C16 | C23×C23 | 1 | 1 | yes | characteristic |
| 214 | C2×C16 | C23×C23 | 1 | 2 | yes | — |
| 215 | C2×C16 | C23×C23 | 1 | 1 | yes | characteristic |
| 216 | #107 | — | 2 | 2 | no | — |
| 217 | #107 | — | 2 | 1 | no | — |
| 218 | #107 | — | 2 | 1 | no | — |
| 219 | #107 | C23×C23 | 1 | 1 | yes | characteristic |
| 220 | #107 | C23×C23 | 1 | 1 | yes | characteristic |
| 221 | #107 | C23×C23 | 1 | 2 | yes | — |
| 222 | #107 | C23×C23 | 1 | 2 | yes | — |
| 223 | #106 | — | 2 | 2 | no | — |
| 224 | C4⋊C8 | D4×C23 | 1 | 1 | yes | characteristic |
| 225 | C4⋊C8 | D4×C23 | 1 | 2 | yes | — |
| 226 | #103 | — | 2 | 2 | no | — |
| 227 | #103 | — | 2 | 2 | no | — |
| 228 | #103 | — | 2 | 1 | no | — |
| 229 | #103 | — | 2 | 1 | no | — |
| 230 | C4×C8 | C23×C23 | 1 | 1 | yes | characteristic |
| 231 | C4×C8 | C23×C23 | 1 | 2 | yes | — |
| 232 | C23 | #33954 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 233 | C2×Q8 | — | 2 | 1 | no | — |
| 234 | C2×Q8 | — | 2 | 1 | no | — |
| 235 | C2×Q8 | — | 4 | 1 | no | — |
| 236 | C2×Q8 | — | 4 | 2 | no | — |
| 237 | C2×Q8 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | D8×C23 | 1 | 1 | yes | characteristic |
| 239 | Q16 | — | 2 | 1 | no | — |
| 240 | Q16 | — | 2 | 1 | no | — |
| 241 | Q16 | C2×Q8×C23 | 1 | 2 | yes | — |
| 242 | Q16 | C2×Q8×C23 | 1 | 2 | yes | — |
| 243 | C2×C8 | C2×D4×C23 | 1 | 2 | yes | — |
| 244 | C2×C8 | C2×D4×C23 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C24×C23 | 1 | 1 | yes | characteristic |
| 246 | C4⋊C4 | — | 2 | 1 | no | — |
| 247 | C4⋊C4 | — | 2 | 2 | no | — |
| 248 | C4⋊C4 | — | 2 | 2 | no | — |
| 249 | C4⋊C4 | — | 2 | 1 | no | — |
| 250 | C4⋊C4 | — | 4 | 1 | no | — |
| 251 | C4⋊C4 | — | 4 | 2 | no | — |
| 252 | C4⋊C4 | — | 4 | 2 | no | — |
| 253 | C4⋊C4 | — | 4 | 1 | no | — |
| 254 | C4⋊C4 | — | 4 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 4 | 1 | no | — |
| 257 | C4⋊C4 | — | 4 | 2 | no | — |
| 258 | C4⋊C4 | — | 4 | 1 | no | — |
| 259 | C4⋊C4 | D8×C23 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | D8×C23 | 1 | 2 | yes | — |
| 261 | C42 | — | 4 | 2 | no | — |
| 262 | C42 | — | 4 | 1 | no | — |
| 263 | C42 | — | 4 | 2 | no | — |
| 264 | C42 | — | 4 | 1 | no | — |
| 265 | C42 | C2×D4×C23 | 1 | 2 | yes | — |
| 266 | C42 | C2×D4×C23 | 1 | 1 | yes | characteristic |
| 267 | C16 | — | 2 | 1 | no | — |
| 268 | C16 | — | 2 | 2 | no | — |
| 269 | C16 | C2×Q8×C23 | 1 | 2 | yes | — |
| 270 | Q8 | — | 2 | 2 | no | — |
| 271 | Q8 | — | 2 | 2 | no | — |
| 272 | Q8 | — | 2 | 2 | no | — |
| 273 | Q8 | — | 4 | 1 | no | — |
| 274 | Q8 | — | 4 | 1 | no | — |
| 275 | Q8 | — | 8 | 2 | no | — |
| 276 | Q8 | — | 16 | 2 | no | — |
| 277 | Q8 | — | 16 | 1 | no | — |
| 278 | C2×C4 | — | 4 | 1 | no | — |
| 279 | C2×C4 | — | 4 | 1 | no | — |
| 280 | C2×C4 | — | 4 | 1 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 4 | 2 | no | — |
| 283 | C2×C4 | — | 4 | 1 | no | — |
| 284 | C2×C4 | C2×D8×C23 | 1 | 2 | yes | — |
| 285 | C2×C4 | C2×D8×C23 | 1 | 2 | yes | — |
| 286 | C2×C4 | C2×D8×C23 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C2×D8×C23 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C22×D4×C23 | 1 | 1 | yes | characteristic |
| 289 | C8 | — | 2 | 2 | no | — |
| 290 | C8 | — | 2 | 1 | no | — |
| 291 | C8 | C22×Q8×C23 | 1 | 1 | yes | characteristic |
| 292 | C8 | C23×2−1+4 | 1 | 1 | yes | characteristic |
| 293 | C22 | C22×D8×C23 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 2 | 2 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 4 | 2 | no | — |
| 298 | C4 | — | 4 | 2 | no | — |
| 299 | C4 | — | 8 | 2 | no | — |
| 300 | C4 | — | 8 | 2 | no | — |
| 301 | C4 | — | 8 | 1 | no | — |
| 302 | C4 | — | 8 | 1 | no | — |
| 303 | C4 | C23×#510 | 1 | 2 | yes | — |
| 304 | C4 | D4×Q8×C23 | 1 | 1 | yes | characteristic |
| 305 | C4 | C23×#578 | 1 | 1 | yes | characteristic |
| 306 | C2 | Q8×D8×C23 | 1 | 1 | yes | characteristic |
| 307 | C2 | C2×C23×#510 | 1 | 1 | yes | characteristic |
| 308 | C2 | C23×#3416 | 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