Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
340 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #20286105 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20276668 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276665 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20276003 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20272708 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272665 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#1342 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #16811423 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16811175 | — | 2 | 1 | no | — |
| 10 | #16811066 | — | 2 | 1 | no | — |
| 11 | #16811064 | — | 2 | 1 | no | — |
| 12 | #16810907 | — | 2 | 1 | no | — |
| 13 | #16810906 | — | 2 | 1 | no | — |
| 14 | #16810881 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16810879 | — | 2 | 1 | no | — |
| 16 | #16810832 | — | 2 | 1 | no | — |
| 17 | #16810831 | — | 2 | 1 | no | — |
| 18 | #16810646 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#13133143 | — | 2 | 1 | no | — |
| 20 | #16810605 | — | 2 | 1 | no | — |
| 21 | #16810434 | — | 2 | 1 | no | — |
| 22 | #16810432 | — | 2 | 1 | no | — |
| 23 | #16810405 | C22 | 1 | 1 | yes | characteristic |
| 24 | C53×#470 | C22 | 1 | 1 | yes | characteristic |
| 25 | C53×#349 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×#338 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#11719945 | — | 2 | 1 | no | — |
| 28 | #13133250 | — | 4 | 1 | no | — |
| 29 | #13133245 | — | 4 | 1 | no | — |
| 30 | #13133239 | — | 2 | 1 | no | — |
| 31 | #13133239 | — | 2 | 1 | no | — |
| 32 | C2×Dic212 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#11719925 | — | 2 | 1 | no | — |
| 34 | C2×#11719917 | D4 | 1 | 1 | yes | characteristic |
| 35 | #13133216 | — | 2 | 1 | no | — |
| 36 | #13133215 | — | 2 | 1 | no | — |
| 37 | #13133207 | — | 4 | 1 | no | — |
| 38 | #13133196 | — | 2 | 1 | no | — |
| 39 | #13133194 | — | 4 | 1 | no | — |
| 40 | Q8×Dic53 | — | 2 | 1 | no | — |
| 41 | #13133182 | — | 2 | 1 | no | — |
| 42 | #13133182 | — | 4 | 1 | no | — |
| 43 | #13133173 | — | 4 | 1 | no | — |
| 44 | C2×C4×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 45 | Dic424 | — | 4 | 1 | no | — |
| 46 | #13133163 | — | 4 | 1 | no | — |
| 47 | #13133162 | — | 4 | 1 | no | — |
| 48 | #13133160 | — | 4 | 1 | no | — |
| 49 | #13133157 | — | 2 | 1 | no | — |
| 50 | #13133149 | D4 | 1 | 1 | yes | characteristic |
| 51 | #13133148 | — | 2 | 1 | no | — |
| 52 | #13133146 | — | 2 | 1 | no | — |
| 53 | #13133146 | — | 2 | 1 | no | — |
| 54 | #13133143 | — | 4 | 1 | no | — |
| 55 | #13133143 | — | 4 | 1 | no | — |
| 56 | #13133142 | — | 2 | 1 | no | — |
| 57 | #13133141 | — | 2 | 1 | no | — |
| 58 | #13133140 | — | 2 | 1 | no | — |
| 59 | #13133135 | — | 4 | 1 | no | — |
| 60 | #13133133 | — | 4 | 1 | no | — |
| 61 | C2×Q16×C53 | D4 | 1 | 1 | yes | characteristic |
| 62 | C2×M16×C53 | — | 2 | 1 | no | — |
| 63 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22⋊Q8×C53 | — | 2 | 1 | no | — |
| 65 | M32×C53 | — | 2 | 1 | no | — |
| 66 | #106×C53 | D4 | 1 | 1 | yes | characteristic |
| 67 | #103×C53 | — | 2 | 1 | no | — |
| 68 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 69 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 71 | C22×Dic53 | — | 2 | 1 | no | — |
| 72 | #11719945 | — | 4 | 1 | no | — |
| 73 | #11719945 | — | 4 | 1 | no | — |
| 74 | #11719945 | — | 4 | 1 | no | — |
| 75 | #11719945 | — | 4 | 1 | no | — |
| 76 | C2×D212 | — | 2 | 1 | no | — |
| 77 | C2×C4×D53 | — | 4 | 1 | no | — |
| 78 | C2×Dic106 | — | 2 | 1 | no | — |
| 79 | C2×Dic106 | — | 2 | 1 | no | — |
| 80 | C2×Dic106 | — | 4 | 1 | no | — |
| 81 | Q16×C53 | — | 4 | 1 | no | — |
| 82 | M16×C53 | — | 4 | 1 | no | — |
| 83 | M16×C53 | — | 4 | 1 | no | — |
| 84 | M16×C53 | — | 4 | 1 | no | — |
| 85 | C2×C8×C53 | — | 4 | 1 | no | — |
| 86 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 89 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 90 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 91 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 92 | #11719925 | — | 8 | 1 | no | — |
| 93 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 94 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 95 | #11719919 | — | 4 | 1 | no | — |
| 96 | #11719919 | — | 4 | 1 | no | — |
| 97 | #11719919 | — | 4 | 1 | no | — |
| 98 | #11719919 | — | 4 | 1 | no | — |
| 99 | #11719919 | — | 4 | 1 | no | — |
| 100 | C4×Dic53 | — | 2 | 1 | no | — |
| 101 | C4×Dic53 | — | 4 | 1 | no | — |
| 102 | C4×Dic53 | — | 4 | 1 | no | — |
| 103 | C4×Dic53 | D8 | 1 | 1 | yes | characteristic |
| 104 | C4×Dic53 | D8 | 1 | 1 | yes | characteristic |
| 105 | #11719917 | — | 4 | 1 | no | — |
| 106 | C2×#88322 | — | 2 | 1 | no | — |
| 107 | Dic212 | — | 2 | 1 | no | — |
| 108 | Dic212 | — | 2 | 1 | no | — |
| 109 | Dic212 | — | 4 | 1 | no | — |
| 110 | D424 | — | 2 | 1 | no | — |
| 111 | D424 | — | 2 | 1 | no | — |
| 112 | #11719913 | — | 4 | 1 | no | — |
| 113 | #11719913 | — | 4 | 1 | no | — |
| 114 | C16×C53 | — | 4 | 1 | no | — |
| 115 | #11719908 | — | 4 | 1 | no | — |
| 116 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 117 | C22×D53 | — | 4 | 1 | no | — |
| 118 | Q8×C53 | — | 4 | 1 | no | — |
| 119 | Q8×C53 | — | 4 | 1 | no | — |
| 120 | C2×C4×C53 | — | 2 | 1 | no | — |
| 121 | C2×C4×C53 | — | 2 | 1 | no | — |
| 122 | C2×C4×C53 | — | 4 | 1 | no | — |
| 123 | C2×C4×C53 | — | 4 | 1 | no | — |
| 124 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 126 | C22⋊D53 | — | 8 | 1 | no | — |
| 127 | C22⋊D53 | — | 8 | 1 | no | — |
| 128 | C2×Dic53 | — | 2 | 1 | no | — |
| 129 | C2×Dic53 | — | 2 | 1 | no | — |
| 130 | C2×Dic53 | — | 4 | 2 | no | — |
| 131 | C2×Dic53 | — | 4 | 1 | no | — |
| 132 | C2×Dic53 | — | 4 | 1 | no | — |
| 133 | C2×Dic53 | — | 4 | 1 | no | — |
| 134 | D212 | — | 4 | 1 | no | — |
| 135 | D212 | — | 4 | 1 | no | — |
| 136 | C4×D53 | — | 8 | 1 | no | — |
| 137 | C4×D53 | — | 8 | 1 | no | — |
| 138 | Dic106 | — | 4 | 1 | no | — |
| 139 | Dic106 | — | 4 | 1 | no | — |
| 140 | Dic106 | — | 4 | 1 | no | — |
| 141 | Dic106 | — | 4 | 1 | no | — |
| 142 | Dic106 | — | 8 | 2 | no | — |
| 143 | C8×C53 | — | 2 | 1 | no | — |
| 144 | C8×C53 | — | 2 | 1 | no | — |
| 145 | C8×C53 | — | 8 | 1 | no | — |
| 146 | #88322 | — | 4 | 1 | no | — |
| 147 | #12174 | — | 53 | 1 | no | maximal, Sylow |
| 148 | C22×C53 | — | 2 | 1 | no | — |
| 149 | C22×C53 | — | 4 | 1 | no | — |
| 150 | C22×C53 | #447 | 1 | 1 | yes | characteristic |
| 151 | C2×D53 | — | 8 | 1 | no | — |
| 152 | C2×D53 | — | 16 | 1 | no | — |
| 153 | C4×C53 | — | 2 | 1 | no | — |
| 154 | C4×C53 | — | 8 | 1 | no | — |
| 155 | C4×C53 | — | 8 | 1 | no | — |
| 156 | C4×C53 | #454 | 1 | 1 | yes | characteristic |
| 157 | C4×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 158 | Dic53 | — | 2 | 2 | no | — |
| 159 | Dic53 | — | 4 | 1 | no | — |
| 160 | Dic53 | — | 8 | 1 | no | — |
| 161 | Dic53 | — | 8 | 1 | no | — |
| 162 | Dic53 | — | 8 | 1 | no | — |
| 163 | #3255 | — | 53 | 1 | no | — |
| 164 | #2191 | — | 53 | 1 | no | — |
| 165 | #2191 | — | 53 | 1 | no | — |
| 166 | #2050 | — | 53 | 1 | no | — |
| 167 | #1342 | — | 53 | 1 | no | — |
| 168 | #1342 | D53 | 1 | 1 | yes | characteristic, complemented |
| 169 | #1327 | — | 53 | 1 | no | — |
| 170 | C2×C53 | — | 2 | 1 | no | — |
| 171 | C2×C53 | — | 4 | 1 | no | — |
| 172 | C2×C53 | #1598 | 1 | 1 | yes | characteristic |
| 173 | D53 | — | 16 | 1 | no | — |
| 174 | #575 | — | 53 | 1 | no | — |
| 175 | #533 | — | 106 | 1 | no | — |
| 176 | #510 | — | 106 | 1 | no | — |
| 177 | #510 | — | 106 | 1 | no | — |
| 178 | #475 | — | 106 | 1 | no | — |
| 179 | #475 | — | 106 | 1 | no | — |
| 180 | #470 | — | 53 | 1 | no | — |
| 181 | #470 | — | 106 | 1 | no | — |
| 182 | #470 | C2×D53 | 1 | 1 | yes | characteristic |
| 183 | #462 | — | 106 | 1 | no | — |
| 184 | #462 | — | 106 | 1 | no | — |
| 185 | #423 | — | 53 | 1 | no | — |
| 186 | C2×#104 | — | 106 | 1 | no | — |
| 187 | #413 | — | 106 | 1 | no | — |
| 188 | #360 | — | 106 | 1 | no | — |
| 189 | #360 | — | 106 | 1 | no | — |
| 190 | #349 | — | 53 | 1 | no | — |
| 191 | #349 | C2×D53 | 1 | 1 | yes | characteristic |
| 192 | #338 | C2×D53 | 1 | 1 | yes | characteristic |
| 193 | C53 | #12174 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | C2×#41 | — | 106 | 1 | no | — |
| 195 | #137 | — | 212 | 1 | no | — |
| 196 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 197 | #135 | — | 106 | 1 | no | — |
| 198 | #135 | — | 106 | 1 | no | — |
| 199 | C2×Q16 | — | 53 | 1 | no | — |
| 200 | C2×Q16 | — | 106 | 1 | no | — |
| 201 | C2×Q16 | D212 | 1 | 1 | yes | characteristic |
| 202 | C2×M16 | — | 2 | 1 | no | — |
| 203 | C2×M16 | — | 53 | 1 | no | — |
| 204 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 205 | C4⋊Q8 | — | 106 | 1 | no | — |
| 206 | C4⋊Q8 | — | 106 | 1 | no | — |
| 207 | #125 | — | 212 | 1 | no | — |
| 208 | C22⋊Q8 | — | 2 | 1 | no | — |
| 209 | C22⋊Q8 | — | 106 | 1 | no | — |
| 210 | C22⋊Q8 | — | 212 | 1 | no | — |
| 211 | C4×Q8 | — | 106 | 1 | no | — |
| 212 | C4×Q8 | — | 106 | 1 | no | — |
| 213 | C4×Q8 | — | 212 | 1 | no | — |
| 214 | #117 | — | 212 | 1 | no | — |
| 215 | C2×C42 | — | 53 | 1 | no | — |
| 216 | Q32 | — | 212 | 1 | no | — |
| 217 | Q32 | — | 212 | 1 | no | — |
| 218 | SD32 | — | 212 | 1 | no | — |
| 219 | SD32 | — | 212 | 1 | no | — |
| 220 | M32 | — | 2 | 1 | no | — |
| 221 | M32 | — | 106 | 1 | no | — |
| 222 | #106 | — | 53 | 1 | no | — |
| 223 | #106 | — | 106 | 1 | no | — |
| 224 | #106 | D212 | 1 | 1 | yes | characteristic |
| 225 | C4⋊C8 | — | 106 | 1 | no | — |
| 226 | C4⋊C8 | — | 106 | 1 | no | — |
| 227 | #104 | — | 212 | 1 | no | — |
| 228 | #104 | — | 212 | 1 | no | — |
| 229 | #103 | — | 2 | 1 | no | — |
| 230 | #103 | — | 106 | 1 | no | — |
| 231 | #103 | — | 106 | 1 | no | — |
| 232 | #103 | — | 106 | 1 | no | — |
| 233 | #101 | — | 212 | 1 | no | — |
| 234 | #100 | — | 212 | 1 | no | — |
| 235 | #41 | — | 212 | 1 | no | — |
| 236 | #41 | — | 212 | 1 | no | — |
| 237 | #41 | — | 212 | 1 | no | — |
| 238 | #41 | — | 212 | 1 | no | — |
| 239 | C2×Q8 | — | 2 | 1 | no | — |
| 240 | C2×Q8 | — | 106 | 1 | no | — |
| 241 | C2×Q8 | — | 106 | 1 | no | — |
| 242 | C2×Q8 | — | 212 | 1 | no | — |
| 243 | C2×D4 | — | 106 | 1 | no | — |
| 244 | C2×D4 | — | 212 | 1 | no | — |
| 245 | C22×C4 | — | 106 | 1 | no | — |
| 246 | C22×C4 | — | 212 | 1 | no | — |
| 247 | C22×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 248 | Q16 | — | 4 | 1 | no | — |
| 249 | Q16 | — | 106 | 1 | no | — |
| 250 | Q16 | — | 106 | 1 | no | — |
| 251 | Q16 | — | 424 | 1 | no | — |
| 252 | Q16 | — | 212 | 1 | no | — |
| 253 | SD16 | — | 212 | 1 | no | — |
| 254 | SD16 | — | 212 | 1 | no | — |
| 255 | D8 | — | 106 | 1 | no | — |
| 256 | D8 | — | 106 | 1 | no | — |
| 257 | M16 | — | 4 | 1 | no | — |
| 258 | M16 | — | 4 | 1 | no | — |
| 259 | M16 | — | 4 | 1 | no | — |
| 260 | M16 | — | 212 | 1 | no | — |
| 261 | C2×C8 | — | 4 | 1 | no | — |
| 262 | C2×C8 | — | 106 | 1 | no | — |
| 263 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 264 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 265 | C4⋊C4 | — | 2 | 1 | no | — |
| 266 | C4⋊C4 | — | 4 | 1 | no | — |
| 267 | C4⋊C4 | — | 106 | 1 | no | — |
| 268 | C4⋊C4 | — | 212 | 1 | no | — |
| 269 | C4⋊C4 | — | 212 | 1 | no | — |
| 270 | C4⋊C4 | — | 212 | 1 | no | — |
| 271 | C4⋊C4 | — | 212 | 1 | no | — |
| 272 | C4⋊C4 | — | 212 | 1 | no | — |
| 273 | C4⋊C4 | — | 212 | 1 | no | — |
| 274 | C22⋊C4 | — | 4 | 1 | no | — |
| 275 | C22⋊C4 | — | 212 | 1 | no | — |
| 276 | C42 | — | 53 | 1 | no | — |
| 277 | C42 | — | 53 | 1 | no | — |
| 278 | C42 | — | 106 | 1 | no | — |
| 279 | C42 | — | 212 | 1 | no | — |
| 280 | C42 | — | 212 | 1 | no | — |
| 281 | C16 | — | 4 | 1 | no | — |
| 282 | C16 | — | 212 | 1 | no | — |
| 283 | C23 | — | 212 | 1 | no | — |
| 284 | C23 | #13133244 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 4 | 1 | no | — |
| 286 | Q8 | — | 4 | 1 | no | — |
| 287 | Q8 | — | 424 | 2 | no | — |
| 288 | Q8 | — | 212 | 1 | no | — |
| 289 | Q8 | — | 212 | 1 | no | — |
| 290 | Q8 | — | 212 | 1 | no | — |
| 291 | Q8 | — | 212 | 1 | no | — |
| 292 | D4 | — | 424 | 1 | no | — |
| 293 | D4 | — | 424 | 1 | no | — |
| 294 | D4 | — | 212 | 1 | no | — |
| 295 | D4 | — | 212 | 1 | no | — |
| 296 | C2×C4 | — | 2 | 1 | no | — |
| 297 | C2×C4 | — | 2 | 1 | no | — |
| 298 | C2×C4 | — | 4 | 1 | no | — |
| 299 | C2×C4 | — | 4 | 1 | no | — |
| 300 | C2×C4 | — | 106 | 1 | no | — |
| 301 | C2×C4 | — | 106 | 1 | no | — |
| 302 | C2×C4 | — | 424 | 1 | no | — |
| 303 | C2×C4 | — | 424 | 1 | no | — |
| 304 | C2×C4 | — | 212 | 2 | no | — |
| 305 | C2×C4 | — | 212 | 1 | no | — |
| 306 | C2×C4 | — | 212 | 1 | no | — |
| 307 | C2×C4 | — | 212 | 1 | no | — |
| 308 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | D8×D53 | 1 | 1 | yes | characteristic |
| 310 | C8 | — | 2 | 1 | no | — |
| 311 | C8 | — | 2 | 1 | no | — |
| 312 | C8 | — | 8 | 1 | no | — |
| 313 | C8 | — | 212 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 4 | 1 | no | — |
| 316 | C22 | — | 848 | 1 | no | — |
| 317 | C22 | — | 424 | 1 | no | — |
| 318 | C22 | #16810758 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 2 | 1 | no | — |
| 320 | C4 | — | 8 | 1 | no | — |
| 321 | C4 | — | 8 | 1 | no | — |
| 322 | C4 | — | 106 | 2 | no | — |
| 323 | C4 | — | 424 | 1 | no | — |
| 324 | C4 | — | 424 | 1 | no | — |
| 325 | C4 | — | 424 | 1 | no | — |
| 326 | C4 | — | 212 | 1 | no | — |
| 327 | C4 | #16810797 | 1 | 1 | yes | characteristic |
| 328 | C4 | #16810811 | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 4 | 1 | no | — |
| 331 | C2 | — | 848 | 1 | no | — |
| 332 | C2 | #20273651 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20286105 | C2 | semidirect |
| 5. #20272708 | C2 | semidirect |
| 6. #20272665 | C2 | semidirect |
| 7. C53×#1342 | C2 | semidirect |
| 168. #1342 | D53 | semidirect |
| 193. C53 | #12174 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20286105 | 2 |
| #20276668 | 2 |
| #20276665 | 2 |
| #20276003 | 2 |
| #20272708 | 2 |
| #20272665 | 2 |
| C53×#1342 | 2 |
| #12174 | 53 |