Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
806 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 155 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 | #20289210 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20285915 | C2 | 1 | 2 | yes | maximal |
| 3 | #20285063 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20274044 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #20274018 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #20273331 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C53×#1537 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C53⋊#564 | C22 | 1 | 4 | yes | — |
| 9 | #16811352 | — | 2 | 2 | no | — |
| 10 | #16811351 | — | 2 | 2 | no | — |
| 11 | #16811348 | — | 2 | 2 | no | — |
| 12 | #16811235 | — | 2 | 2 | no | — |
| 13 | #16811233 | — | 2 | 4 | no | — |
| 14 | #16811180 | C22 | 1 | 2 | yes | — |
| 15 | #16811173 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16811032 | C22 | 1 | 4 | yes | — |
| 17 | #16810993 | C22 | 1 | 2 | yes | — |
| 18 | #16810989 | C22 | 1 | 2 | yes | — |
| 19 | #16810988 | C22 | 1 | 4 | yes | — |
| 20 | #16810985 | — | 2 | 2 | no | — |
| 21 | #16810948 | — | 2 | 4 | no | — |
| 22 | #16810914 | — | 2 | 2 | no | — |
| 23 | #16810904 | — | 2 | 2 | no | — |
| 24 | #16810617 | — | 2 | 2 | no | — |
| 25 | C2×#13133128 | C22 | 1 | 1 | yes | characteristic |
| 26 | #16810364 | C22 | 1 | 8 | yes | — |
| 27 | C53×#533 | C22 | 1 | 1 | yes | characteristic |
| 28 | C53×#432 | C22 | 1 | 2 | yes | — |
| 29 | C53×#330 | C22 | 1 | 4 | yes | — |
| 30 | C2×#11719945 | — | 2 | 1 | no | — |
| 31 | C22×#88322 | — | 2 | 1 | no | — |
| 32 | #13133217 | — | 2 | 2 | no | — |
| 33 | #13133217 | D4 | 1 | 2 | yes | — |
| 34 | #13133217 | D4 | 1 | 2 | yes | — |
| 35 | #13133217 | C23 | 1 | 1 | yes | characteristic |
| 36 | #13133215 | — | 2 | 4 | no | — |
| 37 | #13133214 | C23 | 1 | 1 | yes | characteristic |
| 38 | #13133210 | — | 2 | 8 | no | — |
| 39 | #13133209 | — | 2 | 2 | no | — |
| 40 | #13133209 | C23 | 1 | 4 | yes | — |
| 41 | #13133208 | — | 2 | 4 | no | — |
| 42 | #13133208 | — | 2 | 8 | no | — |
| 43 | #13133207 | — | 2 | 8 | no | — |
| 44 | #13133207 | — | 2 | 8 | no | — |
| 45 | #13133205 | — | 2 | 1 | no | — |
| 46 | #13133196 | — | 2 | 2 | no | — |
| 47 | #13133196 | — | 2 | 2 | no | — |
| 48 | #13133191 | — | 2 | 2 | no | — |
| 49 | Q8×Dic53 | — | 2 | 4 | no | — |
| 50 | C4×Dic106 | — | 2 | 4 | no | — |
| 51 | C4×Dic106 | — | 2 | 2 | no | — |
| 52 | C4×Dic106 | — | 2 | 2 | no | — |
| 53 | #13133182 | — | 2 | 4 | no | — |
| 54 | C4×D212 | — | 2 | 2 | no | — |
| 55 | #13133176 | D4 | 1 | 4 | yes | — |
| 56 | #13133151 | — | 2 | 4 | no | — |
| 57 | #13133149 | — | 2 | 4 | no | — |
| 58 | #13133140 | — | 2 | 4 | no | — |
| 59 | #13133139 | — | 2 | 4 | no | — |
| 60 | #13133139 | — | 2 | 4 | no | — |
| 61 | #13133137 | — | 2 | 4 | no | — |
| 62 | #13133128 | — | 2 | 1 | no | — |
| 63 | #13133128 | C23 | 1 | 2 | yes | — |
| 64 | C4⋊Q8×C53 | D4 | 1 | 2 | yes | — |
| 65 | #125×C53 | C23 | 1 | 2 | yes | — |
| 66 | C22⋊Q8×C53 | — | 2 | 2 | no | — |
| 67 | C4×Q8×C53 | — | 2 | 2 | no | — |
| 68 | #117×C53 | D4 | 1 | 2 | yes | — |
| 69 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 71 | C22⋊C8×C53 | — | 2 | 2 | no | — |
| 72 | #97×C53 | C23 | 1 | 4 | yes | — |
| 73 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 74 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 76 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 77 | #11719945 | — | 4 | 4 | no | — |
| 78 | C2×D212 | — | 2 | 1 | no | — |
| 79 | C2×D212 | — | 2 | 2 | no | — |
| 80 | C2×C4×D53 | — | 4 | 1 | no | — |
| 81 | C2×Dic106 | — | 2 | 2 | no | — |
| 82 | C2×Dic106 | — | 2 | 2 | no | — |
| 83 | C2×Dic106 | — | 2 | 1 | no | — |
| 84 | C2×C8×C53 | — | 2 | 4 | no | — |
| 85 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 86 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 87 | C4⋊C4×C53 | C2×D4 | 1 | 4 | yes | — |
| 88 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 89 | C42×C53 | — | 2 | 2 | no | — |
| 90 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C42×C53 | C2×D4 | 1 | 2 | yes | — |
| 92 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 93 | C22⋊Dic53 | — | 2 | 4 | no | — |
| 94 | #11719921 | — | 4 | 2 | no | — |
| 95 | C4⋊Dic53 | — | 2 | 8 | no | — |
| 96 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 97 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 98 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 99 | C4⋊Dic53 | C2×D4 | 1 | 4 | yes | — |
| 100 | C4⋊Dic53 | C2×D4 | 1 | 4 | yes | — |
| 101 | #11719919 | — | 2 | 8 | no | — |
| 102 | #11719919 | — | 2 | 8 | no | — |
| 103 | #11719919 | — | 2 | 4 | no | — |
| 104 | #11719919 | — | 2 | 4 | no | — |
| 105 | C4×Dic53 | — | 2 | 8 | no | — |
| 106 | C4×Dic53 | — | 2 | 4 | no | — |
| 107 | C2×#88322 | — | 2 | 2 | no | — |
| 108 | C2×#88322 | — | 4 | 2 | no | — |
| 109 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 110 | C22×D53 | — | 4 | 1 | no | — |
| 111 | Q8×C53 | — | 4 | 4 | no | — |
| 112 | C2×C4×C53 | — | 2 | 2 | no | — |
| 113 | C2×C4×C53 | — | 2 | 4 | no | — |
| 114 | C2×C4×C53 | — | 4 | 2 | no | — |
| 115 | C2×C4×C53 | #120 | 1 | 2 | yes | — |
| 116 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 118 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 119 | C22⋊D53 | — | 8 | 2 | no | — |
| 120 | C2×Dic53 | — | 2 | 4 | no | — |
| 121 | C2×Dic53 | — | 2 | 4 | no | — |
| 122 | C2×Dic53 | — | 4 | 1 | no | — |
| 123 | D212 | — | 4 | 4 | no | — |
| 124 | D212 | — | 4 | 2 | no | — |
| 125 | C4×D53 | — | 8 | 2 | no | — |
| 126 | Dic106 | — | 4 | 4 | no | — |
| 127 | Dic106 | — | 4 | 4 | no | — |
| 128 | Dic106 | — | 4 | 2 | no | — |
| 129 | C8×C53 | — | 4 | 4 | no | — |
| 130 | #88322 | — | 4 | 2 | no | — |
| 131 | #19063 | — | 53 | 1 | no | maximal, Sylow |
| 132 | C22×C53 | — | 2 | 1 | no | — |
| 133 | C22×C53 | — | 2 | 2 | no | — |
| 134 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 135 | C2×D53 | — | 8 | 1 | no | — |
| 136 | C2×D53 | — | 8 | 2 | no | — |
| 137 | C4×C53 | — | 2 | 1 | no | — |
| 138 | C4×C53 | — | 2 | 2 | no | — |
| 139 | C4×C53 | — | 2 | 2 | no | — |
| 140 | C4×C53 | — | 4 | 4 | no | — |
| 141 | C4×C53 | #454 | 1 | 2 | yes | — |
| 142 | Dic53 | — | 4 | 4 | no | — |
| 143 | Dic53 | — | 4 | 4 | no | — |
| 144 | Dic53 | — | 8 | 1 | no | — |
| 145 | #3556 | — | 53 | 1 | no | — |
| 146 | #3235 | — | 53 | 2 | no | — |
| 147 | #3135 | — | 53 | 1 | no | — |
| 148 | #1663 | — | 53 | 4 | no | — |
| 149 | #1658 | — | 53 | 4 | no | — |
| 150 | #1537 | D53 | 1 | 1 | yes | characteristic, complemented |
| 151 | #1531 | — | 53 | 2 | no | — |
| 152 | C2×C53 | — | 4 | 1 | no | — |
| 153 | C2×C53 | C2×#454 | 1 | 1 | yes | characteristic |
| 154 | C2×C53 | #3020 | 1 | 2 | yes | — |
| 155 | D53 | — | 16 | 1 | no | — |
| 156 | #564 | — | 53 | 4 | no | — |
| 157 | #557 | — | 106 | 2 | no | — |
| 158 | #557 | — | 106 | 2 | no | — |
| 159 | #557 | — | 106 | 2 | no | — |
| 160 | #541 | — | 106 | 4 | no | — |
| 161 | #541 | — | 106 | 2 | no | — |
| 162 | #533 | — | 53 | 2 | no | — |
| 163 | #533 | C2×D53 | 1 | 1 | yes | characteristic |
| 164 | #532 | — | 53 | 1 | no | — |
| 165 | #501 | — | 53 | 4 | no | — |
| 166 | #491 | — | 53 | 2 | no | — |
| 167 | #490 | — | 53 | 2 | no | — |
| 168 | #490 | — | 53 | 4 | no | — |
| 169 | #489 | — | 106 | 2 | no | — |
| 170 | #482 | — | 106 | 4 | no | — |
| 171 | #476 | — | 106 | 2 | no | — |
| 172 | #474 | — | 106 | 2 | no | — |
| 173 | #432 | C2×D53 | 1 | 2 | yes | — |
| 174 | #416 | — | 106 | 2 | no | — |
| 175 | C2×#97 | — | 53 | 1 | no | — |
| 176 | #330 | — | 53 | 8 | no | — |
| 177 | #330 | C2×D53 | 1 | 4 | yes | — |
| 178 | C53 | #19063 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 179 | C2×#41 | — | 106 | 1 | no | — |
| 180 | C22×C8 | — | 106 | 1 | no | — |
| 181 | C4⋊Q8 | — | 53 | 2 | no | — |
| 182 | C4⋊Q8 | — | 53 | 2 | no | — |
| 183 | C4⋊Q8 | — | 53 | 1 | no | — |
| 184 | C4⋊Q8 | — | 106 | 2 | no | — |
| 185 | C4⋊Q8 | — | 106 | 4 | no | — |
| 186 | C4⋊Q8 | D212 | 1 | 2 | yes | — |
| 187 | #127 | — | 53 | 1 | no | — |
| 188 | #126 | — | 106 | 8 | no | — |
| 189 | #125 | — | 53 | 4 | no | — |
| 190 | #125 | — | 106 | 2 | no | — |
| 191 | #125 | — | 106 | 8 | no | — |
| 192 | #125 | — | 106 | 8 | no | — |
| 193 | #125 | — | 106 | 4 | no | — |
| 194 | #125 | — | 106 | 8 | no | — |
| 195 | #125 | C22×D53 | 1 | 2 | yes | — |
| 196 | #124 | — | 106 | 1 | no | — |
| 197 | C22⋊Q8 | — | 2 | 2 | no | — |
| 198 | C22⋊Q8 | — | 106 | 2 | no | — |
| 199 | C22⋊Q8 | — | 106 | 2 | no | — |
| 200 | #121 | — | 106 | 2 | no | — |
| 201 | C4×Q8 | — | 2 | 2 | no | — |
| 202 | C4×Q8 | — | 106 | 4 | no | — |
| 203 | C4×Q8 | — | 106 | 4 | no | — |
| 204 | C4×Q8 | — | 106 | 4 | no | — |
| 205 | C4×Q8 | — | 106 | 2 | no | — |
| 206 | C4×Q8 | — | 106 | 2 | no | — |
| 207 | C4×D4 | — | 106 | 2 | no | — |
| 208 | #117 | — | 53 | 4 | no | — |
| 209 | #117 | D212 | 1 | 2 | yes | — |
| 210 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 211 | #107 | — | 106 | 4 | no | — |
| 212 | #106 | — | 106 | 4 | no | — |
| 213 | C4⋊C8 | — | 2 | 2 | no | — |
| 214 | #103 | — | 106 | 4 | no | — |
| 215 | #102 | — | 106 | 4 | no | — |
| 216 | #102 | — | 106 | 4 | no | — |
| 217 | #102 | — | 106 | 4 | no | — |
| 218 | C22⋊C8 | — | 2 | 2 | no | — |
| 219 | #97 | — | 53 | 2 | no | — |
| 220 | #97 | — | 106 | 1 | no | — |
| 221 | #97 | C22×D53 | 1 | 4 | yes | — |
| 222 | #41 | — | 212 | 4 | no | — |
| 223 | C2×Q8 | — | 2 | 2 | no | — |
| 224 | C2×Q8 | — | 106 | 2 | no | — |
| 225 | C2×Q8 | — | 106 | 2 | no | — |
| 226 | C2×Q8 | — | 106 | 1 | no | — |
| 227 | C2×D4 | — | 106 | 2 | no | — |
| 228 | C2×D4 | — | 106 | 1 | no | — |
| 229 | C2×D4 | — | 212 | 1 | no | — |
| 230 | C22×C4 | — | 212 | 1 | no | — |
| 231 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | D4×D53 | 1 | 2 | yes | — |
| 233 | C2×C8 | — | 2 | 4 | no | — |
| 234 | C2×C8 | — | 106 | 2 | no | — |
| 235 | C2×C8 | — | 212 | 2 | no | — |
| 236 | C4⋊C4 | — | 2 | 2 | no | — |
| 237 | C4⋊C4 | — | 2 | 4 | no | — |
| 238 | C4⋊C4 | — | 53 | 4 | no | — |
| 239 | C4⋊C4 | — | 53 | 4 | no | — |
| 240 | C4⋊C4 | — | 106 | 8 | no | — |
| 241 | C4⋊C4 | — | 106 | 8 | no | — |
| 242 | C4⋊C4 | — | 106 | 8 | no | — |
| 243 | C4⋊C4 | — | 106 | 2 | no | — |
| 244 | C4⋊C4 | — | 106 | 2 | no | — |
| 245 | C4⋊C4 | — | 106 | 4 | no | — |
| 246 | C4⋊C4 | — | 106 | 4 | no | — |
| 247 | C4⋊C4 | — | 106 | 2 | no | — |
| 248 | C4⋊C4 | C2×D212 | 1 | 4 | yes | — |
| 249 | C22⋊C4 | — | 2 | 2 | no | — |
| 250 | C22⋊C4 | — | 106 | 4 | no | — |
| 251 | C22⋊C4 | — | 212 | 2 | no | — |
| 252 | C42 | — | 2 | 2 | no | — |
| 253 | C42 | — | 106 | 8 | no | — |
| 254 | C42 | — | 106 | 4 | no | — |
| 255 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 256 | C42 | D4×D53 | 1 | 2 | yes | — |
| 257 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 258 | C23 | — | 212 | 1 | no | — |
| 259 | C23 | #13133187 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 4 | 4 | no | — |
| 261 | Q8 | — | 212 | 2 | no | — |
| 262 | Q8 | — | 212 | 4 | no | — |
| 263 | Q8 | — | 212 | 4 | no | — |
| 264 | D4 | — | 424 | 2 | no | — |
| 265 | D4 | — | 212 | 4 | no | — |
| 266 | D4 | — | 212 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 2 | no | — |
| 268 | C2×C4 | — | 2 | 4 | no | — |
| 269 | C2×C4 | — | 4 | 2 | no | — |
| 270 | C2×C4 | — | 106 | 4 | no | — |
| 271 | C2×C4 | — | 106 | 4 | no | — |
| 272 | C2×C4 | — | 424 | 2 | no | — |
| 273 | C2×C4 | — | 212 | 1 | no | — |
| 274 | C2×C4 | #13133187 | 1 | 2 | yes | — |
| 275 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C2×D4×D53 | 1 | 2 | yes | — |
| 277 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 4 | 4 | no | — |
| 279 | C8 | — | 212 | 2 | no | — |
| 280 | C22 | — | 2 | 1 | no | — |
| 281 | C22 | — | 2 | 2 | no | — |
| 282 | C22 | — | 424 | 1 | no | — |
| 283 | C22 | — | 424 | 2 | no | — |
| 284 | C22 | C2×#13133187 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 2 | 2 | no | — |
| 287 | C4 | — | 2 | 2 | no | — |
| 288 | C4 | — | 4 | 4 | no | — |
| 289 | C4 | — | 424 | 1 | no | — |
| 290 | C4 | — | 212 | 4 | no | — |
| 291 | C4 | — | 212 | 4 | no | — |
| 292 | C4 | #16810798 | 1 | 2 | yes | — |
| 293 | C2 | — | 4 | 1 | no | — |
| 294 | C2 | — | 848 | 1 | no | — |
| 295 | C2 | C2×#16810798 | 1 | 1 | yes | characteristic |
| 296 | C2 | #20284081 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20289210 | C2 | semidirect |
| 4. #20274044 | C2 | semidirect |
| 5. #20274018 | C2 | semidirect |
| 6. #20273331 | C2 | semidirect |
| 7. C53×#1537 | C2 | semidirect |
| 150. #1537 | D53 | semidirect |
| 178. C53 | #19063 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289210 | 2 |
| #20285915 | 2 |
| #20285063 | 2 |
| #20274044 | 2 |
| #20274018 | 2 |
| #20273331 | 2 |
| C53×#1537 | 2 |
| #19063 | 53 |