Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
392 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #20276665 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20275819 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20272709 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20272653 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C53×#3202 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16811423 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16811421 | — | 2 | 1 | no | — |
| 8 | #16811064 | — | 2 | 2 | no | — |
| 9 | #16810589 | — | 2 | 1 | no | — |
| 10 | #16810432 | — | 2 | 2 | no | — |
| 11 | #16810405 | C22 | 1 | 2 | yes | — |
| 12 | #16810381 | C22 | 1 | 1 | yes | characteristic |
| 13 | C53×#536 | — | 2 | 1 | no | — |
| 14 | C53×#485 | — | 2 | 2 | no | — |
| 15 | C53×#470 | C22 | 1 | 2 | yes | — |
| 16 | C53×#467 | — | 2 | 1 | no | — |
| 17 | C53×#451 | — | 2 | 2 | no | — |
| 18 | C53×#408 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#11719945 | — | 2 | 1 | no | — |
| 20 | #13133250 | — | 4 | 1 | no | — |
| 21 | #13133248 | — | 4 | 1 | no | — |
| 22 | #13133247 | — | 4 | 1 | no | — |
| 23 | #13133245 | — | 4 | 1 | no | — |
| 24 | #13133243 | — | 4 | 1 | no | — |
| 25 | #13133242 | — | 4 | 1 | no | — |
| 26 | #13133239 | — | 2 | 2 | no | — |
| 27 | #13133237 | — | 4 | 2 | no | — |
| 28 | C2×#11719917 | — | 2 | 1 | no | — |
| 29 | #13133163 | — | 4 | 2 | no | — |
| 30 | #13133160 | — | 4 | 2 | no | — |
| 31 | #13133157 | — | 2 | 2 | no | — |
| 32 | #13133129 | — | 4 | 2 | no | — |
| 33 | C2×#41×C53 | D4 | 1 | 1 | yes | characteristic |
| 34 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 35 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 36 | C2×Q16×C53 | D4 | 1 | 2 | yes | — |
| 37 | C2×M16×C53 | C23 | 1 | 1 | yes | characteristic |
| 38 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 39 | C4⋊Q8×C53 | — | 4 | 1 | no | — |
| 40 | #124×C53 | — | 4 | 1 | no | — |
| 41 | #123×C53 | — | 4 | 1 | no | — |
| 42 | #123×C53 | — | 4 | 1 | no | — |
| 43 | C22⋊Q8×C53 | — | 2 | 2 | no | — |
| 44 | C22⋊Q8×C53 | — | 2 | 2 | no | — |
| 45 | #117×C53 | — | 2 | 1 | no | — |
| 46 | #107×C53 | — | 2 | 1 | no | — |
| 47 | #106×C53 | D4 | 1 | 2 | yes | — |
| 48 | #103×C53 | — | 2 | 2 | no | — |
| 49 | #103×C53 | — | 2 | 2 | no | — |
| 50 | C22⋊C8×C53 | — | 2 | 2 | no | — |
| 51 | #41×C53 | — | 4 | 1 | no | — |
| 52 | #41×C53 | — | 4 | 1 | no | — |
| 53 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 54 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 55 | C2×Q8×C53 | — | 8 | 1 | no | — |
| 56 | C2×Q8×C53 | — | 8 | 1 | no | — |
| 57 | C2×Q8×C53 | D8 | 1 | 1 | yes | characteristic |
| 58 | C2×D4×C53 | — | 2 | 1 | no | — |
| 59 | C2×D4×C53 | D8 | 1 | 1 | yes | characteristic |
| 60 | C22×C4×C53 | — | 2 | 1 | no | — |
| 61 | C22×C4×C53 | — | 4 | 1 | no | — |
| 62 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 64 | #11719945 | — | 4 | 1 | no | — |
| 65 | #11719945 | — | 4 | 1 | no | — |
| 66 | #11719945 | — | 4 | 1 | no | — |
| 67 | #11719945 | — | 4 | 1 | no | — |
| 68 | C2×D212 | — | 2 | 1 | no | — |
| 69 | C2×C4×D53 | — | 4 | 1 | no | — |
| 70 | C2×Dic106 | — | 2 | 1 | no | — |
| 71 | Q16×C53 | — | 4 | 2 | no | — |
| 72 | Q16×C53 | — | 8 | 1 | no | — |
| 73 | M16×C53 | — | 4 | 1 | no | — |
| 74 | C2×C8×C53 | — | 2 | 1 | no | — |
| 75 | C2×C8×C53 | — | 4 | 1 | no | — |
| 76 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 77 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 78 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 79 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 80 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 81 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 82 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 83 | C42×C53 | — | 4 | 1 | no | — |
| 84 | #11719925 | — | 4 | 2 | no | — |
| 85 | #11719924 | — | 4 | 2 | no | — |
| 86 | #11719923 | — | 4 | 2 | no | — |
| 87 | #11719922 | — | 4 | 2 | no | — |
| 88 | #11719917 | — | 4 | 1 | no | — |
| 89 | #11719917 | — | 4 | 1 | no | — |
| 90 | C2×#88322 | — | 2 | 2 | no | — |
| 91 | Dic212 | — | 2 | 2 | no | — |
| 92 | D424 | — | 2 | 2 | no | — |
| 93 | #11719913 | — | 4 | 2 | no | — |
| 94 | #11719908 | — | 4 | 2 | no | — |
| 95 | C23×C53 | — | 2 | 1 | no | — |
| 96 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 97 | C22×D53 | — | 4 | 1 | no | — |
| 98 | Q8×C53 | — | 4 | 1 | no | — |
| 99 | Q8×C53 | — | 4 | 2 | no | — |
| 100 | Q8×C53 | — | 4 | 2 | no | — |
| 101 | Q8×C53 | — | 8 | 2 | no | — |
| 102 | D4×C53 | — | 4 | 1 | no | — |
| 103 | D4×C53 | — | 8 | 1 | no | — |
| 104 | C2×C4×C53 | — | 2 | 1 | no | — |
| 105 | C2×C4×C53 | — | 2 | 1 | no | — |
| 106 | C2×C4×C53 | — | 2 | 1 | no | — |
| 107 | C2×C4×C53 | — | 4 | 1 | no | — |
| 108 | C2×C4×C53 | — | 4 | 1 | no | — |
| 109 | C2×C4×C53 | — | 4 | 2 | no | — |
| 110 | C2×C4×C53 | — | 4 | 2 | no | — |
| 111 | C2×C4×C53 | — | 8 | 2 | no | — |
| 112 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 114 | C22⋊D53 | — | 8 | 1 | no | — |
| 115 | C22⋊D53 | — | 8 | 1 | no | — |
| 116 | C2×Dic53 | — | 4 | 1 | no | — |
| 117 | D212 | — | 4 | 1 | no | — |
| 118 | D212 | — | 4 | 1 | no | — |
| 119 | C4×D53 | — | 8 | 1 | no | — |
| 120 | C4×D53 | — | 8 | 1 | no | — |
| 121 | Dic106 | — | 4 | 1 | no | — |
| 122 | Dic106 | — | 4 | 1 | no | — |
| 123 | C8×C53 | — | 2 | 2 | no | — |
| 124 | C8×C53 | — | 4 | 1 | no | — |
| 125 | #88322 | — | 4 | 2 | no | — |
| 126 | #12118 | — | 53 | 1 | no | maximal, Sylow |
| 127 | C22×C53 | — | 2 | 1 | no | — |
| 128 | C22×C53 | — | 4 | 1 | no | — |
| 129 | C22×C53 | — | 4 | 2 | no | — |
| 130 | C22×C53 | — | 4 | 1 | no | — |
| 131 | C22×C53 | #447 | 1 | 1 | yes | characteristic |
| 132 | C2×D53 | — | 8 | 2 | no | — |
| 133 | C2×D53 | — | 8 | 1 | no | — |
| 134 | C4×C53 | — | 2 | 1 | no | — |
| 135 | C4×C53 | — | 4 | 1 | no | — |
| 136 | C4×C53 | — | 8 | 2 | no | — |
| 137 | C4×C53 | — | 8 | 2 | no | — |
| 138 | C4×C53 | #453 | 1 | 1 | yes | characteristic |
| 139 | C4×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 140 | Dic53 | — | 8 | 1 | no | — |
| 141 | #3202 | D53 | 1 | 1 | yes | characteristic, complemented |
| 142 | #2191 | — | 53 | 2 | no | — |
| 143 | #2017 | — | 53 | 1 | no | — |
| 144 | #1342 | — | 53 | 2 | no | — |
| 145 | #1321 | — | 53 | 1 | no | — |
| 146 | C2×C53 | — | 2 | 1 | no | — |
| 147 | C2×C53 | — | 4 | 1 | no | — |
| 148 | C2×C53 | — | 8 | 1 | no | — |
| 149 | C2×C53 | #1595 | 1 | 1 | yes | characteristic |
| 150 | D53 | — | 16 | 1 | no | — |
| 151 | #575 | — | 53 | 1 | no | — |
| 152 | #575 | — | 106 | 1 | no | — |
| 153 | #536 | — | 2 | 1 | no | — |
| 154 | #510 | — | 106 | 2 | no | — |
| 155 | #485 | — | 2 | 2 | no | — |
| 156 | #470 | C2×D53 | 1 | 2 | yes | — |
| 157 | #467 | — | 2 | 1 | no | — |
| 158 | #451 | — | 2 | 2 | no | — |
| 159 | #408 | — | 106 | 1 | no | — |
| 160 | #408 | C2×D53 | 1 | 1 | yes | characteristic |
| 161 | #360 | — | 106 | 2 | no | — |
| 162 | #349 | — | 53 | 2 | no | — |
| 163 | #338 | — | 53 | 1 | no | — |
| 164 | C53 | #12118 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 165 | C2×#41 | — | 106 | 1 | no | — |
| 166 | C2×#41 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 167 | C22×Q8 | — | 2 | 1 | no | — |
| 168 | #137 | — | 212 | 1 | no | — |
| 169 | #137 | — | 212 | 1 | no | — |
| 170 | #137 | — | 212 | 1 | no | — |
| 171 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 172 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 173 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 174 | #135 | — | 106 | 2 | no | — |
| 175 | #135 | — | 212 | 2 | no | — |
| 176 | C2×Q16 | — | 2 | 1 | no | — |
| 177 | C2×Q16 | C22⋊D53 | 1 | 2 | yes | — |
| 178 | C2×M16 | — | 106 | 1 | no | — |
| 179 | C2×M16 | C22×D53 | 1 | 1 | yes | characteristic |
| 180 | C22×C8 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 181 | C4⋊Q8 | — | 4 | 1 | no | — |
| 182 | #124 | — | 4 | 1 | no | — |
| 183 | #123 | — | 4 | 1 | no | — |
| 184 | #123 | — | 4 | 1 | no | — |
| 185 | C22⋊Q8 | — | 2 | 2 | no | — |
| 186 | C22⋊Q8 | — | 2 | 2 | no | — |
| 187 | #117 | — | 2 | 1 | no | — |
| 188 | Q32 | — | 212 | 2 | no | — |
| 189 | SD32 | — | 212 | 2 | no | — |
| 190 | M32 | — | 106 | 2 | no | — |
| 191 | #107 | — | 2 | 1 | no | — |
| 192 | #106 | C22⋊D53 | 1 | 2 | yes | — |
| 193 | #103 | — | 2 | 2 | no | — |
| 194 | #103 | — | 2 | 2 | no | — |
| 195 | C22⋊C8 | — | 2 | 2 | no | — |
| 196 | C22⋊C8 | — | 212 | 2 | no | — |
| 197 | #41 | — | 4 | 1 | no | — |
| 198 | #41 | — | 4 | 1 | no | — |
| 199 | #41 | — | 212 | 1 | no | — |
| 200 | #41 | — | 212 | 1 | no | — |
| 201 | #41 | — | 212 | 1 | no | — |
| 202 | #41 | — | 212 | 1 | no | — |
| 203 | C2×Q8 | — | 2 | 2 | no | — |
| 204 | C2×Q8 | — | 4 | 1 | no | — |
| 205 | C2×Q8 | — | 8 | 1 | no | — |
| 206 | C2×Q8 | — | 8 | 1 | no | — |
| 207 | C2×Q8 | — | 106 | 1 | no | — |
| 208 | C2×Q8 | #11719922 | 1 | 1 | yes | characteristic |
| 209 | C2×D4 | — | 2 | 1 | no | — |
| 210 | C2×D4 | — | 106 | 1 | no | — |
| 211 | C2×D4 | — | 212 | 1 | no | — |
| 212 | C2×D4 | #11719922 | 1 | 1 | yes | characteristic |
| 213 | C22×C4 | — | 2 | 1 | no | — |
| 214 | C22×C4 | — | 4 | 1 | no | — |
| 215 | C22×C4 | — | 212 | 1 | no | — |
| 216 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 217 | Q16 | — | 4 | 2 | no | — |
| 218 | Q16 | — | 8 | 1 | no | — |
| 219 | Q16 | — | 106 | 2 | no | — |
| 220 | Q16 | — | 212 | 2 | no | — |
| 221 | SD16 | — | 212 | 2 | no | — |
| 222 | SD16 | — | 212 | 2 | no | — |
| 223 | SD16 | — | 212 | 2 | no | — |
| 224 | D8 | — | 106 | 2 | no | — |
| 225 | D8 | — | 212 | 2 | no | — |
| 226 | M16 | — | 4 | 1 | no | — |
| 227 | M16 | — | 212 | 1 | no | — |
| 228 | M16 | — | 212 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 1 | no | — |
| 230 | C2×C8 | — | 4 | 1 | no | — |
| 231 | C2×C8 | — | 106 | 2 | no | — |
| 232 | C2×C8 | C2×C22⋊D53 | 1 | 2 | yes | — |
| 233 | C4⋊C4 | — | 2 | 2 | no | — |
| 234 | C4⋊C4 | — | 4 | 2 | no | — |
| 235 | C4⋊C4 | — | 4 | 2 | no | — |
| 236 | C22⋊C4 | — | 4 | 1 | no | — |
| 237 | C22⋊C4 | — | 4 | 2 | no | — |
| 238 | C22⋊C4 | — | 4 | 2 | no | — |
| 239 | C42 | — | 4 | 1 | no | — |
| 240 | C16 | — | 212 | 2 | no | — |
| 241 | C23 | — | 2 | 1 | no | — |
| 242 | C23 | — | 212 | 1 | no | — |
| 243 | C23 | #13133243 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 4 | 1 | no | — |
| 245 | Q8 | — | 4 | 2 | no | — |
| 246 | Q8 | — | 4 | 2 | no | — |
| 247 | Q8 | — | 8 | 2 | no | — |
| 248 | Q8 | — | 212 | 1 | no | — |
| 249 | Q8 | — | 212 | 1 | no | — |
| 250 | D4 | — | 4 | 1 | no | — |
| 251 | D4 | — | 8 | 1 | no | — |
| 252 | D4 | — | 424 | 1 | no | — |
| 253 | D4 | — | 424 | 1 | no | — |
| 254 | D4 | — | 212 | 1 | no | — |
| 255 | D4 | — | 212 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 4 | 2 | no | — |
| 262 | C2×C4 | — | 4 | 2 | no | — |
| 263 | C2×C4 | — | 8 | 2 | no | — |
| 264 | C2×C4 | — | 424 | 1 | no | — |
| 265 | C2×C4 | — | 424 | 1 | no | — |
| 266 | C2×C4 | — | 212 | 1 | no | — |
| 267 | C2×C4 | #13133186 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 269 | C8 | — | 2 | 2 | no | — |
| 270 | C8 | — | 4 | 1 | no | — |
| 271 | C8 | — | 212 | 2 | no | — |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 4 | 1 | no | — |
| 274 | C22 | — | 4 | 2 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 424 | 2 | no | — |
| 277 | C22 | — | 424 | 1 | no | — |
| 278 | C22 | #16810757 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 1 | no | — |
| 280 | C4 | — | 4 | 1 | no | — |
| 281 | C4 | — | 8 | 2 | no | — |
| 282 | C4 | — | 8 | 2 | no | — |
| 283 | C4 | — | 424 | 1 | no | — |
| 284 | C4 | #16810791 | 1 | 1 | yes | characteristic |
| 285 | C4 | #16810810 | 1 | 1 | yes | characteristic |
| 286 | C2 | — | 2 | 1 | no | — |
| 287 | C2 | — | 4 | 1 | no | — |
| 288 | C2 | — | 8 | 1 | no | — |
| 289 | C2 | — | 848 | 1 | no | — |
| 290 | C2 | #20273633 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20276665 | C2 | semidirect |
| 3. #20272709 | C2 | semidirect |
| 4. #20272653 | C2 | semidirect |
| 5. C53×#3202 | C2 | semidirect |
| 141. #3202 | D53 | semidirect |
| 164. C53 | #12118 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20276665 | 2 |
| #20275819 | 2 |
| #20272709 | 2 |
| #20272653 | 2 |
| C53×#3202 | 2 |
| #12118 | 53 |