Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
494 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #20291006 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20290169 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20289837 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #20284068 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16811609 | — | 2 | 1 | no | — |
| 6 | C4×#11719940 | — | 2 | 1 | no | — |
| 7 | #16811533 | — | 2 | 1 | no | — |
| 8 | #16811531 | C22 | 1 | 2 | yes | complemented |
| 9 | #16811479 | — | 2 | 1 | no | — |
| 10 | #16811478 | C22 | 1 | 2 | yes | complemented |
| 11 | #16811459 | C4 | 1 | 1 | yes | characteristic |
| 12 | #16811423 | — | 2 | 1 | no | — |
| 13 | #16811169 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16810808 | — | 2 | 1 | no | — |
| 15 | #16810803 | C22 | 1 | 2 | yes | — |
| 16 | #16810798 | — | 2 | 1 | no | — |
| 17 | #16810793 | C4 | 1 | 2 | yes | — |
| 18 | C2×#13133143 | — | 2 | 1 | no | — |
| 19 | C53×#413 | C4 | 1 | 1 | yes | characteristic |
| 20 | D53⋊M16 | — | 4 | 1 | no | — |
| 21 | C2×#11719940 | — | 4 | 1 | no | — |
| 22 | #13133278 | — | 2 | 2 | no | — |
| 23 | #13133277 | — | 4 | 1 | no | — |
| 24 | #13133272 | — | 2 | 2 | no | — |
| 25 | C4×#88324 | — | 4 | 1 | no | — |
| 26 | #13133265 | — | 2 | 1 | no | — |
| 27 | #13133264 | — | 4 | 1 | no | — |
| 28 | #13133263 | — | 4 | 1 | no | — |
| 29 | #13133262 | — | 4 | 1 | no | — |
| 30 | #13133262 | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×#11719945 | — | 2 | 1 | no | — |
| 32 | C2×#11719945 | — | 4 | 1 | no | — |
| 33 | C2×#11719945 | D4 | 1 | 1 | yes | characteristic |
| 34 | #41×D53 | — | 4 | 2 | no | — |
| 35 | #13133250 | — | 2 | 2 | no | — |
| 36 | #13133250 | — | 4 | 1 | no | — |
| 37 | #13133245 | — | 2 | 2 | no | — |
| 38 | #13133245 | — | 4 | 1 | no | — |
| 39 | #13133239 | — | 4 | 2 | no | — |
| 40 | #13133215 | D4 | 1 | 1 | yes | characteristic |
| 41 | #13133213 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #13133206 | D4 | 1 | 1 | yes | characteristic |
| 43 | #13133204 | C23 | 1 | 1 | yes | characteristic |
| 44 | #13133189 | — | 4 | 1 | no | — |
| 45 | C4×C22⋊D53 | — | 4 | 1 | no | — |
| 46 | C2×C4×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 47 | #13133143 | — | 2 | 2 | no | — |
| 48 | #13133143 | — | 2 | 2 | no | — |
| 49 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C2×M16×C53 | — | 2 | 1 | no | — |
| 51 | #101×C53 | — | 2 | 1 | no | — |
| 52 | #100×C53 | C2×C4 | 1 | 2 | yes | — |
| 53 | #41×C53 | — | 4 | 1 | no | — |
| 54 | #41×C53 | — | 4 | 1 | no | — |
| 55 | C2×Q8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C2×D4×C53 | — | 2 | 1 | no | — |
| 57 | C2×D4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C22×C4×C53 | — | 2 | 1 | no | — |
| 59 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 61 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 62 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 63 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 64 | C22×Dic53 | — | 2 | 1 | no | — |
| 65 | #11719949 | — | 4 | 2 | no | — |
| 66 | Q8×D53 | — | 4 | 2 | no | — |
| 67 | #11719947 | — | 4 | 2 | no | — |
| 68 | #11719947 | — | 4 | 4 | no | — |
| 69 | D4×D53 | — | 4 | 2 | no | — |
| 70 | D4×D53 | — | 4 | 4 | no | — |
| 71 | #11719945 | — | 2 | 2 | no | — |
| 72 | #11719945 | — | 2 | 2 | no | — |
| 73 | #11719945 | — | 4 | 1 | no | — |
| 74 | #11719945 | — | 4 | 2 | no | — |
| 75 | #11719945 | — | 4 | 2 | no | — |
| 76 | #11719945 | — | 4 | 2 | no | — |
| 77 | #11719945 | — | 4 | 1 | no | — |
| 78 | #11719945 | — | 4 | 1 | no | — |
| 79 | #11719945 | — | 4 | 1 | no | — |
| 80 | C2×D212 | — | 2 | 1 | no | — |
| 81 | C2×D212 | — | 4 | 1 | no | — |
| 82 | C2×D212 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×D53 | — | 2 | 1 | no | — |
| 84 | C2×C4×D53 | — | 4 | 1 | no | — |
| 85 | C2×C4×D53 | — | 4 | 2 | no | — |
| 86 | C2×Dic106 | — | 2 | 1 | no | — |
| 87 | C2×Dic106 | — | 4 | 1 | no | — |
| 88 | C2×Dic106 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | #11719940 | — | 4 | 4 | no | — |
| 90 | #11719940 | — | 4 | 2 | no | — |
| 91 | C2×#88324 | — | 4 | 2 | no | — |
| 92 | #11719936 | — | 8 | 1 | no | — |
| 93 | D53⋊C8 | — | 8 | 1 | no | — |
| 94 | M16×C53 | — | 2 | 2 | no | — |
| 95 | M16×C53 | — | 4 | 1 | no | — |
| 96 | C2×C8×C53 | — | 4 | 1 | no | — |
| 97 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 98 | #11719921 | — | 4 | 1 | no | — |
| 99 | #11719919 | — | 4 | 1 | no | — |
| 100 | C4×Dic53 | — | 2 | 1 | no | — |
| 101 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C4×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | Dic212 | — | 4 | 2 | no | — |
| 104 | D424 | — | 4 | 2 | no | — |
| 105 | #11719913 | — | 4 | 2 | no | — |
| 106 | #11719913 | — | 4 | 2 | no | — |
| 107 | C23×C53 | — | 2 | 1 | no | — |
| 108 | C23×C53 | #99 | 1 | 1 | yes | characteristic |
| 109 | C22×D53 | — | 2 | 1 | no | — |
| 110 | C22×D53 | — | 4 | 1 | no | — |
| 111 | C22×D53 | — | 4 | 2 | no | — |
| 112 | Q8×C53 | — | 4 | 1 | no | — |
| 113 | D4×C53 | — | 4 | 1 | no | — |
| 114 | D4×C53 | — | 4 | 2 | no | — |
| 115 | C2×C4×C53 | — | 2 | 1 | no | — |
| 116 | C2×C4×C53 | — | 2 | 1 | no | — |
| 117 | C2×C4×C53 | — | 2 | 1 | no | — |
| 118 | C2×C4×C53 | — | 4 | 1 | no | — |
| 119 | C2×C4×C53 | — | 4 | 1 | no | — |
| 120 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C22⋊D53 | — | 4 | 2 | no | — |
| 123 | C22⋊D53 | — | 4 | 4 | no | — |
| 124 | C22⋊D53 | — | 4 | 2 | no | — |
| 125 | C22⋊D53 | — | 8 | 2 | no | — |
| 126 | C22⋊D53 | — | 8 | 1 | no | — |
| 127 | C2×Dic53 | — | 2 | 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 | 2 | no | — |
| 133 | D212 | — | 2 | 2 | no | — |
| 134 | D212 | — | 4 | 1 | no | — |
| 135 | D212 | — | 4 | 2 | no | — |
| 136 | D212 | — | 4 | 1 | no | — |
| 137 | C4×D53 | — | 4 | 2 | no | — |
| 138 | C4×D53 | — | 4 | 4 | no | — |
| 139 | C4×D53 | — | 4 | 1 | no | — |
| 140 | C4×D53 | — | 4 | 1 | no | — |
| 141 | C4×D53 | — | 4 | 2 | no | — |
| 142 | Dic106 | — | 2 | 2 | no | — |
| 143 | Dic106 | — | 4 | 2 | no | — |
| 144 | Dic106 | — | 4 | 1 | no | — |
| 145 | Dic106 | — | 4 | 1 | no | — |
| 146 | #88324 | — | 4 | 4 | no | — |
| 147 | #88324 | — | 8 | 2 | no | — |
| 148 | C8×C53 | — | 4 | 2 | no | — |
| 149 | #13564 | — | 53 | 1 | no | maximal, Sylow |
| 150 | C22×C53 | — | 2 | 1 | no | — |
| 151 | C22×C53 | — | 4 | 1 | no | — |
| 152 | C22×C53 | — | 4 | 1 | no | — |
| 153 | C22×C53 | — | 8 | 1 | no | — |
| 154 | C22×C53 | C2×#99 | 1 | 1 | yes | characteristic |
| 155 | C2×D53 | — | 4 | 2 | no | — |
| 156 | C2×D53 | — | 4 | 1 | no | — |
| 157 | C2×D53 | — | 8 | 1 | no | — |
| 158 | C2×D53 | — | 16 | 1 | no | — |
| 159 | C2×D53 | — | 16 | 2 | no | — |
| 160 | C4×C53 | — | 2 | 1 | no | — |
| 161 | C4×C53 | — | 4 | 1 | no | — |
| 162 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 163 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 164 | Dic53 | — | 2 | 2 | no | — |
| 165 | Dic53 | — | 4 | 1 | no | — |
| 166 | Dic53 | — | 4 | 1 | no | — |
| 167 | Dic53 | — | 4 | 2 | no | — |
| 168 | #3018 | — | 53 | 1 | no | — |
| 169 | #2132 | — | 53 | 1 | no | — |
| 170 | #1525 | — | 53 | 1 | no | — |
| 171 | #1403 | — | 53 | 4 | no | — |
| 172 | C2×C53 | — | 2 | 1 | no | — |
| 173 | C2×C53 | — | 4 | 1 | no | — |
| 174 | C2×C53 | — | 8 | 1 | no | — |
| 175 | C2×C53 | C2×AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 176 | D53 | — | 8 | 1 | no | — |
| 177 | D53 | — | 8 | 2 | no | — |
| 178 | #585 | — | 53 | 1 | no | — |
| 179 | #575 | — | 106 | 1 | no | — |
| 180 | #532 | — | 53 | 1 | no | — |
| 181 | #456 | — | 106 | 1 | no | — |
| 182 | #455 | — | 53 | 2 | no | — |
| 183 | #454 | — | 106 | 1 | no | — |
| 184 | #453 | — | 53 | 2 | no | — |
| 185 | C2×#104 | — | 106 | 1 | no | — |
| 186 | #413 | — | 106 | 1 | no | — |
| 187 | #413 | #6157 | 1 | 1 | yes | characteristic |
| 188 | C4×M16 | — | 106 | 1 | no | — |
| 189 | #356 | — | 106 | 1 | no | — |
| 190 | #355 | — | 53 | 2 | no | — |
| 191 | #329 | — | 53 | 2 | no | — |
| 192 | #329 | — | 106 | 1 | no | — |
| 193 | C53 | #13564 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | 2−1+4 | — | 106 | 1 | no | — |
| 195 | 2−1+4 | — | 212 | 1 | no | — |
| 196 | H(5,𝔽2) | — | 53 | 2 | no | — |
| 197 | H(5,𝔽2) | — | 212 | 1 | no | — |
| 198 | H(5,𝔽2) | — | 212 | 1 | no | — |
| 199 | C2×#41 | — | 53 | 1 | no | — |
| 200 | C2×#41 | — | 106 | 1 | no | — |
| 201 | C2×#41 | — | 212 | 1 | no | — |
| 202 | C2×#41 | — | 212 | 2 | no | — |
| 203 | C2×#41 | C2×#6157 | 1 | 1 | yes | characteristic |
| 204 | #137 | — | 106 | 2 | no | — |
| 205 | #137 | — | 212 | 1 | no | — |
| 206 | AGL(1,ℤ/8) | — | 106 | 2 | no | — |
| 207 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 208 | #135 | — | 212 | 2 | no | — |
| 209 | C2×M16 | — | 2 | 1 | no | — |
| 210 | C2×M16 | — | 212 | 1 | no | — |
| 211 | C2×M16 | — | 212 | 1 | no | — |
| 212 | C4⋊Q8 | — | 53 | 1 | no | — |
| 213 | #127 | — | 53 | 1 | no | — |
| 214 | #124 | — | 53 | 1 | no | — |
| 215 | #124 | — | 53 | 1 | no | — |
| 216 | #121 | — | 212 | 1 | no | — |
| 217 | C4×D4 | — | 212 | 1 | no | — |
| 218 | C2×C42 | — | 53 | 1 | no | — |
| 219 | #104 | — | 106 | 2 | no | — |
| 220 | #104 | — | 106 | 2 | no | — |
| 221 | #101 | — | 2 | 1 | no | — |
| 222 | #101 | — | 106 | 2 | no | — |
| 223 | #100 | — | 212 | 1 | no | — |
| 224 | #100 | C2×#6157 | 1 | 2 | yes | — |
| 225 | #97 | — | 106 | 2 | no | — |
| 226 | C4×C8 | — | 212 | 1 | no | — |
| 227 | #41 | — | 4 | 1 | no | — |
| 228 | #41 | — | 4 | 1 | no | — |
| 229 | #41 | — | 106 | 2 | no | — |
| 230 | #41 | — | 106 | 2 | no | — |
| 231 | #41 | — | 212 | 1 | no | — |
| 232 | #41 | — | 212 | 2 | no | — |
| 233 | #41 | — | 212 | 1 | no | — |
| 234 | #41 | — | 212 | 1 | no | — |
| 235 | #41 | — | 212 | 2 | no | — |
| 236 | #41 | — | 212 | 2 | no | — |
| 237 | #41 | — | 212 | 4 | no | — |
| 238 | #41 | — | 212 | 2 | no | — |
| 239 | #41 | — | 212 | 1 | no | — |
| 240 | #41 | — | 212 | 2 | no | — |
| 241 | C2×Q8 | — | 53 | 1 | no | — |
| 242 | C2×Q8 | — | 106 | 1 | no | — |
| 243 | C2×Q8 | — | 212 | 1 | no | — |
| 244 | C2×Q8 | — | 212 | 2 | no | — |
| 245 | C2×Q8 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 246 | C2×D4 | — | 2 | 1 | no | — |
| 247 | C2×D4 | — | 53 | 1 | no | — |
| 248 | C2×D4 | — | 106 | 1 | no | — |
| 249 | C2×D4 | — | 106 | 1 | no | — |
| 250 | C2×D4 | — | 212 | 1 | no | — |
| 251 | C2×D4 | — | 212 | 1 | no | — |
| 252 | C2×D4 | — | 212 | 2 | no | — |
| 253 | C2×D4 | — | 212 | 4 | no | — |
| 254 | C2×D4 | — | 212 | 1 | no | — |
| 255 | C2×D4 | — | 212 | 2 | no | — |
| 256 | C2×D4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 257 | C22×C4 | — | 2 | 1 | no | — |
| 258 | C22×C4 | — | 106 | 1 | no | — |
| 259 | C22×C4 | — | 106 | 1 | no | — |
| 260 | C22×C4 | — | 212 | 1 | no | — |
| 261 | C22×C4 | — | 212 | 2 | no | — |
| 262 | C22×C4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 263 | Q16 | — | 212 | 2 | no | — |
| 264 | SD16 | — | 212 | 2 | no | — |
| 265 | SD16 | — | 212 | 2 | no | — |
| 266 | D8 | — | 212 | 2 | no | — |
| 267 | M16 | — | 2 | 2 | no | — |
| 268 | M16 | — | 4 | 1 | no | — |
| 269 | M16 | — | 424 | 1 | no | — |
| 270 | M16 | — | 212 | 2 | no | — |
| 271 | M16 | — | 212 | 4 | no | — |
| 272 | C2×C8 | — | 4 | 1 | no | — |
| 273 | C2×C8 | — | 424 | 1 | no | — |
| 274 | C2×C8 | — | 212 | 2 | no | — |
| 275 | C4⋊C4 | — | 212 | 1 | no | — |
| 276 | C22⋊C4 | — | 212 | 1 | no | — |
| 277 | C22⋊C4 | — | 212 | 1 | no | — |
| 278 | C42 | — | 53 | 1 | no | — |
| 279 | C42 | — | 53 | 1 | no | — |
| 280 | C42 | — | 106 | 1 | no | — |
| 281 | C23 | — | 2 | 1 | no | — |
| 282 | C23 | — | 106 | 1 | no | — |
| 283 | C23 | — | 212 | 1 | no | — |
| 284 | C23 | — | 212 | 2 | no | — |
| 285 | C23 | #13133276 | 1 | 1 | yes | characteristic |
| 286 | Q8 | — | 4 | 1 | no | — |
| 287 | Q8 | — | 106 | 2 | no | — |
| 288 | Q8 | — | 212 | 1 | no | — |
| 289 | Q8 | — | 212 | 2 | no | — |
| 290 | Q8 | — | 212 | 1 | no | — |
| 291 | D4 | — | 4 | 1 | no | — |
| 292 | D4 | — | 4 | 2 | no | — |
| 293 | D4 | — | 106 | 2 | no | — |
| 294 | D4 | — | 424 | 2 | no | — |
| 295 | D4 | — | 424 | 1 | no | — |
| 296 | D4 | — | 212 | 1 | no | — |
| 297 | D4 | — | 212 | 2 | no | — |
| 298 | D4 | — | 212 | 2 | no | — |
| 299 | D4 | — | 212 | 4 | no | — |
| 300 | D4 | — | 212 | 2 | no | — |
| 301 | D4 | — | 212 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 106 | 1 | no | — |
| 308 | C2×C4 | — | 106 | 1 | no | — |
| 309 | C2×C4 | — | 106 | 1 | no | — |
| 310 | C2×C4 | — | 212 | 2 | no | — |
| 311 | C2×C4 | — | 212 | 4 | no | — |
| 312 | C2×C4 | — | 212 | 2 | no | — |
| 313 | C2×C4 | — | 212 | 1 | no | — |
| 314 | C2×C4 | — | 212 | 1 | no | — |
| 315 | C2×C4 | — | 212 | 1 | no | — |
| 316 | C2×C4 | — | 212 | 2 | no | — |
| 317 | C2×C4 | — | 212 | 2 | no | — |
| 318 | C2×C4 | #13133276 | 1 | 1 | yes | characteristic |
| 319 | C2×C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 320 | C8 | — | 4 | 2 | no | — |
| 321 | C8 | — | 424 | 2 | no | — |
| 322 | C8 | — | 212 | 4 | no | — |
| 323 | C22 | — | 2 | 1 | no | — |
| 324 | C22 | — | 4 | 1 | no | — |
| 325 | C22 | — | 4 | 1 | no | — |
| 326 | C22 | — | 8 | 1 | no | — |
| 327 | C22 | — | 848 | 1 | no | — |
| 328 | C22 | — | 848 | 2 | no | — |
| 329 | C22 | — | 424 | 1 | no | — |
| 330 | C22 | — | 212 | 2 | no | — |
| 331 | C22 | — | 212 | 1 | no | — |
| 332 | C22 | C2×#13133276 | 1 | 1 | yes | characteristic |
| 333 | C4 | — | 2 | 1 | no | — |
| 334 | C4 | — | 4 | 1 | no | — |
| 335 | C4 | — | 106 | 2 | no | — |
| 336 | C4 | — | 212 | 1 | no | — |
| 337 | C4 | — | 212 | 1 | no | — |
| 338 | C4 | — | 212 | 2 | no | — |
| 339 | C4 | #16811524 | 1 | 1 | yes | characteristic |
| 340 | C4 | #16811524 | 1 | 1 | yes | characteristic |
| 341 | C2 | — | 2 | 1 | no | — |
| 342 | C2 | — | 4 | 1 | no | — |
| 343 | C2 | — | 8 | 1 | no | — |
| 344 | C2 | — | 424 | 2 | no | — |
| 345 | C2 | — | 424 | 1 | no | — |
| 346 | C2 | C2×#16811524 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20291006 | C2 | semidirect |
| 2. #20290169 | C2 | semidirect |
| 3. #20289837 | C2 | semidirect |
| 8. #16811531 | C22 | semidirect |
| 10. #16811478 | C22 | semidirect |
| 193. C53 | #13564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291006 | 2 |
| #20290169 | 2 |
| #20289837 | 2 |
| #20284068 | 2 |
| #13564 | 53 |