Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
416 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #20291601 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20290716 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20290687 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20289802 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276379 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16811701 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16811655 | — | 2 | 2 | no | — |
| 8 | #16811652 | — | 2 | 2 | no | — |
| 9 | #16811645 | — | 2 | 1 | no | — |
| 10 | #16811629 | — | 2 | 2 | no | — |
| 11 | #16811622 | — | 2 | 1 | no | — |
| 12 | #16811603 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16811598 | — | 2 | 1 | no | — |
| 14 | #16811596 | C22 | 1 | 2 | yes | — |
| 15 | #16811504 | — | 2 | 2 | no | — |
| 16 | #16811494 | — | 2 | 1 | no | — |
| 17 | #16811172 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16810595 | C22 | 1 | 2 | yes | — |
| 19 | #16810418 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #16810415 | C4 | 1 | 2 | yes | complemented |
| 21 | C53×#354 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #13133304 | — | 4 | 1 | no | — |
| 23 | #13133304 | — | 4 | 1 | no | — |
| 24 | C2×#11719936 | — | 2 | 1 | no | — |
| 25 | D53⋊M16 | — | 2 | 1 | no | — |
| 26 | C2×#11719940 | — | 2 | 2 | no | — |
| 27 | C2×#11719940 | D4 | 1 | 2 | yes | — |
| 28 | C2×D53⋊C8 | — | 2 | 1 | no | — |
| 29 | C22×#88324 | D4 | 1 | 2 | yes | — |
| 30 | #13133296 | — | 2 | 1 | no | — |
| 31 | #13133284 | — | 4 | 2 | no | — |
| 32 | C4⋊#88324 | — | 2 | 1 | no | — |
| 33 | C4⋊#88324 | — | 2 | 1 | no | — |
| 34 | C4⋊#88324 | — | 4 | 2 | no | — |
| 35 | #13133282 | — | 4 | 1 | no | — |
| 36 | #13133282 | — | 4 | 1 | no | — |
| 37 | #13133275 | — | 2 | 2 | no | — |
| 38 | #13133274 | — | 2 | 2 | no | — |
| 39 | #13133274 | — | 2 | 2 | no | — |
| 40 | C22⋊#88324 | — | 2 | 2 | no | — |
| 41 | #13133272 | — | 2 | 2 | no | — |
| 42 | C2×#11719949 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×#11719947 | C23 | 1 | 1 | yes | characteristic |
| 44 | #13133215 | Q8 | 1 | 1 | yes | characteristic |
| 45 | #13133213 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #13133205 | Q8 | 1 | 1 | yes | characteristic |
| 47 | #13133190 | — | 2 | 2 | no | — |
| 48 | #13133177 | — | 2 | 2 | no | — |
| 49 | #13133173 | — | 2 | 2 | no | — |
| 50 | C42×D53 | D4 | 1 | 1 | yes | characteristic |
| 51 | #13133132 | C2×C4 | 1 | 2 | yes | — |
| 52 | #13133131 | — | 2 | 2 | no | — |
| 53 | #124×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #99×C53 | C2×C4 | 1 | 2 | yes | — |
| 55 | C2×Q8×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C2×D4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 57 | C2×C22⋊D53 | — | 2 | 2 | no | — |
| 58 | C2×C22⋊D53 | #41 | 1 | 2 | yes | — |
| 59 | C22×Dic53 | C2×D4 | 1 | 2 | yes | — |
| 60 | #11719949 | — | 4 | 1 | no | — |
| 61 | #11719949 | — | 4 | 1 | no | — |
| 62 | #11719947 | — | 4 | 1 | no | — |
| 63 | #11719947 | — | 4 | 1 | no | — |
| 64 | C2×D212 | — | 2 | 1 | no | — |
| 65 | C2×D212 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×D53 | — | 2 | 1 | no | — |
| 67 | C2×C4×D53 | — | 2 | 1 | no | — |
| 68 | C2×C4×D53 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C2×Dic106 | C2×Q8 | 1 | 1 | yes | characteristic |
| 70 | C22⋊#6157 | — | 4 | 1 | no | — |
| 71 | #11719940 | — | 4 | 2 | no | — |
| 72 | #11719940 | — | 4 | 1 | no | — |
| 73 | #11719940 | — | 4 | 2 | no | — |
| 74 | C2×#88324 | — | 2 | 2 | no | — |
| 75 | C2×#88324 | — | 2 | 2 | no | — |
| 76 | C2×#88324 | — | 2 | 2 | no | — |
| 77 | C2×#88324 | — | 2 | 2 | no | — |
| 78 | C2×#88324 | — | 4 | 2 | no | — |
| 79 | C4⋊#6157 | — | 4 | 1 | no | — |
| 80 | C4×#6157 | — | 4 | 1 | no | — |
| 81 | #11719936 | — | 2 | 1 | no | — |
| 82 | #11719936 | — | 2 | 1 | no | — |
| 83 | #11719936 | — | 8 | 1 | no | — |
| 84 | D53⋊C8 | — | 4 | 1 | no | — |
| 85 | D53⋊C8 | — | 8 | 1 | no | — |
| 86 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 87 | C22⋊C4×C53 | — | 2 | 2 | no | — |
| 88 | C42×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 90 | #11719921 | — | 2 | 2 | no | — |
| 91 | #11719919 | — | 2 | 2 | no | — |
| 92 | #11719919 | — | 4 | 2 | no | — |
| 93 | C4×Dic53 | — | 2 | 1 | no | — |
| 94 | C4×Dic53 | — | 4 | 2 | no | — |
| 95 | C4×Dic53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C23×C53 | C4×D4 | 1 | 2 | yes | — |
| 97 | C22×D53 | — | 2 | 1 | no | — |
| 98 | C22×D53 | #123 | 1 | 1 | yes | characteristic |
| 99 | C2×#6157 | — | 8 | 1 | no | — |
| 100 | Q8×C53 | — | 4 | 1 | no | — |
| 101 | D4×C53 | — | 4 | 1 | no | — |
| 102 | C2×C4×C53 | — | 2 | 1 | no | — |
| 103 | C2×C4×C53 | — | 2 | 1 | no | — |
| 104 | C2×C4×C53 | — | 4 | 2 | no | — |
| 105 | C2×C4×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 106 | C22⋊D53 | — | 4 | 2 | no | — |
| 107 | C22⋊D53 | — | 8 | 2 | no | — |
| 108 | C2×Dic53 | — | 2 | 2 | no | — |
| 109 | C2×Dic53 | — | 2 | 2 | no | — |
| 110 | C2×Dic53 | — | 2 | 1 | no | — |
| 111 | C2×Dic53 | — | 4 | 2 | no | — |
| 112 | C2×Dic53 | #120 | 1 | 1 | yes | characteristic |
| 113 | C2×Dic53 | C22⋊Q8 | 1 | 2 | yes | — |
| 114 | D212 | — | 4 | 1 | no | — |
| 115 | D212 | — | 8 | 1 | no | — |
| 116 | C4×D53 | — | 2 | 1 | no | — |
| 117 | C4×D53 | — | 2 | 1 | no | — |
| 118 | C4×D53 | — | 4 | 1 | no | — |
| 119 | C4×D53 | — | 4 | 1 | no | — |
| 120 | C4×D53 | — | 8 | 1 | no | — |
| 121 | Dic106 | — | 4 | 1 | no | — |
| 122 | #88324 | — | 4 | 1 | no | — |
| 123 | #88324 | — | 4 | 1 | no | — |
| 124 | #88324 | — | 4 | 2 | no | — |
| 125 | #88324 | — | 4 | 2 | no | — |
| 126 | #12969 | — | 53 | 1 | no | maximal, Sylow |
| 127 | C22×C53 | — | 2 | 2 | no | — |
| 128 | C22×C53 | — | 4 | 2 | no | — |
| 129 | C22×C53 | #385 | 1 | 1 | yes | characteristic |
| 130 | C2×D53 | — | 2 | 1 | no | — |
| 131 | C2×D53 | — | 4 | 1 | no | — |
| 132 | C2×D53 | — | 4 | 2 | no | — |
| 133 | C2×D53 | — | 4 | 1 | no | — |
| 134 | #6157 | — | 16 | 1 | no | — |
| 135 | C4×C53 | — | 2 | 1 | no | — |
| 136 | C4×C53 | — | 4 | 1 | no | — |
| 137 | C4×C53 | — | 4 | 1 | no | — |
| 138 | C4×C53 | — | 8 | 2 | no | — |
| 139 | Dic53 | — | 2 | 2 | no | — |
| 140 | Dic53 | — | 4 | 1 | no | — |
| 141 | Dic53 | — | 8 | 2 | no | — |
| 142 | Dic53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 143 | Dic53 | #458 | 1 | 1 | yes | characteristic |
| 144 | #2980 | — | 53 | 1 | no | — |
| 145 | #2126 | — | 53 | 1 | no | — |
| 146 | #1884 | — | 53 | 1 | no | — |
| 147 | #1868 | — | 53 | 2 | no | — |
| 148 | #1391 | — | 53 | 2 | no | — |
| 149 | C2×C53 | — | 2 | 1 | no | — |
| 150 | C2×C53 | — | 4 | 2 | no | — |
| 151 | C2×C53 | #1867 | 1 | 1 | yes | characteristic |
| 152 | D53 | — | 4 | 1 | no | — |
| 153 | D53 | — | 8 | 1 | no | — |
| 154 | #568 | — | 53 | 1 | no | — |
| 155 | #532 | — | 53 | 1 | no | — |
| 156 | #435 | — | 106 | 2 | no | — |
| 157 | #435 | — | 106 | 2 | no | — |
| 158 | #432 | — | 106 | 1 | no | — |
| 159 | #424 | — | 106 | 2 | no | — |
| 160 | #421 | — | 106 | 1 | no | — |
| 161 | #410 | — | 53 | 2 | no | — |
| 162 | #410 | — | 53 | 1 | no | — |
| 163 | #408 | — | 53 | 2 | no | — |
| 164 | #408 | — | 106 | 1 | no | — |
| 165 | #354 | — | 53 | 1 | no | — |
| 166 | #354 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 167 | #353 | — | 53 | 2 | no | — |
| 168 | #344 | — | 106 | 2 | no | — |
| 169 | #339 | — | 106 | 1 | no | — |
| 170 | C53 | #12969 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C2×#41 | — | 53 | 1 | no | — |
| 172 | C2×#41 | — | 53 | 1 | no | — |
| 173 | #131 | — | 212 | 1 | no | — |
| 174 | #131 | — | 212 | 1 | no | — |
| 175 | C2×M16 | — | 53 | 2 | no | — |
| 176 | C2×M16 | — | 106 | 1 | no | — |
| 177 | C2×M16 | — | 106 | 2 | no | — |
| 178 | C2×M16 | — | 106 | 1 | no | — |
| 179 | C22×C8 | — | 53 | 2 | no | — |
| 180 | C22×C8 | — | 106 | 1 | no | — |
| 181 | C4⋊Q8 | — | 53 | 1 | no | — |
| 182 | #127 | — | 53 | 1 | no | — |
| 183 | #124 | — | 53 | 1 | no | — |
| 184 | #124 | C2×#6157 | 1 | 1 | yes | characteristic |
| 185 | #121 | — | 106 | 2 | no | — |
| 186 | C4×D4 | — | 106 | 2 | no | — |
| 187 | #117 | — | 106 | 2 | no | — |
| 188 | #117 | — | 106 | 1 | no | — |
| 189 | C2×C42 | — | 53 | 1 | no | — |
| 190 | C4⋊C8 | — | 106 | 1 | no | — |
| 191 | C4⋊C8 | — | 106 | 1 | no | — |
| 192 | C4⋊C8 | — | 212 | 2 | no | — |
| 193 | C4⋊C8 | — | 212 | 2 | no | — |
| 194 | #104 | — | 212 | 1 | no | — |
| 195 | #104 | — | 212 | 1 | no | — |
| 196 | #99 | — | 53 | 2 | no | — |
| 197 | #99 | — | 106 | 2 | no | — |
| 198 | #99 | — | 106 | 2 | no | — |
| 199 | #99 | C2×#6157 | 1 | 2 | yes | — |
| 200 | C22⋊C8 | — | 106 | 2 | no | — |
| 201 | C22⋊C8 | — | 106 | 2 | no | — |
| 202 | C22⋊C8 | — | 106 | 2 | no | — |
| 203 | #97 | — | 106 | 2 | no | — |
| 204 | #41 | — | 212 | 1 | no | — |
| 205 | #41 | — | 212 | 1 | no | — |
| 206 | #41 | — | 212 | 1 | no | — |
| 207 | #41 | — | 212 | 1 | no | — |
| 208 | C2×Q8 | — | 53 | 1 | no | — |
| 209 | C2×Q8 | C4⋊#6157 | 1 | 1 | yes | characteristic |
| 210 | C2×D4 | — | 53 | 2 | no | — |
| 211 | C2×D4 | — | 53 | 1 | no | — |
| 212 | C2×D4 | — | 106 | 1 | no | — |
| 213 | C2×D4 | — | 106 | 2 | no | — |
| 214 | C2×D4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 215 | C22×C4 | — | 53 | 2 | no | — |
| 216 | C22×C4 | — | 53 | 1 | no | — |
| 217 | C22×C4 | — | 106 | 1 | no | — |
| 218 | C22×C4 | — | 106 | 1 | no | — |
| 219 | M16 | — | 106 | 1 | no | — |
| 220 | M16 | — | 106 | 1 | no | — |
| 221 | M16 | — | 424 | 1 | no | — |
| 222 | M16 | — | 212 | 1 | no | — |
| 223 | M16 | — | 212 | 2 | no | — |
| 224 | M16 | — | 212 | 2 | no | — |
| 225 | C2×C8 | — | 106 | 2 | no | — |
| 226 | C2×C8 | — | 106 | 2 | no | — |
| 227 | C2×C8 | — | 106 | 2 | no | — |
| 228 | C2×C8 | — | 106 | 2 | no | — |
| 229 | C2×C8 | — | 424 | 1 | no | — |
| 230 | C2×C8 | — | 212 | 1 | no | — |
| 231 | C2×C8 | — | 212 | 2 | no | — |
| 232 | C4⋊C4 | — | 106 | 2 | no | — |
| 233 | C4⋊C4 | — | 212 | 2 | no | — |
| 234 | C4⋊C4 | — | 212 | 1 | no | — |
| 235 | C22⋊C4 | — | 2 | 2 | no | — |
| 236 | C22⋊C4 | — | 2 | 2 | no | — |
| 237 | C22⋊C4 | — | 106 | 2 | no | — |
| 238 | C22⋊C4 | — | 106 | 2 | no | — |
| 239 | C22⋊C4 | — | 212 | 1 | no | — |
| 240 | C42 | — | 53 | 1 | no | — |
| 241 | C42 | — | 106 | 1 | no | — |
| 242 | C42 | — | 212 | 2 | no | — |
| 243 | C42 | — | 212 | 1 | no | — |
| 244 | C42 | C4⋊#6157 | 1 | 1 | yes | characteristic |
| 245 | C23 | — | 53 | 1 | no | — |
| 246 | C23 | — | 106 | 1 | no | — |
| 247 | C23 | D4×#6157 | 1 | 2 | yes | — |
| 248 | Q8 | — | 4 | 1 | no | — |
| 249 | Q8 | — | 212 | 1 | no | — |
| 250 | D4 | — | 4 | 1 | no | — |
| 251 | D4 | — | 424 | 1 | no | — |
| 252 | D4 | — | 424 | 2 | no | — |
| 253 | D4 | — | 212 | 1 | no | — |
| 254 | D4 | — | 212 | 2 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 4 | 2 | no | — |
| 258 | C2×C4 | — | 53 | 1 | no | — |
| 259 | C2×C4 | — | 53 | 2 | no | — |
| 260 | C2×C4 | — | 106 | 1 | no | — |
| 261 | C2×C4 | — | 106 | 1 | no | — |
| 262 | C2×C4 | — | 106 | 2 | no | — |
| 263 | C2×C4 | — | 106 | 2 | no | — |
| 264 | C2×C4 | — | 106 | 1 | no | — |
| 265 | C2×C4 | — | 424 | 1 | no | — |
| 266 | C2×C4 | — | 424 | 1 | no | — |
| 267 | C2×C4 | — | 212 | 1 | no | — |
| 268 | C2×C4 | — | 212 | 1 | no | — |
| 269 | C2×C4 | — | 212 | 2 | no | — |
| 270 | C2×C4 | C2×C4⋊#6157 | 1 | 1 | yes | characteristic |
| 271 | C8 | — | 212 | 1 | no | — |
| 272 | C8 | — | 212 | 1 | no | — |
| 273 | C8 | — | 212 | 2 | no | — |
| 274 | C8 | — | 212 | 2 | no | — |
| 275 | C22 | — | 2 | 2 | no | — |
| 276 | C22 | — | 4 | 2 | no | — |
| 277 | C22 | — | 106 | 1 | no | — |
| 278 | C22 | — | 212 | 2 | no | — |
| 279 | C22 | — | 212 | 1 | no | — |
| 280 | C22 | — | 212 | 1 | no | — |
| 281 | C22 | #16811570 | 1 | 1 | yes | characteristic |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 4 | 1 | no | — |
| 284 | C4 | — | 4 | 1 | no | — |
| 285 | C4 | — | 8 | 2 | no | — |
| 286 | C4 | — | 53 | 1 | no | — |
| 287 | C4 | — | 53 | 1 | no | — |
| 288 | C4 | — | 848 | 1 | no | — |
| 289 | C4 | — | 106 | 2 | no | — |
| 290 | C4 | — | 424 | 2 | no | — |
| 291 | C4 | — | 212 | 1 | no | — |
| 292 | C2 | — | 2 | 1 | no | — |
| 293 | C2 | — | 4 | 2 | no | — |
| 294 | C2 | — | 424 | 1 | no | — |
| 295 | C2 | — | 212 | 1 | no | — |
| 296 | C2 | #20290686 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20290687 | C2 | semidirect |
| 4. #20289802 | C2 | semidirect |
| 19. #16810418 | C4 | semidirect |
| 20. #16810415 | C4 | semidirect |
| 21. C53×#354 | C4 | semidirect |
| 166. #354 | #6157 | semidirect |
| 170. C53 | #12969 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291601 | 2 |
| #20290716 | 2 |
| #20290687 | 2 |
| #20289802 | 2 |
| #20276379 | 2 |
| #12969 | 53 |