Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
570 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 135 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 | #20287621 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20287338 | C2 | 1 | 2 | yes | maximal |
| 3 | #20274331 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20274241 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20274237 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20273858 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #20273852 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #20273602 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C53×#1633 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16811339 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16811333 | C22 | 1 | 2 | yes | — |
| 12 | #16811029 | — | 2 | 2 | no | — |
| 13 | #16811026 | C22 | 1 | 2 | yes | — |
| 14 | #16811026 | C22 | 1 | 2 | yes | — |
| 15 | #16811003 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16810962 | — | 2 | 2 | no | — |
| 17 | #16810938 | — | 2 | 2 | no | — |
| 18 | #16810930 | — | 2 | 2 | no | — |
| 19 | #16810918 | C22 | 1 | 2 | yes | — |
| 20 | #16810914 | C22 | 1 | 2 | yes | — |
| 21 | #16810910 | C22 | 1 | 2 | yes | — |
| 22 | #16810900 | — | 2 | 2 | no | — |
| 23 | #16810872 | — | 2 | 2 | no | — |
| 24 | #16810754 | C22 | 1 | 2 | yes | — |
| 25 | #16810721 | — | 2 | 2 | no | — |
| 26 | C4×#11719924 | C22 | 1 | 2 | yes | — |
| 27 | #16810700 | C22 | 1 | 2 | yes | — |
| 28 | #16810664 | — | 2 | 2 | no | — |
| 29 | #16810373 | C22 | 1 | 2 | yes | — |
| 30 | #16810372 | C22 | 1 | 1 | yes | characteristic |
| 31 | #16810371 | C22 | 1 | 2 | yes | — |
| 32 | #16810352 | C22 | 1 | 2 | yes | — |
| 33 | C8×#88322 | C22 | 1 | 1 | yes | characteristic |
| 34 | Q82×C53 | C22 | 1 | 1 | yes | characteristic |
| 35 | C53×#494 | C22 | 1 | 1 | yes | characteristic |
| 36 | C53×#477 | C22 | 1 | 2 | yes | — |
| 37 | C53×#333 | C22 | 1 | 1 | yes | characteristic |
| 38 | C53×Q8⋊C8 | C22 | 1 | 2 | yes | — |
| 39 | C2×Dic212 | — | 4 | 1 | no | — |
| 40 | C2×#11719924 | D4 | 1 | 2 | yes | — |
| 41 | C2×#11719912 | — | 2 | 2 | no | — |
| 42 | #13133217 | C23 | 1 | 2 | yes | — |
| 43 | #13133209 | — | 2 | 2 | no | — |
| 44 | #13133206 | — | 4 | 1 | no | — |
| 45 | #13133195 | — | 4 | 2 | no | — |
| 46 | #13133193 | — | 2 | 2 | no | — |
| 47 | #13133192 | — | 2 | 2 | no | — |
| 48 | C4×D212 | — | 2 | 2 | no | — |
| 49 | C4×D212 | C23 | 1 | 2 | yes | — |
| 50 | #13133175 | — | 2 | 2 | no | — |
| 51 | C4⋊C4×D53 | — | 2 | 2 | no | — |
| 52 | #13133150 | — | 2 | 2 | no | — |
| 53 | #13133150 | — | 2 | 2 | no | — |
| 54 | #13133150 | — | 2 | 2 | no | — |
| 55 | #13133150 | — | 2 | 2 | no | — |
| 56 | #13133150 | — | 2 | 2 | no | — |
| 57 | #13133149 | — | 2 | 2 | no | — |
| 58 | #13133148 | — | 2 | 2 | no | — |
| 59 | #13133148 | D4 | 1 | 2 | yes | — |
| 60 | C4⋊#88322 | — | 2 | 2 | no | — |
| 61 | C4⋊#88322 | C23 | 1 | 2 | yes | — |
| 62 | #13133141 | — | 2 | 2 | no | — |
| 63 | #13133141 | D4 | 1 | 2 | yes | — |
| 64 | #13133139 | — | 2 | 2 | no | — |
| 65 | #13133137 | — | 2 | 2 | no | — |
| 66 | #13133137 | D4 | 1 | 2 | yes | — |
| 67 | #13133129 | — | 2 | 2 | no | — |
| 68 | #13133128 | — | 2 | 2 | no | — |
| 69 | C4×#88322 | C23 | 1 | 2 | yes | — |
| 70 | C2×Q16×C53 | — | 4 | 1 | no | — |
| 71 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 72 | C4⋊Q8×C53 | — | 4 | 1 | no | — |
| 73 | C4⋊Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 74 | C4⋊Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4×Q8×C53 | — | 2 | 2 | no | — |
| 76 | C4×Q8×C53 | C23 | 1 | 2 | yes | — |
| 77 | #107×C53 | — | 2 | 2 | no | — |
| 78 | C4⋊C8×C53 | C23 | 1 | 2 | yes | — |
| 79 | #103×C53 | — | 2 | 2 | no | — |
| 80 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 81 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 82 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 83 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 84 | C2×Q8×C53 | C2×D4 | 1 | 2 | yes | — |
| 85 | C2×D212 | — | 4 | 1 | no | — |
| 86 | C2×D212 | C2×D4 | 1 | 2 | yes | — |
| 87 | C2×C4×D53 | — | 2 | 2 | no | — |
| 88 | C2×C4×D53 | — | 4 | 2 | no | — |
| 89 | C2×Dic106 | — | 2 | 2 | no | — |
| 90 | Q16×C53 | — | 8 | 2 | no | — |
| 91 | C2×C8×C53 | — | 2 | 1 | no | — |
| 92 | C2×C8×C53 | — | 2 | 2 | no | — |
| 93 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 94 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 95 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 96 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 97 | C4⋊C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 98 | C42×C53 | — | 2 | 2 | no | — |
| 99 | C42×C53 | — | 4 | 1 | no | — |
| 100 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 101 | #11719924 | — | 2 | 4 | no | — |
| 102 | #11719921 | — | 2 | 2 | no | — |
| 103 | #11719921 | — | 4 | 2 | no | — |
| 104 | #11719921 | — | 4 | 2 | no | — |
| 105 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 106 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 107 | C4⋊Dic53 | C2×D4 | 1 | 2 | yes | — |
| 108 | #11719919 | — | 4 | 2 | no | — |
| 109 | #11719919 | — | 4 | 2 | no | — |
| 110 | C4×Dic53 | — | 4 | 1 | no | — |
| 111 | C2×#88322 | — | 2 | 2 | no | — |
| 112 | C2×#88322 | C2×D4 | 1 | 2 | yes | — |
| 113 | C2×#88322 | C2×D4 | 1 | 2 | yes | — |
| 114 | Dic212 | — | 8 | 2 | no | — |
| 115 | #11719912 | — | 4 | 4 | no | — |
| 116 | C22×D53 | — | 2 | 2 | no | — |
| 117 | Q8×C53 | — | 2 | 2 | no | — |
| 118 | Q8×C53 | — | 2 | 2 | no | — |
| 119 | Q8×C53 | — | 8 | 1 | no | — |
| 120 | Q8×C53 | — | 8 | 1 | no | — |
| 121 | Q8×C53 | — | 16 | 1 | no | — |
| 122 | C2×C4×C53 | — | 2 | 2 | no | — |
| 123 | C2×C4×C53 | — | 2 | 2 | no | — |
| 124 | C2×C4×C53 | — | 4 | 1 | no | — |
| 125 | C2×C4×C53 | — | 4 | 1 | no | — |
| 126 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 127 | C2×C4×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 128 | C2×Dic53 | — | 2 | 2 | no | — |
| 129 | C2×Dic53 | — | 4 | 2 | no | — |
| 130 | D212 | — | 2 | 2 | no | — |
| 131 | D212 | — | 2 | 2 | no | — |
| 132 | D212 | — | 16 | 1 | no | — |
| 133 | C4×D53 | — | 4 | 2 | no | — |
| 134 | C4×D53 | — | 4 | 2 | no | — |
| 135 | C4×D53 | — | 16 | 2 | no | — |
| 136 | Dic106 | — | 8 | 2 | no | — |
| 137 | C8×C53 | — | 2 | 2 | no | — |
| 138 | C8×C53 | — | 4 | 2 | no | — |
| 139 | #88322 | — | 2 | 2 | no | — |
| 140 | #88322 | — | 4 | 2 | no | — |
| 141 | #88322 | #137 | 1 | 4 | yes | — |
| 142 | #20324 | — | 53 | 1 | no | maximal, Sylow |
| 143 | C22×C53 | D42 | 1 | 1 | yes | characteristic |
| 144 | C2×D53 | — | 4 | 2 | no | — |
| 145 | C2×D53 | — | 4 | 2 | no | — |
| 146 | C2×D53 | — | 4 | 2 | no | — |
| 147 | C4×C53 | — | 2 | 1 | no | — |
| 148 | C4×C53 | — | 4 | 2 | no | — |
| 149 | C4×C53 | — | 4 | 2 | no | — |
| 150 | C4×C53 | — | 8 | 1 | no | — |
| 151 | C4×C53 | — | 8 | 1 | no | — |
| 152 | C4×C53 | C2×#137 | 1 | 2 | yes | — |
| 153 | C4×C53 | #578 | 1 | 2 | yes | — |
| 154 | Dic53 | — | 4 | 2 | no | — |
| 155 | Dic53 | — | 8 | 2 | no | — |
| 156 | #3385 | — | 53 | 2 | no | — |
| 157 | #3357 | — | 53 | 2 | no | — |
| 158 | #1713 | — | 53 | 1 | no | — |
| 159 | #1696 | — | 53 | 2 | no | — |
| 160 | #1695 | — | 53 | 2 | no | — |
| 161 | #1634 | — | 53 | 1 | no | — |
| 162 | #1633 | D53 | 1 | 1 | yes | characteristic, complemented |
| 163 | #1632 | — | 53 | 2 | no | — |
| 164 | #1590 | — | 53 | 2 | no | — |
| 165 | C2×C53 | #3276 | 1 | 2 | yes | — |
| 166 | C2×C53 | #3292 | 1 | 1 | yes | characteristic |
| 167 | D53 | — | 8 | 2 | no | — |
| 168 | Q82 | C2×D53 | 1 | 1 | yes | characteristic |
| 169 | #555 | — | 53 | 1 | no | — |
| 170 | #554 | — | 53 | 2 | no | — |
| 171 | #501 | — | 106 | 2 | no | — |
| 172 | #500 | — | 53 | 2 | no | — |
| 173 | #500 | — | 53 | 2 | no | — |
| 174 | #494 | — | 53 | 1 | no | — |
| 175 | #494 | C2×D53 | 1 | 1 | yes | characteristic |
| 176 | #485 | — | 106 | 2 | no | — |
| 177 | #480 | — | 106 | 2 | no | — |
| 178 | #479 | — | 106 | 2 | no | — |
| 179 | #477 | — | 53 | 2 | no | — |
| 180 | #477 | C2×D53 | 1 | 2 | yes | — |
| 181 | #476 | — | 53 | 2 | no | — |
| 182 | #476 | — | 53 | 2 | no | — |
| 183 | #474 | — | 106 | 2 | no | — |
| 184 | #468 | — | 106 | 2 | no | — |
| 185 | #446 | — | 53 | 2 | no | — |
| 186 | #440 | — | 106 | 2 | no | — |
| 187 | C4×SD16 | — | 53 | 2 | no | — |
| 188 | #436 | — | 53 | 2 | no | — |
| 189 | #428 | — | 106 | 2 | no | — |
| 190 | #334 | — | 53 | 2 | no | — |
| 191 | #333 | — | 53 | 1 | no | — |
| 192 | #333 | — | 53 | 2 | no | — |
| 193 | #333 | C2×D53 | 1 | 1 | yes | characteristic |
| 194 | Q8⋊C8 | C2×D53 | 1 | 2 | yes | — |
| 195 | D4⋊C8 | — | 53 | 2 | no | — |
| 196 | C82 | — | 53 | 1 | no | — |
| 197 | C53 | #20324 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 198 | C2×Q16 | — | 4 | 1 | no | — |
| 199 | C2×Q16 | — | 212 | 1 | no | — |
| 200 | C2×SD16 | — | 53 | 2 | no | — |
| 201 | C2×M16 | — | 106 | 2 | no | — |
| 202 | C4⋊Q8 | — | 2 | 2 | no | — |
| 203 | C4⋊Q8 | — | 4 | 1 | no | — |
| 204 | C4⋊Q8 | — | 53 | 2 | no | — |
| 205 | C4⋊Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 206 | C4⋊Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 207 | #125 | — | 106 | 2 | no | — |
| 208 | #124 | — | 212 | 1 | no | — |
| 209 | C22⋊Q8 | — | 106 | 2 | no | — |
| 210 | C22⋊Q8 | — | 212 | 2 | no | — |
| 211 | #121 | — | 106 | 2 | no | — |
| 212 | C4×Q8 | — | 2 | 2 | no | — |
| 213 | C4×Q8 | C22×D53 | 1 | 2 | yes | — |
| 214 | C4×D4 | — | 53 | 2 | no | — |
| 215 | C4×D4 | — | 106 | 2 | no | — |
| 216 | #117 | — | 106 | 2 | no | — |
| 217 | C2×C4⋊C4 | — | 106 | 2 | no | — |
| 218 | #107 | — | 2 | 2 | no | — |
| 219 | #107 | — | 106 | 2 | no | — |
| 220 | #107 | — | 106 | 2 | no | — |
| 221 | #107 | — | 106 | 2 | no | — |
| 222 | #107 | — | 106 | 2 | no | — |
| 223 | #107 | — | 106 | 2 | no | — |
| 224 | #106 | — | 53 | 2 | no | — |
| 225 | #106 | — | 106 | 2 | no | — |
| 226 | #106 | — | 106 | 2 | no | — |
| 227 | C4⋊C8 | — | 53 | 2 | no | — |
| 228 | C4⋊C8 | — | 106 | 2 | no | — |
| 229 | C4⋊C8 | C22×D53 | 1 | 2 | yes | — |
| 230 | #103 | — | 2 | 2 | no | — |
| 231 | #103 | — | 53 | 2 | no | — |
| 232 | #103 | — | 106 | 2 | no | — |
| 233 | #102 | — | 53 | 2 | no | — |
| 234 | #102 | — | 106 | 2 | no | — |
| 235 | #102 | — | 106 | 2 | no | — |
| 236 | C22⋊C8 | — | 106 | 2 | no | — |
| 237 | #97 | — | 106 | 2 | no | — |
| 238 | C4×C8 | — | 53 | 2 | no | — |
| 239 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 240 | C2×Q8 | — | 2 | 1 | no | — |
| 241 | C2×Q8 | — | 2 | 1 | no | — |
| 242 | C2×Q8 | — | 4 | 1 | no | — |
| 243 | C2×Q8 | — | 106 | 2 | no | — |
| 244 | C2×Q8 | D4×D53 | 1 | 2 | yes | — |
| 245 | C2×D4 | — | 53 | 2 | no | — |
| 246 | C2×D4 | — | 212 | 1 | no | — |
| 247 | C22×C4 | — | 106 | 2 | no | — |
| 248 | C22×C4 | — | 212 | 2 | no | — |
| 249 | Q16 | — | 8 | 2 | no | — |
| 250 | Q16 | — | 424 | 2 | no | — |
| 251 | SD16 | — | 106 | 4 | no | — |
| 252 | M16 | — | 212 | 4 | no | — |
| 253 | C2×C8 | — | 2 | 1 | no | — |
| 254 | C2×C8 | — | 2 | 2 | no | — |
| 255 | C2×C8 | — | 53 | 2 | no | — |
| 256 | C2×C8 | — | 53 | 2 | no | — |
| 257 | C2×C8 | — | 106 | 2 | no | — |
| 258 | C4⋊C4 | — | 2 | 2 | no | — |
| 259 | C4⋊C4 | — | 2 | 2 | no | — |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 4 | 2 | no | — |
| 262 | C4⋊C4 | — | 53 | 2 | no | — |
| 263 | C4⋊C4 | — | 106 | 2 | no | — |
| 264 | C4⋊C4 | — | 106 | 2 | no | — |
| 265 | C4⋊C4 | — | 212 | 2 | no | — |
| 266 | C4⋊C4 | — | 212 | 2 | no | — |
| 267 | C4⋊C4 | D4×D53 | 1 | 2 | yes | — |
| 268 | C22⋊C4 | — | 106 | 2 | no | — |
| 269 | C22⋊C4 | — | 212 | 2 | no | — |
| 270 | C22⋊C4 | — | 212 | 2 | no | — |
| 271 | C42 | — | 2 | 2 | no | — |
| 272 | C42 | — | 4 | 1 | no | — |
| 273 | C42 | — | 212 | 1 | no | — |
| 274 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 275 | C23 | — | 106 | 2 | no | — |
| 276 | Q8 | — | 2 | 2 | no | — |
| 277 | Q8 | — | 2 | 2 | no | — |
| 278 | Q8 | — | 8 | 1 | no | — |
| 279 | Q8 | — | 8 | 1 | no | — |
| 280 | Q8 | — | 16 | 1 | no | — |
| 281 | Q8 | — | 424 | 2 | no | — |
| 282 | D4 | — | 848 | 1 | no | — |
| 283 | D4 | — | 106 | 2 | no | — |
| 284 | D4 | — | 106 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 1 | no | — |
| 288 | C2×C4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 848 | 2 | no | — |
| 290 | C2×C4 | — | 106 | 2 | no | — |
| 291 | C2×C4 | — | 212 | 2 | no | — |
| 292 | C2×C4 | — | 212 | 2 | no | — |
| 293 | C2×C4 | — | 212 | 2 | no | — |
| 294 | C2×C4 | C2×D4×D53 | 1 | 2 | yes | — |
| 295 | C2×C4 | #13133262 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 2 | no | — |
| 297 | C8 | — | 4 | 2 | no | — |
| 298 | C8 | — | 53 | 4 | no | — |
| 299 | C8 | — | 106 | 2 | no | — |
| 300 | C8 | — | 212 | 2 | no | — |
| 301 | C22 | — | 212 | 2 | no | — |
| 302 | C22 | — | 212 | 2 | no | — |
| 303 | C22 | — | 212 | 2 | no | — |
| 304 | C22 | #16811255 | 1 | 1 | yes | characteristic |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 4 | 2 | no | — |
| 307 | C4 | — | 4 | 2 | no | — |
| 308 | C4 | — | 8 | 1 | no | — |
| 309 | C4 | — | 8 | 1 | no | — |
| 310 | C4 | — | 424 | 2 | no | — |
| 311 | C4 | — | 212 | 2 | no | — |
| 312 | C4 | #137×D53 | 1 | 2 | yes | — |
| 313 | C4 | #16811436 | 1 | 2 | yes | — |
| 314 | C2 | — | 424 | 2 | no | — |
| 315 | C2 | #20286354 | 1 | 2 | yes | — |
| 316 | C2 | #20286511 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20287621 | C2 | semidirect |
| 3. #20274331 | C2 | semidirect |
| 4. #20274241 | C2 | semidirect |
| 5. #20274237 | C2 | semidirect |
| 7. #20273852 | C2 | semidirect |
| 8. #20273602 | C2 | semidirect |
| 9. C53×#1633 | C2 | semidirect |
| 162. #1633 | D53 | semidirect |
| 197. C53 | #20324 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20287621 | 2 |
| #20287338 | 2 |
| #20274331 | 2 |
| #20274241 | 2 |
| #20274237 | 2 |
| #20273858 | 2 |
| #20273852 | 2 |
| #20273602 | 2 |
| C53×#1633 | 2 |
| #20324 | 53 |