Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
868 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 163 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 | #20289129 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20286109 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20285037 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20273937 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #20273902 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #20273169 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C53×#1560 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811373 | — | 2 | 2 | no | — |
| 9 | #16811359 | C22 | 1 | 2 | yes | — |
| 10 | #16811353 | C22 | 1 | 2 | yes | — |
| 11 | #16811343 | — | 2 | 2 | no | — |
| 12 | D4×Dic106 | — | 2 | 2 | no | — |
| 13 | #16811240 | — | 2 | 4 | no | — |
| 14 | #16811194 | — | 2 | 2 | no | — |
| 15 | #16811180 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16811180 | C22 | 1 | 2 | yes | — |
| 17 | #16810977 | — | 2 | 2 | no | — |
| 18 | #16810973 | C22 | 1 | 2 | yes | — |
| 19 | #16810969 | C22 | 1 | 2 | yes | — |
| 20 | #16810909 | — | 2 | 2 | no | — |
| 21 | #16810909 | C22 | 1 | 4 | yes | — |
| 22 | #16810881 | — | 2 | 4 | no | — |
| 23 | #16810834 | — | 2 | 2 | no | — |
| 24 | #16810834 | C22 | 1 | 4 | yes | — |
| 25 | #16810619 | — | 2 | 2 | no | — |
| 26 | C2×C4×#88322 | C22 | 1 | 1 | yes | characteristic |
| 27 | #16810353 | C22 | 1 | 8 | yes | — |
| 28 | C53×#532 | C22 | 1 | 1 | yes | characteristic |
| 29 | C53×#423 | C22 | 1 | 2 | yes | — |
| 30 | C53×#333 | C22 | 1 | 2 | yes | — |
| 31 | C53×#331 | C22 | 1 | 2 | yes | — |
| 32 | C22×Dic106 | — | 2 | 2 | no | — |
| 33 | C2×Dic212 | — | 2 | 4 | no | — |
| 34 | C22×#88322 | — | 2 | 1 | no | — |
| 35 | #13133217 | D4 | 1 | 2 | yes | — |
| 36 | #13133217 | D4 | 1 | 2 | yes | — |
| 37 | #13133217 | C23 | 1 | 2 | yes | — |
| 38 | #13133216 | — | 2 | 4 | no | — |
| 39 | #13133215 | — | 2 | 4 | no | — |
| 40 | #13133210 | — | 2 | 8 | no | — |
| 41 | #13133209 | — | 2 | 2 | no | — |
| 42 | #13133209 | — | 2 | 1 | no | — |
| 43 | #13133204 | — | 2 | 4 | no | — |
| 44 | #13133204 | — | 2 | 4 | no | — |
| 45 | #13133198 | — | 2 | 4 | no | — |
| 46 | #13133198 | — | 2 | 4 | no | — |
| 47 | #13133196 | — | 2 | 4 | no | — |
| 48 | #13133196 | — | 2 | 2 | no | — |
| 49 | #13133196 | — | 2 | 2 | no | — |
| 50 | #13133196 | D4 | 1 | 4 | yes | — |
| 51 | #13133194 | — | 2 | 8 | no | — |
| 52 | C4×Dic106 | — | 2 | 2 | no | — |
| 53 | C4×Dic106 | — | 2 | 2 | no | — |
| 54 | C4×Dic106 | C23 | 1 | 4 | yes | — |
| 55 | #13133182 | — | 2 | 4 | no | — |
| 56 | D4×Dic53 | — | 2 | 4 | no | — |
| 57 | #13133176 | — | 2 | 2 | no | — |
| 58 | #13133176 | — | 2 | 2 | no | — |
| 59 | #13133176 | — | 2 | 1 | no | — |
| 60 | #13133149 | — | 2 | 4 | no | — |
| 61 | #13133142 | — | 2 | 4 | no | — |
| 62 | #13133142 | — | 2 | 4 | no | — |
| 63 | #13133141 | — | 2 | 4 | no | — |
| 64 | #13133138 | — | 2 | 4 | no | — |
| 65 | C4×#88322 | — | 2 | 1 | no | — |
| 66 | C4×#88322 | C23 | 1 | 2 | yes | — |
| 67 | C2×#41×C53 | D4 | 1 | 2 | yes | — |
| 68 | C2×M16×C53 | — | 2 | 2 | no | — |
| 69 | C4⋊Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 70 | #127×C53 | C23 | 1 | 1 | yes | characteristic |
| 71 | #124×C53 | D4 | 1 | 2 | yes | — |
| 72 | #121×C53 | — | 2 | 2 | no | — |
| 73 | C4×D4×C53 | — | 2 | 2 | no | — |
| 74 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 76 | C4⋊C8×C53 | C23 | 1 | 4 | yes | — |
| 77 | #41×C53 | — | 4 | 4 | no | — |
| 78 | C2×Q8×C53 | C2×D4 | 1 | 2 | yes | — |
| 79 | C2×D4×C53 | — | 2 | 2 | no | — |
| 80 | C2×D4×C53 | — | 2 | 2 | no | — |
| 81 | C2×D4×C53 | C2×D4 | 1 | 2 | yes | — |
| 82 | C22×C4×C53 | — | 2 | 2 | no | — |
| 83 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C22×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 85 | C22×Dic53 | — | 2 | 4 | no | — |
| 86 | C2×Dic106 | — | 2 | 2 | no | — |
| 87 | C2×Dic106 | — | 2 | 2 | no | — |
| 88 | C2×Dic106 | — | 2 | 4 | no | — |
| 89 | C2×Dic106 | — | 8 | 4 | no | — |
| 90 | C2×Dic106 | C2×D4 | 1 | 4 | yes | — |
| 91 | M16×C53 | — | 8 | 2 | no | — |
| 92 | C2×C8×C53 | — | 2 | 4 | no | — |
| 93 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 94 | C22⋊C4×C53 | — | 2 | 4 | no | — |
| 95 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C42×C53 | C2×D4 | 1 | 2 | yes | — |
| 97 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 98 | C22⋊Dic53 | — | 2 | 4 | no | — |
| 99 | C22⋊Dic53 | — | 2 | 8 | no | — |
| 100 | C22⋊Dic53 | — | 2 | 4 | no | — |
| 101 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 104 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 105 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 106 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 107 | C4⋊Dic53 | C2×D4 | 1 | 4 | yes | — |
| 108 | #11719919 | — | 2 | 8 | no | — |
| 109 | #11719919 | — | 2 | 4 | no | — |
| 110 | #11719919 | — | 2 | 4 | no | — |
| 111 | #11719919 | — | 4 | 2 | no | — |
| 112 | C4×Dic53 | — | 2 | 4 | no | — |
| 113 | C4×Dic53 | — | 2 | 4 | no | — |
| 114 | C4×Dic53 | — | 4 | 1 | no | — |
| 115 | C2×#88322 | — | 2 | 2 | no | — |
| 116 | C2×#88322 | — | 4 | 2 | no | — |
| 117 | Dic212 | — | 8 | 4 | no | — |
| 118 | C23×C53 | — | 2 | 2 | no | — |
| 119 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 120 | Q8×C53 | — | 4 | 2 | no | — |
| 121 | D4×C53 | — | 4 | 2 | no | — |
| 122 | D4×C53 | — | 4 | 2 | no | — |
| 123 | D4×C53 | — | 4 | 2 | no | — |
| 124 | D4×C53 | — | 8 | 2 | no | — |
| 125 | C2×C4×C53 | — | 2 | 2 | no | — |
| 126 | C2×C4×C53 | — | 2 | 2 | no | — |
| 127 | C2×C4×C53 | — | 2 | 2 | no | — |
| 128 | C2×C4×C53 | — | 2 | 2 | no | — |
| 129 | C2×C4×C53 | — | 4 | 4 | no | — |
| 130 | C2×C4×C53 | #120 | 1 | 2 | yes | — |
| 131 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 133 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 134 | C2×Dic53 | — | 2 | 4 | no | — |
| 135 | C2×Dic53 | — | 2 | 4 | no | — |
| 136 | C2×Dic53 | — | 4 | 2 | no | — |
| 137 | C2×Dic53 | — | 8 | 4 | no | — |
| 138 | Dic106 | — | 2 | 4 | no | — |
| 139 | Dic106 | — | 2 | 4 | no | — |
| 140 | Dic106 | — | 4 | 4 | no | — |
| 141 | Dic106 | — | 4 | 2 | no | — |
| 142 | Dic106 | — | 4 | 2 | no | — |
| 143 | Dic106 | — | 8 | 4 | no | — |
| 144 | C8×C53 | — | 4 | 4 | no | — |
| 145 | #88322 | — | 4 | 2 | no | — |
| 146 | #18693 | — | 53 | 1 | no | maximal, Sylow |
| 147 | C22×C53 | — | 2 | 2 | no | — |
| 148 | C22×C53 | — | 2 | 1 | no | — |
| 149 | C22×C53 | — | 4 | 4 | no | — |
| 150 | C22×C53 | — | 4 | 2 | no | — |
| 151 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 152 | C4×C53 | — | 2 | 2 | no | — |
| 153 | C4×C53 | — | 2 | 1 | no | — |
| 154 | C4×C53 | — | 4 | 2 | no | — |
| 155 | C4×C53 | #455 | 1 | 2 | yes | — |
| 156 | C4×C53 | #577 | 1 | 2 | yes | — |
| 157 | C4×C53 | #578 | 1 | 2 | yes | — |
| 158 | Dic53 | — | 4 | 4 | no | — |
| 159 | Dic53 | — | 4 | 4 | no | — |
| 160 | Dic53 | — | 8 | 2 | no | — |
| 161 | #3550 | — | 53 | 1 | no | — |
| 162 | #3255 | — | 53 | 2 | no | — |
| 163 | #3131 | — | 53 | 1 | no | — |
| 164 | #1645 | — | 53 | 4 | no | — |
| 165 | #1640 | — | 53 | 4 | no | — |
| 166 | #1560 | D53 | 1 | 1 | yes | characteristic |
| 167 | #1493 | — | 53 | 2 | no | — |
| 168 | C2×C53 | — | 4 | 1 | no | — |
| 169 | C2×C53 | — | 8 | 2 | no | — |
| 170 | C2×C53 | #3013 | 1 | 2 | yes | — |
| 171 | C2×C53 | C2×#455 | 1 | 1 | yes | characteristic |
| 172 | #563 | — | 106 | 2 | no | — |
| 173 | #559 | — | 53 | 2 | no | — |
| 174 | Q82 | — | 53 | 2 | no | — |
| 175 | #556 | — | 106 | 2 | no | — |
| 176 | D4×Q8 | — | 106 | 2 | no | — |
| 177 | #542 | — | 106 | 4 | no | — |
| 178 | #536 | — | 106 | 2 | no | — |
| 179 | #533 | — | 53 | 2 | no | — |
| 180 | #533 | — | 53 | 1 | no | — |
| 181 | #532 | C2×D53 | 1 | 1 | yes | characteristic |
| 182 | #488 | — | 106 | 2 | no | — |
| 183 | #487 | — | 53 | 2 | no | — |
| 184 | #486 | — | 53 | 2 | no | — |
| 185 | #475 | — | 53 | 4 | no | — |
| 186 | #475 | — | 106 | 2 | no | — |
| 187 | #470 | — | 106 | 4 | no | — |
| 188 | #462 | — | 53 | 4 | no | — |
| 189 | #462 | — | 106 | 2 | no | — |
| 190 | #423 | C2×D53 | 1 | 2 | yes | — |
| 191 | #416 | — | 106 | 2 | no | — |
| 192 | C2×C4×C8 | — | 53 | 1 | no | — |
| 193 | #333 | C2×D53 | 1 | 2 | yes | — |
| 194 | #331 | C2×D53 | 1 | 2 | yes | — |
| 195 | Q8⋊C8 | — | 53 | 8 | no | — |
| 196 | C53 | #18693 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 197 | C2×#41 | D212 | 1 | 2 | yes | — |
| 198 | C22×Q8 | — | 106 | 2 | no | — |
| 199 | C2×Q16 | — | 106 | 4 | no | — |
| 200 | C2×M16 | — | 2 | 2 | no | — |
| 201 | C22×C8 | — | 106 | 1 | no | — |
| 202 | C4⋊Q8 | — | 53 | 2 | no | — |
| 203 | C4⋊Q8 | — | 53 | 2 | no | — |
| 204 | C4⋊Q8 | — | 53 | 2 | no | — |
| 205 | C4⋊Q8 | — | 106 | 4 | no | — |
| 206 | C4⋊Q8 | — | 106 | 4 | no | — |
| 207 | C4⋊Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 208 | #127 | C22×D53 | 1 | 1 | yes | characteristic |
| 209 | #126 | — | 106 | 8 | no | — |
| 210 | #125 | — | 106 | 2 | no | — |
| 211 | #125 | — | 106 | 1 | no | — |
| 212 | #124 | — | 106 | 4 | no | — |
| 213 | #124 | — | 106 | 4 | no | — |
| 214 | #124 | D212 | 1 | 2 | yes | — |
| 215 | #123 | — | 106 | 4 | no | — |
| 216 | #123 | — | 106 | 4 | no | — |
| 217 | C22⋊Q8 | — | 53 | 4 | no | — |
| 218 | C22⋊Q8 | — | 106 | 8 | no | — |
| 219 | C22⋊Q8 | — | 106 | 4 | no | — |
| 220 | C22⋊Q8 | — | 106 | 2 | no | — |
| 221 | C22⋊Q8 | — | 106 | 2 | no | — |
| 222 | #121 | — | 2 | 2 | no | — |
| 223 | C4×Q8 | — | 53 | 4 | no | — |
| 224 | C4×Q8 | — | 106 | 4 | no | — |
| 225 | C4×Q8 | — | 106 | 2 | no | — |
| 226 | C4×Q8 | — | 106 | 2 | no | — |
| 227 | C4×D4 | — | 2 | 2 | no | — |
| 228 | C4×D4 | — | 106 | 4 | no | — |
| 229 | #117 | — | 106 | 2 | no | — |
| 230 | #117 | — | 106 | 2 | no | — |
| 231 | #117 | — | 106 | 1 | no | — |
| 232 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 233 | #106 | — | 106 | 4 | no | — |
| 234 | C4⋊C8 | — | 2 | 2 | no | — |
| 235 | C4⋊C8 | C22×D53 | 1 | 4 | yes | — |
| 236 | #103 | — | 106 | 4 | no | — |
| 237 | #103 | — | 106 | 4 | no | — |
| 238 | #103 | — | 106 | 4 | no | — |
| 239 | #102 | — | 106 | 4 | no | — |
| 240 | C4×C8 | — | 53 | 2 | no | — |
| 241 | C4×C8 | — | 106 | 1 | no | — |
| 242 | #41 | — | 4 | 4 | no | — |
| 243 | C2×Q8 | — | 53 | 4 | no | — |
| 244 | C2×Q8 | — | 106 | 4 | no | — |
| 245 | C2×Q8 | — | 106 | 2 | no | — |
| 246 | C2×Q8 | — | 106 | 2 | no | — |
| 247 | C2×Q8 | — | 424 | 4 | no | — |
| 248 | C2×Q8 | C2×D212 | 1 | 2 | yes | — |
| 249 | C2×D4 | — | 2 | 2 | no | — |
| 250 | C2×D4 | — | 2 | 2 | no | — |
| 251 | C2×D4 | C2×D212 | 1 | 2 | yes | — |
| 252 | C22×C4 | — | 2 | 2 | no | — |
| 253 | C22×C4 | — | 106 | 4 | no | — |
| 254 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 255 | C22×C4 | D4×D53 | 1 | 2 | yes | — |
| 256 | Q16 | — | 424 | 4 | no | — |
| 257 | M16 | — | 8 | 2 | no | — |
| 258 | C2×C8 | — | 2 | 4 | no | — |
| 259 | C2×C8 | — | 106 | 2 | no | — |
| 260 | C2×C8 | — | 212 | 2 | no | — |
| 261 | C4⋊C4 | — | 2 | 2 | no | — |
| 262 | C4⋊C4 | — | 53 | 4 | no | — |
| 263 | C4⋊C4 | — | 106 | 8 | no | — |
| 264 | C4⋊C4 | — | 106 | 2 | no | — |
| 265 | C4⋊C4 | — | 106 | 2 | no | — |
| 266 | C4⋊C4 | — | 106 | 2 | no | — |
| 267 | C4⋊C4 | — | 106 | 4 | no | — |
| 268 | C4⋊C4 | — | 106 | 4 | no | — |
| 269 | C4⋊C4 | — | 106 | 2 | no | — |
| 270 | C4⋊C4 | — | 106 | 2 | no | — |
| 271 | C4⋊C4 | — | 212 | 2 | no | — |
| 272 | C22⋊C4 | — | 2 | 4 | no | — |
| 273 | C22⋊C4 | — | 106 | 4 | no | — |
| 274 | C22⋊C4 | — | 106 | 8 | no | — |
| 275 | C22⋊C4 | — | 106 | 4 | no | — |
| 276 | C22⋊C4 | — | 212 | 1 | no | — |
| 277 | C42 | — | 106 | 4 | no | — |
| 278 | C42 | — | 106 | 4 | no | — |
| 279 | C42 | — | 212 | 1 | no | — |
| 280 | C42 | C2×D212 | 1 | 1 | yes | characteristic |
| 281 | C42 | D4×D53 | 1 | 2 | yes | — |
| 282 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 283 | C23 | — | 2 | 2 | no | — |
| 284 | C23 | #13133187 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 4 | 2 | no | — |
| 286 | Q8 | — | 106 | 4 | no | — |
| 287 | Q8 | — | 106 | 4 | no | — |
| 288 | Q8 | — | 424 | 4 | no | — |
| 289 | Q8 | — | 212 | 2 | no | — |
| 290 | Q8 | — | 212 | 2 | no | — |
| 291 | Q8 | — | 212 | 4 | no | — |
| 292 | D4 | — | 4 | 2 | no | — |
| 293 | D4 | — | 4 | 2 | no | — |
| 294 | D4 | — | 4 | 2 | no | — |
| 295 | D4 | — | 8 | 2 | no | — |
| 296 | C2×C4 | — | 2 | 2 | no | — |
| 297 | C2×C4 | — | 2 | 2 | no | — |
| 298 | C2×C4 | — | 2 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 2 | no | — |
| 300 | C2×C4 | — | 4 | 4 | no | — |
| 301 | C2×C4 | — | 106 | 4 | no | — |
| 302 | C2×C4 | — | 106 | 4 | no | — |
| 303 | C2×C4 | — | 424 | 4 | no | — |
| 304 | C2×C4 | — | 212 | 2 | no | — |
| 305 | C2×C4 | #13133187 | 1 | 2 | yes | — |
| 306 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | C2×D4×D53 | 1 | 2 | yes | — |
| 308 | C2×C4 | C22×D212 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 4 | 4 | no | — |
| 310 | C8 | — | 212 | 2 | no | — |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 2 | 1 | no | — |
| 313 | C22 | — | 4 | 4 | no | — |
| 314 | C22 | — | 4 | 2 | no | — |
| 315 | C22 | C2×#13133187 | 1 | 1 | yes | characteristic |
| 316 | C4 | — | 2 | 2 | no | — |
| 317 | C4 | — | 2 | 1 | no | — |
| 318 | C4 | — | 4 | 2 | no | — |
| 319 | C4 | — | 424 | 2 | no | — |
| 320 | C4 | — | 212 | 4 | no | — |
| 321 | C4 | — | 212 | 4 | no | — |
| 322 | C4 | #16810803 | 1 | 2 | yes | — |
| 323 | C4 | #16811433 | 1 | 2 | yes | — |
| 324 | C4 | #16811439 | 1 | 2 | yes | — |
| 325 | C2 | — | 4 | 1 | no | — |
| 326 | C2 | — | 8 | 2 | no | — |
| 327 | C2 | #20284032 | 1 | 2 | yes | — |
| 328 | C2 | C2×#16810803 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20286109 | C2 | semidirect |
| 4. #20273937 | C2 | semidirect |
| 5. #20273902 | C2 | semidirect |
| 6. #20273169 | C2 | semidirect |
| 196. C53 | #18693 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289129 | 2 |
| #20286109 | 2 |
| #20285037 | 2 |
| #20273937 | 2 |
| #20273902 | 2 |
| #20273169 | 2 |
| C53×#1560 | 2 |
| #18693 | 53 |