Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1108 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 228 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 | #20291647 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20290252 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #20290228 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #20290218 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20288496 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | Q8×#88324 | — | 2 | 2 | no | — |
| 7 | #16811658 | — | 2 | 6 | no | — |
| 8 | #16811640 | C22 | 1 | 6 | yes | — |
| 9 | #16811626 | C22 | 1 | 6 | yes | — |
| 10 | #16811483 | C22 | 1 | 6 | yes | — |
| 11 | #16811483 | C22 | 1 | 6 | yes | — |
| 12 | #16811481 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16811481 | C22 | 1 | 3 | yes | — |
| 14 | C2×#13133264 | C4 | 1 | 1 | yes | characteristic |
| 15 | Q8×C22⋊D53 | — | 2 | 2 | no | — |
| 16 | #16811272 | — | 2 | 6 | no | — |
| 17 | #16811197 | C4 | 1 | 3 | yes | — |
| 18 | #16811194 | C4 | 1 | 3 | yes | — |
| 19 | #16811169 | C22 | 1 | 3 | yes | — |
| 20 | C2×#13133215 | C22 | 1 | 3 | yes | — |
| 21 | #41×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×2−1+4×C53 | C4 | 1 | 1 | yes | characteristic |
| 23 | C2×#11719940 | — | 2 | 6 | no | — |
| 24 | #13133284 | C23 | 1 | 6 | yes | — |
| 25 | #13133284 | C23 | 1 | 2 | yes | — |
| 26 | C4⋊#88324 | — | 2 | 6 | no | — |
| 27 | C22⋊#88324 | — | 2 | 6 | no | — |
| 28 | C4×#88324 | — | 2 | 6 | no | — |
| 29 | #13133264 | — | 4 | 4 | no | — |
| 30 | C2×#11719945 | C2×C4 | 1 | 3 | yes | — |
| 31 | C2×#11719947 | — | 2 | 3 | no | — |
| 32 | C22×Dic106 | C2×C4 | 1 | 3 | yes | — |
| 33 | C2×Q8×D53 | — | 2 | 1 | no | — |
| 34 | #13133215 | — | 2 | 6 | no | — |
| 35 | #13133215 | C2×C4 | 1 | 6 | yes | — |
| 36 | #13133215 | C23 | 1 | 3 | yes | — |
| 37 | #13133213 | C23 | 1 | 1 | yes | characteristic |
| 38 | #13133206 | C2×C4 | 1 | 3 | yes | — |
| 39 | #13133204 | C2×C4 | 1 | 3 | yes | — |
| 40 | #13133200 | — | 2 | 6 | no | — |
| 41 | #13133198 | — | 2 | 6 | no | — |
| 42 | #13133196 | — | 2 | 6 | no | — |
| 43 | #13133193 | — | 2 | 6 | no | — |
| 44 | #13133189 | — | 2 | 6 | no | — |
| 45 | Q8×Dic53 | D4 | 1 | 2 | yes | — |
| 46 | D4×Dic53 | D4 | 1 | 6 | yes | — |
| 47 | C4×C22⋊D53 | — | 2 | 6 | no | — |
| 48 | #13133176 | C2×C4 | 1 | 3 | yes | — |
| 49 | C2×#11719919 | — | 2 | 6 | no | — |
| 50 | C2×C4×Dic53 | C23 | 1 | 3 | yes | — |
| 51 | 2−1+4×C53 | — | 4 | 4 | no | — |
| 52 | C2×#41×C53 | — | 2 | 3 | no | — |
| 53 | C2×#41×C53 | C2×C4 | 1 | 3 | yes | — |
| 54 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 56 | C22×Q8×C53 | C2×C4 | 1 | 3 | yes | — |
| 57 | #41×C53 | — | 2 | 4 | no | — |
| 58 | #41×C53 | — | 4 | 6 | no | — |
| 59 | #41×C53 | — | 4 | 12 | no | — |
| 60 | C2×Q8×C53 | — | 2 | 3 | no | — |
| 61 | C2×Q8×C53 | — | 4 | 6 | no | — |
| 62 | C2×Q8×C53 | — | 4 | 4 | no | — |
| 63 | C2×Q8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C2×Q8×C53 | C22×C4 | 1 | 3 | yes | — |
| 65 | C2×D4×C53 | — | 2 | 3 | no | — |
| 66 | C2×D4×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 67 | C2×D4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C53 | — | 2 | 3 | no | — |
| 69 | C22×C4×C53 | — | 2 | 3 | no | — |
| 70 | C22×C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 71 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 72 | C2×C22⋊D53 | — | 2 | 2 | no | — |
| 73 | C22×Dic53 | — | 2 | 3 | no | — |
| 74 | C22×Dic53 | C2×D4 | 1 | 6 | yes | — |
| 75 | Q8×D53 | — | 4 | 4 | no | — |
| 76 | #11719947 | — | 4 | 12 | no | — |
| 77 | #11719945 | — | 4 | 6 | no | — |
| 78 | C2×D212 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×D53 | — | 2 | 3 | no | — |
| 80 | C2×Dic106 | — | 2 | 3 | no | — |
| 81 | C2×Dic106 | — | 4 | 6 | no | — |
| 82 | C2×Dic106 | C22×C4 | 1 | 3 | yes | — |
| 83 | #11719940 | — | 8 | 6 | no | — |
| 84 | C2×#88324 | — | 2 | 2 | no | — |
| 85 | C2×#88324 | — | 2 | 6 | no | — |
| 86 | C22⋊Dic53 | — | 2 | 6 | no | — |
| 87 | C22⋊Dic53 | C22⋊C4 | 1 | 6 | yes | — |
| 88 | #11719921 | — | 2 | 6 | no | — |
| 89 | C4⋊Dic53 | C22⋊C4 | 1 | 6 | yes | — |
| 90 | #11719919 | — | 2 | 12 | no | — |
| 91 | #11719919 | — | 2 | 6 | no | — |
| 92 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 93 | C4×Dic53 | C2×D4 | 1 | 6 | yes | — |
| 94 | C4×Dic53 | C24 | 1 | 1 | yes | characteristic |
| 95 | C23×C53 | — | 2 | 1 | no | — |
| 96 | C23×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 97 | C22×D53 | — | 2 | 1 | no | — |
| 98 | Q8×C53 | — | 4 | 3 | no | — |
| 99 | Q8×C53 | — | 4 | 6 | no | — |
| 100 | Q8×C53 | #100 | 1 | 4 | yes | — |
| 101 | D4×C53 | — | 2 | 6 | no | — |
| 102 | D4×C53 | — | 4 | 1 | no | — |
| 103 | D4×C53 | — | 4 | 6 | no | — |
| 104 | C2×C4×C53 | — | 2 | 6 | no | — |
| 105 | C2×C4×C53 | — | 2 | 3 | no | — |
| 106 | C2×C4×C53 | — | 4 | 6 | no | — |
| 107 | C2×C4×C53 | — | 4 | 6 | no | — |
| 108 | C2×C4×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 109 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 110 | C22⋊D53 | — | 4 | 6 | no | — |
| 111 | C22⋊D53 | — | 4 | 2 | no | — |
| 112 | C22⋊D53 | — | 4 | 2 | no | — |
| 113 | C2×Dic53 | — | 2 | 6 | no | — |
| 114 | C2×Dic53 | — | 2 | 6 | no | — |
| 115 | C2×Dic53 | — | 2 | 3 | no | — |
| 116 | C2×Dic53 | — | 4 | 6 | no | — |
| 117 | C2×Dic53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 118 | C2×Dic53 | C22×D4 | 1 | 2 | yes | — |
| 119 | D212 | — | 4 | 1 | no | — |
| 120 | C4×D53 | — | 4 | 6 | no | — |
| 121 | Dic106 | — | 4 | 6 | no | — |
| 122 | Dic106 | — | 4 | 3 | no | — |
| 123 | #88324 | — | 4 | 6 | no | — |
| 124 | #88324 | — | 4 | 2 | no | — |
| 125 | #17773 | — | 53 | 1 | no | maximal, Sylow |
| 126 | C22×C53 | — | 2 | 3 | no | — |
| 127 | C22×C53 | — | 2 | 6 | no | — |
| 128 | C22×C53 | — | 4 | 1 | no | — |
| 129 | C22×C53 | — | 4 | 2 | no | — |
| 130 | C22×C53 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 131 | C2×D53 | — | 4 | 1 | no | — |
| 132 | C2×D53 | — | 4 | 2 | no | — |
| 133 | C4×C53 | — | 2 | 1 | no | — |
| 134 | C4×C53 | — | 4 | 3 | no | — |
| 135 | C4×C53 | C2×#100 | 1 | 6 | yes | — |
| 136 | Dic53 | — | 2 | 6 | no | — |
| 137 | Dic53 | — | 4 | 3 | no | — |
| 138 | Dic53 | #577 | 1 | 2 | yes | — |
| 139 | Dic53 | #578 | 1 | 2 | yes | — |
| 140 | #3499 | — | 53 | 1 | no | — |
| 141 | #2991 | — | 53 | 2 | no | — |
| 142 | #1560 | — | 53 | 3 | no | — |
| 143 | #1548 | — | 53 | 6 | no | — |
| 144 | #1543 | — | 53 | 3 | no | — |
| 145 | C2×C53 | — | 4 | 3 | no | — |
| 146 | C2×C53 | — | 8 | 1 | no | — |
| 147 | C2×C53 | C22×#100 | 1 | 1 | yes | characteristic |
| 148 | C2×C53 | #2898 | 1 | 2 | yes | — |
| 149 | D53 | — | 8 | 1 | no | — |
| 150 | C2×2−1+4 | — | 53 | 1 | no | — |
| 151 | C2×2−1+4 | #6157 | 1 | 1 | yes | characteristic |
| 152 | D4×Q8 | — | 106 | 2 | no | — |
| 153 | #547 | — | 106 | 6 | no | — |
| 154 | #536 | — | 53 | 3 | no | — |
| 155 | #536 | — | 53 | 3 | no | — |
| 156 | #532 | — | 53 | 3 | no | — |
| 157 | C2×C4⋊Q8 | — | 53 | 3 | no | — |
| 158 | C4×#41 | — | 53 | 1 | no | — |
| 159 | C8×Q8 | — | 106 | 2 | no | — |
| 160 | #436 | — | 106 | 6 | no | — |
| 161 | #431 | — | 53 | 6 | no | — |
| 162 | #423 | — | 53 | 6 | no | — |
| 163 | #333 | — | 53 | 6 | no | — |
| 164 | #333 | — | 53 | 6 | no | — |
| 165 | #331 | — | 53 | 1 | no | — |
| 166 | #331 | — | 53 | 3 | no | — |
| 167 | C53 | #17773 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 168 | 2−1+4 | — | 4 | 4 | no | — |
| 169 | 2−1+4 | — | 212 | 4 | no | — |
| 170 | C2×#41 | — | 2 | 3 | no | — |
| 171 | C2×#41 | — | 53 | 3 | no | — |
| 172 | C2×#41 | — | 106 | 3 | no | — |
| 173 | C2×#41 | C2×#6157 | 1 | 3 | yes | — |
| 174 | C2×#41 | C2×#6157 | 1 | 1 | yes | characteristic |
| 175 | C22×Q8 | — | 2 | 1 | no | — |
| 176 | C22×Q8 | — | 53 | 3 | no | — |
| 177 | C22×Q8 | — | 106 | 1 | no | — |
| 178 | C22×Q8 | C2×#6157 | 1 | 3 | yes | — |
| 179 | C2×M16 | — | 106 | 6 | no | — |
| 180 | C4⋊Q8 | — | 53 | 3 | no | — |
| 181 | C4⋊Q8 | — | 53 | 6 | no | — |
| 182 | C4⋊Q8 | — | 106 | 6 | no | — |
| 183 | #127 | — | 53 | 1 | no | — |
| 184 | #124 | — | 53 | 3 | no | — |
| 185 | #124 | — | 53 | 3 | no | — |
| 186 | #123 | — | 106 | 6 | no | — |
| 187 | #123 | — | 106 | 6 | no | — |
| 188 | C22⋊Q8 | — | 106 | 6 | no | — |
| 189 | C22⋊Q8 | — | 106 | 6 | no | — |
| 190 | #121 | — | 106 | 6 | no | — |
| 191 | C4×Q8 | — | 53 | 2 | no | — |
| 192 | C4×D4 | — | 53 | 6 | no | — |
| 193 | C4×D4 | — | 106 | 6 | no | — |
| 194 | #117 | — | 53 | 3 | no | — |
| 195 | C2×C4⋊C4 | — | 106 | 6 | no | — |
| 196 | C2×C42 | — | 53 | 3 | no | — |
| 197 | C4⋊C8 | — | 53 | 6 | no | — |
| 198 | C4⋊C8 | — | 53 | 2 | no | — |
| 199 | C4⋊C8 | — | 106 | 6 | no | — |
| 200 | C22⋊C8 | — | 106 | 6 | no | — |
| 201 | C4×C8 | — | 106 | 6 | no | — |
| 202 | #41 | — | 2 | 4 | no | — |
| 203 | #41 | — | 4 | 6 | no | — |
| 204 | #41 | — | 4 | 12 | no | — |
| 205 | #41 | — | 212 | 6 | no | — |
| 206 | #41 | — | 212 | 12 | no | — |
| 207 | C2×Q8 | — | 2 | 3 | no | — |
| 208 | C2×Q8 | — | 4 | 6 | no | — |
| 209 | C2×Q8 | — | 4 | 4 | no | — |
| 210 | C2×Q8 | — | 53 | 3 | no | — |
| 211 | C2×Q8 | — | 106 | 3 | no | — |
| 212 | C2×Q8 | — | 212 | 6 | no | — |
| 213 | C2×Q8 | — | 212 | 4 | no | — |
| 214 | C2×Q8 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 215 | C2×Q8 | C22×#6157 | 1 | 3 | yes | — |
| 216 | C2×D4 | — | 2 | 3 | no | — |
| 217 | C2×D4 | — | 53 | 1 | no | — |
| 218 | C2×D4 | — | 106 | 2 | no | — |
| 219 | C2×D4 | — | 106 | 3 | no | — |
| 220 | C2×D4 | C22⋊#6157 | 1 | 3 | yes | — |
| 221 | C2×D4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | — | 2 | 3 | no | — |
| 223 | C22×C4 | — | 2 | 3 | no | — |
| 224 | C22×C4 | — | 53 | 6 | no | — |
| 225 | C22×C4 | — | 106 | 3 | no | — |
| 226 | C22×C4 | — | 106 | 3 | no | — |
| 227 | C22×C4 | C22×#6157 | 1 | 3 | yes | — |
| 228 | M16 | — | 424 | 6 | no | — |
| 229 | C2×C8 | — | 106 | 2 | no | — |
| 230 | C2×C8 | — | 106 | 6 | no | — |
| 231 | C4⋊C4 | — | 53 | 6 | no | — |
| 232 | C4⋊C4 | — | 106 | 6 | no | — |
| 233 | C4⋊C4 | — | 106 | 12 | no | — |
| 234 | C22⋊C4 | — | 53 | 6 | no | — |
| 235 | C22⋊C4 | — | 106 | 6 | no | — |
| 236 | C22⋊C4 | — | 106 | 6 | no | — |
| 237 | C42 | — | 53 | 6 | no | — |
| 238 | C42 | — | 53 | 1 | no | — |
| 239 | C42 | — | 53 | 3 | no | — |
| 240 | C23 | — | 2 | 1 | no | — |
| 241 | C23 | — | 106 | 1 | no | — |
| 242 | C23 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 243 | Q8 | — | 4 | 3 | no | — |
| 244 | Q8 | — | 4 | 6 | no | — |
| 245 | Q8 | — | 212 | 3 | no | — |
| 246 | Q8 | — | 212 | 6 | no | — |
| 247 | Q8 | #13133277 | 1 | 4 | yes | — |
| 248 | D4 | — | 2 | 6 | no | — |
| 249 | D4 | — | 4 | 1 | no | — |
| 250 | D4 | — | 4 | 6 | no | — |
| 251 | D4 | — | 212 | 1 | no | — |
| 252 | D4 | — | 212 | 2 | no | — |
| 253 | D4 | — | 212 | 2 | no | — |
| 254 | D4 | — | 212 | 6 | no | — |
| 255 | C2×C4 | — | 2 | 6 | no | — |
| 256 | C2×C4 | — | 2 | 3 | no | — |
| 257 | C2×C4 | — | 4 | 6 | no | — |
| 258 | C2×C4 | — | 4 | 6 | no | — |
| 259 | C2×C4 | — | 53 | 2 | no | — |
| 260 | C2×C4 | — | 53 | 6 | no | — |
| 261 | C2×C4 | — | 106 | 6 | no | — |
| 262 | C2×C4 | — | 106 | 6 | no | — |
| 263 | C2×C4 | — | 106 | 3 | no | — |
| 264 | C2×C4 | — | 212 | 6 | no | — |
| 265 | C2×C4 | — | 212 | 6 | no | — |
| 266 | C2×C4 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 267 | C2×C4 | C23×#6157 | 1 | 1 | yes | characteristic |
| 268 | C8 | — | 212 | 2 | no | — |
| 269 | C8 | — | 212 | 6 | no | — |
| 270 | C22 | — | 2 | 6 | no | — |
| 271 | C22 | — | 2 | 3 | no | — |
| 272 | C22 | — | 4 | 2 | no | — |
| 273 | C22 | — | 4 | 1 | no | — |
| 274 | C22 | — | 212 | 2 | no | — |
| 275 | C22 | — | 212 | 1 | no | — |
| 276 | C22 | C22×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 277 | C4 | — | 2 | 1 | no | — |
| 278 | C4 | — | 4 | 3 | no | — |
| 279 | C4 | — | 53 | 2 | no | — |
| 280 | C4 | — | 53 | 2 | no | — |
| 281 | C4 | — | 106 | 6 | no | — |
| 282 | C4 | — | 212 | 3 | no | — |
| 283 | C4 | C2×#13133277 | 1 | 6 | yes | — |
| 284 | C2 | — | 4 | 3 | no | — |
| 285 | C2 | — | 8 | 1 | no | — |
| 286 | C2 | — | 424 | 1 | no | — |
| 287 | C2 | C22×#13133277 | 1 | 1 | yes | characteristic |
| 288 | C2 | #20291337 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20291647 | C2 | semidirect |
| 2. #20290252 | C2 | semidirect |
| 3. #20290228 | C2 | semidirect |
| 4. #20290218 | C2 | semidirect |
| 167. C53 | #17773 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291647 | 2 |
| #20290252 | 2 |
| #20290228 | 2 |
| #20290218 | 2 |
| #20288496 | 2 |
| #17773 | 53 |