Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
376 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 43 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 | #20274668 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20272647 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C53×#1270 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#13133262 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#13133145 | — | 2 | 1 | no | — |
| 6 | #16810622 | — | 2 | 1 | no | — |
| 7 | C2×#13133134 | C4 | 1 | 1 | yes | characteristic |
| 8 | #16810349 | — | 2 | 1 | no | — |
| 9 | C53×#390 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#336 | — | 2 | 1 | no | — |
| 11 | #13133262 | — | 2 | 1 | no | — |
| 12 | #13133262 | — | 4 | 2 | no | — |
| 13 | #13133262 | — | 4 | 1 | no | — |
| 14 | #13133262 | C8 | 1 | 2 | yes | complemented |
| 15 | C2×#11719945 | — | 2 | 1 | no | — |
| 16 | C2×#11719947 | — | 4 | 1 | no | — |
| 17 | C22×C22⋊D53 | — | 4 | 1 | no | — |
| 18 | C2×D4×D53 | — | 4 | 1 | no | — |
| 19 | C2×#11719917 | — | 2 | 1 | no | — |
| 20 | #13133145 | — | 4 | 2 | no | — |
| 21 | #13133140 | — | 4 | 1 | no | — |
| 22 | #13133137 | — | 4 | 1 | no | — |
| 23 | #13133134 | — | 4 | 1 | no | — |
| 24 | C22⋊#88322 | — | 4 | 1 | no | — |
| 25 | C22×D4×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C22×C8×C53 | — | 4 | 1 | no | — |
| 27 | C2×C4⋊C4×C53 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×C22⋊C4×C53 | — | 4 | 1 | no | — |
| 29 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 30 | C24×C53 | — | 2 | 1 | no | — |
| 31 | C23×D53 | — | 4 | 1 | no | — |
| 32 | C2×D4×C53 | — | 2 | 1 | no | — |
| 33 | C2×D4×C53 | — | 4 | 1 | no | — |
| 34 | C2×D4×C53 | — | 4 | 1 | no | — |
| 35 | C2×D4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 36 | C2×D4×C53 | M16 | 1 | 1 | yes | characteristic |
| 37 | C22×C4×C53 | — | 2 | 1 | no | — |
| 38 | C22×C4×C53 | — | 2 | 1 | no | — |
| 39 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 40 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 41 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 42 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 43 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 44 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 45 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 46 | C22×Dic53 | — | 4 | 1 | no | — |
| 47 | #11719947 | — | 4 | 2 | no | — |
| 48 | #11719947 | — | 4 | 2 | no | — |
| 49 | #11719947 | — | 4 | 4 | no | — |
| 50 | D4×D53 | — | 4 | 2 | no | — |
| 51 | D4×D53 | — | 4 | 2 | no | — |
| 52 | D4×D53 | — | 4 | 4 | no | — |
| 53 | #11719945 | — | 2 | 2 | no | — |
| 54 | #11719945 | — | 2 | 2 | no | — |
| 55 | #11719945 | — | 4 | 1 | no | — |
| 56 | #11719945 | — | 4 | 1 | no | — |
| 57 | C2×D212 | — | 2 | 1 | no | — |
| 58 | C2×C4×D53 | — | 4 | 1 | no | — |
| 59 | C2×Dic106 | — | 2 | 1 | no | — |
| 60 | C2×C8×C53 | — | 4 | 2 | no | — |
| 61 | C2×C8×C53 | — | 8 | 1 | no | — |
| 62 | C2×C8×C53 | — | 8 | 1 | no | — |
| 63 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 64 | C22⋊C4×C53 | — | 8 | 1 | no | — |
| 65 | C22⋊C4×C53 | — | 8 | 1 | no | — |
| 66 | C42×C53 | — | 4 | 1 | no | — |
| 67 | #11719917 | — | 4 | 1 | no | — |
| 68 | #11719917 | — | 4 | 1 | no | — |
| 69 | C2×#88322 | — | 4 | 1 | no | — |
| 70 | C2×#88322 | — | 8 | 1 | no | — |
| 71 | C23×C53 | — | 2 | 1 | no | — |
| 72 | C23×C53 | — | 2 | 1 | no | — |
| 73 | C23×C53 | — | 4 | 1 | no | — |
| 74 | C23×C53 | — | 4 | 1 | no | — |
| 75 | C23×C53 | — | 8 | 1 | no | — |
| 76 | C23×C53 | — | 8 | 1 | no | — |
| 77 | C23×C53 | #99 | 1 | 1 | yes | characteristic |
| 78 | C22×D53 | — | 4 | 1 | no | — |
| 79 | C22×D53 | — | 4 | 2 | no | — |
| 80 | C22×D53 | — | 4 | 2 | no | — |
| 81 | C22×D53 | — | 16 | 1 | no | — |
| 82 | C22×D53 | — | 16 | 1 | no | — |
| 83 | D4×C53 | — | 4 | 1 | no | — |
| 84 | D4×C53 | — | 4 | 1 | no | — |
| 85 | D4×C53 | — | 4 | 2 | no | — |
| 86 | C2×C4×C53 | — | 2 | 1 | no | — |
| 87 | C2×C4×C53 | — | 2 | 1 | no | — |
| 88 | C2×C4×C53 | — | 2 | 1 | no | — |
| 89 | C2×C4×C53 | — | 2 | 1 | no | — |
| 90 | C2×C4×C53 | — | 4 | 1 | no | — |
| 91 | C2×C4×C53 | — | 4 | 1 | no | — |
| 92 | C2×C4×C53 | — | 4 | 1 | no | — |
| 93 | C2×C4×C53 | — | 8 | 1 | no | — |
| 94 | C2×C4×C53 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C53 | #100 | 1 | 1 | yes | characteristic |
| 96 | C22⋊D53 | — | 4 | 4 | no | — |
| 97 | C22⋊D53 | — | 4 | 4 | no | — |
| 98 | C22⋊D53 | — | 4 | 2 | no | — |
| 99 | C22⋊D53 | — | 4 | 2 | no | — |
| 100 | C2×Dic53 | — | 4 | 2 | no | — |
| 101 | C2×Dic53 | — | 4 | 1 | no | — |
| 102 | C2×Dic53 | — | 4 | 2 | no | — |
| 103 | D212 | — | 2 | 2 | no | — |
| 104 | D212 | — | 4 | 1 | no | — |
| 105 | C4×D53 | — | 4 | 2 | no | — |
| 106 | C4×D53 | — | 4 | 2 | no | — |
| 107 | Dic106 | — | 2 | 2 | no | — |
| 108 | Dic106 | — | 4 | 1 | no | — |
| 109 | C8×C53 | — | 8 | 2 | no | — |
| 110 | #88322 | — | 8 | 1 | no | — |
| 111 | #88322 | — | 16 | 1 | no | — |
| 112 | #7344 | — | 53 | 1 | no | maximal, Sylow |
| 113 | C22×C53 | — | 2 | 1 | no | — |
| 114 | C22×C53 | — | 2 | 1 | no | — |
| 115 | C22×C53 | — | 4 | 1 | no | — |
| 116 | C22×C53 | — | 4 | 1 | no | — |
| 117 | C22×C53 | — | 8 | 1 | no | — |
| 118 | C22×C53 | — | 8 | 1 | no | — |
| 119 | C22×C53 | — | 8 | 1 | no | — |
| 120 | C22×C53 | — | 8 | 1 | no | — |
| 121 | C22×C53 | — | 16 | 1 | no | — |
| 122 | C22×C53 | #323 | 1 | 1 | yes | characteristic |
| 123 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 124 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 125 | C2×D53 | — | 4 | 2 | no | — |
| 126 | C2×D53 | — | 8 | 1 | no | — |
| 127 | C2×D53 | — | 8 | 1 | no | — |
| 128 | C2×D53 | — | 16 | 2 | no | — |
| 129 | C2×D53 | — | 16 | 2 | no | — |
| 130 | C4×C53 | — | 2 | 1 | no | — |
| 131 | C4×C53 | — | 2 | 1 | no | — |
| 132 | C4×C53 | — | 4 | 1 | no | — |
| 133 | C4×C53 | — | 4 | 1 | no | — |
| 134 | C4×C53 | — | 8 | 1 | no | — |
| 135 | Dic53 | — | 4 | 2 | no | — |
| 136 | #1792 | — | 53 | 1 | no | — |
| 137 | #1318 | — | 53 | 1 | no | — |
| 138 | #1270 | D53 | 1 | 1 | yes | characteristic, complemented |
| 139 | C2×C53 | — | 2 | 1 | no | — |
| 140 | C2×C53 | — | 2 | 1 | no | — |
| 141 | C2×C53 | — | 8 | 1 | no | — |
| 142 | C2×C53 | — | 8 | 1 | no | — |
| 143 | C2×C53 | #1316 | 1 | 1 | yes | characteristic |
| 144 | C2×C53 | #1402 | 1 | 1 | yes | characteristic |
| 145 | C2×C53 | #1403 | 1 | 1 | yes | characteristic |
| 146 | D53 | — | 8 | 2 | no | — |
| 147 | C2×H(5,𝔽2) | — | 53 | 1 | no | — |
| 148 | C2×#104 | — | 106 | 1 | no | — |
| 149 | #417 | — | 106 | 1 | no | — |
| 150 | C2×#100 | — | 53 | 1 | no | — |
| 151 | #390 | C2×D53 | 1 | 1 | yes | characteristic |
| 152 | #336 | — | 2 | 1 | no | — |
| 153 | #324 | — | 106 | 1 | no | — |
| 154 | C53 | #7344 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | H(5,𝔽2) | — | 53 | 2 | no | — |
| 156 | H(5,𝔽2) | — | 106 | 1 | no | — |
| 157 | H(5,𝔽2) | — | 212 | 1 | no | — |
| 158 | H(5,𝔽2) | — | 212 | 2 | no | — |
| 159 | C2×#41 | — | 106 | 1 | no | — |
| 160 | C2×#41 | — | 212 | 1 | no | — |
| 161 | C22×D4 | — | 212 | 1 | no | — |
| 162 | C22×D4 | — | 212 | 1 | no | — |
| 163 | C22×D4 | C4×D53 | 1 | 1 | yes | characteristic |
| 164 | C2×M16 | — | 106 | 1 | no | — |
| 165 | C22×C8 | — | 4 | 1 | no | — |
| 166 | C2×C4⋊C4 | D212 | 1 | 1 | yes | characteristic |
| 167 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 168 | C2×C42 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 169 | #104 | — | 212 | 2 | no | — |
| 170 | #103 | — | 212 | 1 | no | — |
| 171 | #102 | — | 212 | 1 | no | — |
| 172 | #100 | — | 212 | 1 | no | — |
| 173 | C22⋊C8 | — | 212 | 1 | no | — |
| 174 | C24 | — | 2 | 1 | no | — |
| 175 | C24 | — | 212 | 1 | no | — |
| 176 | #41 | — | 106 | 2 | no | — |
| 177 | #41 | — | 106 | 2 | no | — |
| 178 | #41 | — | 212 | 1 | no | — |
| 179 | #41 | — | 212 | 2 | no | — |
| 180 | #41 | — | 212 | 4 | no | — |
| 181 | #41 | — | 212 | 1 | no | — |
| 182 | #41 | — | 212 | 2 | no | — |
| 183 | C2×Q8 | — | 106 | 1 | no | — |
| 184 | C2×D4 | — | 2 | 1 | no | — |
| 185 | C2×D4 | — | 4 | 1 | no | — |
| 186 | C2×D4 | — | 4 | 1 | no | — |
| 187 | C2×D4 | — | 106 | 1 | no | — |
| 188 | C2×D4 | — | 212 | 2 | no | — |
| 189 | C2×D4 | — | 212 | 2 | no | — |
| 190 | C2×D4 | — | 212 | 4 | no | — |
| 191 | C2×D4 | — | 212 | 2 | no | — |
| 192 | C2×D4 | — | 212 | 2 | no | — |
| 193 | C2×D4 | — | 212 | 2 | no | — |
| 194 | C2×D4 | — | 212 | 1 | no | — |
| 195 | C2×D4 | — | 212 | 2 | no | — |
| 196 | C2×D4 | — | 212 | 2 | no | — |
| 197 | C2×D4 | C8×D53 | 1 | 1 | yes | characteristic |
| 198 | C2×D4 | #11719912 | 1 | 1 | yes | characteristic |
| 199 | C22×C4 | — | 2 | 1 | no | — |
| 200 | C22×C4 | — | 2 | 1 | no | — |
| 201 | C22×C4 | — | 212 | 1 | no | — |
| 202 | C22×C4 | — | 212 | 1 | no | — |
| 203 | C22×C4 | #11719921 | 1 | 1 | yes | characteristic |
| 204 | M16 | — | 212 | 1 | no | — |
| 205 | M16 | — | 212 | 1 | no | — |
| 206 | C2×C8 | — | 4 | 2 | no | — |
| 207 | C2×C8 | — | 8 | 1 | no | — |
| 208 | C2×C8 | — | 8 | 1 | no | — |
| 209 | C2×C8 | — | 424 | 1 | no | — |
| 210 | C2×C8 | — | 212 | 1 | no | — |
| 211 | C4⋊C4 | — | 4 | 1 | no | — |
| 212 | C22⋊C4 | — | 8 | 1 | no | — |
| 213 | C22⋊C4 | — | 8 | 1 | no | — |
| 214 | C42 | — | 4 | 1 | no | — |
| 215 | C23 | — | 2 | 1 | no | — |
| 216 | C23 | — | 2 | 1 | no | — |
| 217 | C23 | — | 4 | 1 | no | — |
| 218 | C23 | — | 4 | 1 | no | — |
| 219 | C23 | — | 8 | 1 | no | — |
| 220 | C23 | — | 8 | 1 | no | — |
| 221 | C23 | — | 848 | 1 | no | — |
| 222 | C23 | — | 848 | 1 | no | — |
| 223 | C23 | — | 212 | 2 | no | — |
| 224 | C23 | — | 212 | 2 | no | — |
| 225 | C23 | — | 212 | 1 | no | — |
| 226 | C23 | #13133131 | 1 | 1 | yes | characteristic |
| 227 | Q8 | — | 106 | 2 | no | — |
| 228 | Q8 | — | 212 | 1 | no | — |
| 229 | D4 | — | 4 | 1 | no | — |
| 230 | D4 | — | 4 | 1 | no | — |
| 231 | D4 | — | 4 | 2 | no | — |
| 232 | D4 | — | 106 | 2 | no | — |
| 233 | D4 | — | 212 | 1 | no | — |
| 234 | D4 | — | 212 | 4 | no | — |
| 235 | D4 | — | 212 | 4 | no | — |
| 236 | D4 | — | 212 | 2 | no | — |
| 237 | D4 | — | 212 | 2 | no | — |
| 238 | C2×C4 | — | 2 | 1 | no | — |
| 239 | C2×C4 | — | 2 | 1 | no | — |
| 240 | C2×C4 | — | 2 | 1 | no | — |
| 241 | C2×C4 | — | 2 | 1 | no | — |
| 242 | C2×C4 | — | 4 | 1 | no | — |
| 243 | C2×C4 | — | 4 | 1 | no | — |
| 244 | C2×C4 | — | 4 | 1 | no | — |
| 245 | C2×C4 | — | 8 | 1 | no | — |
| 246 | C2×C4 | — | 212 | 2 | no | — |
| 247 | C2×C4 | — | 212 | 2 | no | — |
| 248 | C2×C4 | — | 212 | 2 | no | — |
| 249 | C2×C4 | — | 212 | 1 | no | — |
| 250 | C2×C4 | — | 212 | 2 | no | — |
| 251 | C2×C4 | #13133129 | 1 | 1 | yes | characteristic |
| 252 | C2×C4 | #13133133 | 1 | 1 | yes | characteristic |
| 253 | C8 | — | 8 | 2 | no | — |
| 254 | C8 | — | 848 | 1 | no | — |
| 255 | C8 | — | 424 | 1 | no | — |
| 256 | C22 | — | 2 | 1 | no | — |
| 257 | C22 | — | 2 | 1 | no | — |
| 258 | C22 | — | 4 | 1 | no | — |
| 259 | C22 | — | 4 | 1 | no | — |
| 260 | C22 | — | 8 | 1 | no | — |
| 261 | C22 | — | 8 | 1 | no | — |
| 262 | C22 | — | 8 | 1 | no | — |
| 263 | C22 | — | 8 | 1 | no | — |
| 264 | C22 | — | 16 | 1 | no | — |
| 265 | C22 | — | 848 | 2 | no | — |
| 266 | C22 | — | 848 | 2 | no | — |
| 267 | C22 | — | 424 | 1 | no | — |
| 268 | C22 | — | 424 | 1 | no | — |
| 269 | C22 | — | 212 | 2 | no | — |
| 270 | C22 | #16810346 | 1 | 1 | yes | characteristic |
| 271 | C22 | #16810409 | 1 | 1 | yes | characteristic |
| 272 | C22 | #16810411 | 1 | 1 | yes | characteristic |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 2 | 1 | no | — |
| 275 | C4 | — | 4 | 1 | no | — |
| 276 | C4 | — | 4 | 1 | no | — |
| 277 | C4 | — | 8 | 1 | no | — |
| 278 | C4 | — | 212 | 2 | no | — |
| 279 | C2 | — | 2 | 1 | no | — |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 8 | 1 | no | — |
| 282 | C2 | — | 8 | 1 | no | — |
| 283 | C2 | — | 424 | 2 | no | — |
| 284 | C2 | #20272642 | 1 | 1 | yes | characteristic |
| 285 | C2 | #20272861 | 1 | 1 | yes | characteristic |
| 286 | C2 | #20272864 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20272647 | C2 | semidirect |
| 3. C53×#1270 | C2 | semidirect |
| 14. #13133262 | C8 | semidirect |
| 138. #1270 | D53 | semidirect |
| 154. C53 | #7344 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20274668 | 2 |
| #20272647 | 2 |
| C53×#1270 | 2 |
| #7344 | 53 |