Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
332 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 37 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 | #20275879 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20272619 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C53×#1316 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811249 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #16810479 | — | 2 | 1 | no | — |
| 6 | #16810479 | — | 2 | 1 | no | — |
| 7 | #16810390 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #16810389 | — | 2 | 1 | no | — |
| 9 | C53×#379 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#323 | — | 2 | 1 | no | — |
| 11 | #13133196 | — | 4 | 1 | no | — |
| 12 | #13133194 | — | 4 | 2 | no | — |
| 13 | C2×C22⋊Dic53 | — | 2 | 1 | no | — |
| 14 | C2×C22⋊Dic53 | — | 2 | 2 | no | — |
| 15 | C2×C22⋊Dic53 | — | 2 | 1 | no | — |
| 16 | C2×C22⋊Dic53 | — | 4 | 2 | no | — |
| 17 | C2×C22⋊Dic53 | — | 4 | 1 | no | — |
| 18 | C2×C22⋊Dic53 | — | 4 | 1 | no | — |
| 19 | C25×C53 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C22⋊C4×C53 | — | 4 | 1 | no | — |
| 21 | C2×C22⋊C4×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C22⋊C4×C53 | D4 | 1 | 1 | yes | characteristic |
| 23 | C22⋊C8×C53 | — | 4 | 1 | no | — |
| 24 | C24×C53 | — | 2 | 1 | no | — |
| 25 | C24×C53 | — | 2 | 1 | no | — |
| 26 | C24×C53 | — | 4 | 1 | no | — |
| 27 | C24×C53 | — | 4 | 1 | no | — |
| 28 | C24×C53 | — | 8 | 2 | no | — |
| 29 | C24×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 30 | C24×C53 | SD16 | 1 | 1 | yes | characteristic |
| 31 | C24×C53 | Q16 | 1 | 1 | yes | characteristic |
| 32 | C22×C4×C53 | — | 2 | 1 | no | — |
| 33 | C22×C4×C53 | — | 2 | 1 | no | — |
| 34 | C22×Dic53 | — | 2 | 2 | no | — |
| 35 | C22×Dic53 | — | 4 | 1 | no | — |
| 36 | C22×Dic53 | — | 4 | 2 | no | — |
| 37 | C2×Dic106 | — | 4 | 1 | no | — |
| 38 | C2×C8×C53 | — | 8 | 1 | no | — |
| 39 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 40 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 41 | C22⋊C4×C53 | — | 8 | 2 | no | — |
| 42 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 43 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 44 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 45 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 46 | C22⋊Dic53 | — | 8 | 4 | no | — |
| 47 | C22⋊Dic53 | — | 8 | 2 | no | — |
| 48 | C22⋊Dic53 | — | 8 | 1 | no | — |
| 49 | C22⋊Dic53 | — | 8 | 1 | no | — |
| 50 | C22⋊Dic53 | — | 8 | 2 | no | — |
| 51 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 52 | #11719919 | — | 4 | 2 | no | — |
| 53 | C23×C53 | — | 2 | 1 | no | — |
| 54 | C23×C53 | — | 2 | 1 | no | — |
| 55 | C23×C53 | — | 2 | 1 | no | — |
| 56 | C23×C53 | — | 2 | 1 | no | — |
| 57 | C23×C53 | — | 4 | 1 | no | — |
| 58 | C23×C53 | — | 4 | 1 | no | — |
| 59 | C23×C53 | — | 4 | 2 | no | — |
| 60 | C23×C53 | — | 4 | 2 | no | — |
| 61 | C23×C53 | — | 4 | 2 | no | — |
| 62 | C23×C53 | — | 4 | 4 | no | — |
| 63 | C23×C53 | — | 4 | 1 | no | — |
| 64 | C23×C53 | — | 4 | 1 | no | — |
| 65 | C23×C53 | — | 8 | 1 | no | — |
| 66 | C23×C53 | — | 8 | 1 | no | — |
| 67 | C23×C53 | — | 8 | 1 | no | — |
| 68 | C23×C53 | — | 8 | 1 | no | — |
| 69 | C23×C53 | — | 8 | 1 | no | — |
| 70 | C23×C53 | — | 8 | 2 | no | — |
| 71 | C23×C53 | — | 8 | 2 | no | — |
| 72 | C23×C53 | — | 8 | 2 | no | — |
| 73 | C23×C53 | #99 | 1 | 1 | yes | characteristic |
| 74 | C23×C53 | #103 | 1 | 1 | yes | characteristic |
| 75 | C23×C53 | #104 | 1 | 1 | yes | characteristic |
| 76 | C2×C4×C53 | — | 4 | 1 | no | — |
| 77 | C2×C4×C53 | — | 4 | 1 | no | — |
| 78 | C2×C4×C53 | — | 8 | 1 | no | — |
| 79 | C2×C4×C53 | — | 8 | 1 | no | — |
| 80 | C2×Dic53 | — | 4 | 2 | no | — |
| 81 | C2×Dic53 | — | 8 | 2 | no | — |
| 82 | C2×Dic53 | — | 8 | 1 | no | — |
| 83 | C2×Dic53 | — | 8 | 2 | no | — |
| 84 | C2×Dic53 | — | 8 | 2 | no | — |
| 85 | C2×Dic53 | — | 8 | 2 | no | — |
| 86 | C2×Dic53 | — | 8 | 2 | no | — |
| 87 | Dic106 | — | 16 | 1 | no | — |
| 88 | C8×C53 | — | 16 | 1 | no | — |
| 89 | #7426 | — | 53 | 1 | no | maximal, Sylow |
| 90 | C22×C53 | — | 2 | 2 | no | — |
| 91 | C22×C53 | — | 2 | 1 | no | — |
| 92 | C22×C53 | — | 2 | 1 | no | — |
| 93 | C22×C53 | — | 4 | 1 | no | — |
| 94 | C22×C53 | — | 4 | 1 | no | — |
| 95 | C22×C53 | — | 4 | 1 | no | — |
| 96 | C22×C53 | — | 4 | 2 | no | — |
| 97 | C22×C53 | — | 4 | 1 | no | — |
| 98 | C22×C53 | — | 4 | 1 | no | — |
| 99 | C22×C53 | — | 4 | 1 | no | — |
| 100 | C22×C53 | — | 4 | 2 | no | — |
| 101 | C22×C53 | — | 4 | 1 | no | — |
| 102 | C22×C53 | — | 4 | 2 | no | — |
| 103 | C22×C53 | — | 4 | 1 | no | — |
| 104 | C22×C53 | — | 8 | 1 | no | — |
| 105 | C22×C53 | — | 8 | 2 | no | — |
| 106 | C22×C53 | — | 8 | 2 | no | — |
| 107 | C22×C53 | — | 8 | 1 | no | — |
| 108 | C22×C53 | — | 8 | 4 | no | — |
| 109 | C22×C53 | — | 8 | 1 | no | — |
| 110 | C22×C53 | #328 | 1 | 1 | yes | characteristic |
| 111 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 112 | C22×C53 | #354 | 1 | 1 | yes | characteristic |
| 113 | C4×C53 | — | 8 | 1 | no | — |
| 114 | C4×C53 | — | 8 | 1 | no | — |
| 115 | Dic53 | — | 8 | 2 | no | — |
| 116 | Dic53 | — | 16 | 1 | no | — |
| 117 | Dic53 | — | 16 | 2 | no | — |
| 118 | #2029 | — | 53 | 1 | no | — |
| 119 | #1316 | D53 | 1 | 1 | yes | characteristic |
| 120 | #1304 | — | 53 | 1 | no | — |
| 121 | C2×C53 | — | 2 | 1 | no | — |
| 122 | C2×C53 | — | 2 | 1 | no | — |
| 123 | C2×C53 | — | 2 | 1 | no | — |
| 124 | C2×C53 | — | 4 | 2 | no | — |
| 125 | C2×C53 | — | 4 | 1 | no | — |
| 126 | C2×C53 | — | 4 | 1 | no | — |
| 127 | C2×C53 | — | 8 | 1 | no | — |
| 128 | C2×C53 | #1351 | 1 | 1 | yes | characteristic |
| 129 | #543 | — | 53 | 1 | no | — |
| 130 | #379 | — | 106 | 1 | no | — |
| 131 | #379 | — | 106 | 1 | no | — |
| 132 | #379 | C2×D53 | 1 | 1 | yes | characteristic |
| 133 | #342 | — | 53 | 1 | no | — |
| 134 | #342 | — | 106 | 1 | no | — |
| 135 | #323 | — | 2 | 1 | no | — |
| 136 | C53 | #7426 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 137 | C25 | D212 | 1 | 1 | yes | characteristic |
| 138 | C22⋊Q8 | — | 212 | 2 | no | — |
| 139 | C22⋊Q8 | — | 212 | 1 | no | — |
| 140 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 141 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 142 | C2×C22⋊C4 | — | 106 | 1 | no | — |
| 143 | C2×C22⋊C4 | — | 106 | 1 | no | — |
| 144 | C2×C22⋊C4 | — | 212 | 2 | no | — |
| 145 | C2×C22⋊C4 | — | 212 | 1 | no | — |
| 146 | C2×C22⋊C4 | — | 212 | 1 | no | — |
| 147 | C2×C22⋊C4 | C4×D53 | 1 | 1 | yes | characteristic |
| 148 | C2×C22⋊C4 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 149 | C22⋊C8 | — | 4 | 1 | no | — |
| 150 | C24 | — | 2 | 1 | no | — |
| 151 | C24 | — | 2 | 1 | no | — |
| 152 | C24 | — | 4 | 1 | no | — |
| 153 | C24 | — | 4 | 1 | no | — |
| 154 | C24 | — | 8 | 2 | no | — |
| 155 | C24 | #11719913 | 1 | 1 | yes | characteristic |
| 156 | C24 | Dic212 | 1 | 1 | yes | characteristic |
| 157 | C24 | #11719921 | 1 | 1 | yes | characteristic |
| 158 | C2×Q8 | — | 212 | 1 | no | — |
| 159 | C22×C4 | — | 2 | 1 | no | — |
| 160 | C22×C4 | — | 2 | 1 | no | — |
| 161 | C22×C4 | — | 106 | 2 | no | — |
| 162 | C22×C4 | — | 212 | 2 | no | — |
| 163 | C22×C4 | — | 212 | 1 | no | — |
| 164 | C2×C8 | — | 8 | 1 | no | — |
| 165 | C4⋊C4 | — | 212 | 2 | no | — |
| 166 | C4⋊C4 | — | 212 | 1 | no | — |
| 167 | C22⋊C4 | — | 4 | 1 | no | — |
| 168 | C22⋊C4 | — | 4 | 1 | no | — |
| 169 | C22⋊C4 | — | 8 | 2 | no | — |
| 170 | C22⋊C4 | — | 424 | 2 | no | — |
| 171 | C22⋊C4 | — | 424 | 1 | no | — |
| 172 | C22⋊C4 | — | 424 | 1 | no | — |
| 173 | C22⋊C4 | — | 424 | 4 | no | — |
| 174 | C22⋊C4 | — | 424 | 2 | no | — |
| 175 | C22⋊C4 | — | 212 | 1 | no | — |
| 176 | C22⋊C4 | — | 212 | 1 | no | — |
| 177 | C22⋊C4 | — | 212 | 1 | no | — |
| 178 | C22⋊C4 | — | 212 | 1 | no | — |
| 179 | C23 | — | 2 | 1 | no | — |
| 180 | C23 | — | 2 | 1 | no | — |
| 181 | C23 | — | 2 | 1 | no | — |
| 182 | C23 | — | 2 | 1 | no | — |
| 183 | C23 | — | 4 | 2 | no | — |
| 184 | C23 | — | 4 | 1 | no | — |
| 185 | C23 | — | 4 | 1 | no | — |
| 186 | C23 | — | 4 | 2 | no | — |
| 187 | C23 | — | 4 | 2 | no | — |
| 188 | C23 | — | 4 | 4 | no | — |
| 189 | C23 | — | 4 | 1 | no | — |
| 190 | C23 | — | 4 | 1 | no | — |
| 191 | C23 | — | 8 | 1 | no | — |
| 192 | C23 | — | 8 | 1 | no | — |
| 193 | C23 | — | 8 | 1 | no | — |
| 194 | C23 | — | 8 | 1 | no | — |
| 195 | C23 | — | 8 | 1 | no | — |
| 196 | C23 | — | 8 | 2 | no | — |
| 197 | C23 | — | 8 | 2 | no | — |
| 198 | C23 | — | 8 | 2 | no | — |
| 199 | C23 | #13133131 | 1 | 1 | yes | characteristic |
| 200 | C23 | #13133142 | 1 | 1 | yes | characteristic |
| 201 | C23 | #13133143 | 1 | 1 | yes | characteristic |
| 202 | Q8 | — | 848 | 1 | no | — |
| 203 | C2×C4 | — | 4 | 1 | no | — |
| 204 | C2×C4 | — | 4 | 1 | no | — |
| 205 | C2×C4 | — | 8 | 1 | no | — |
| 206 | C2×C4 | — | 8 | 1 | no | — |
| 207 | C2×C4 | — | 424 | 2 | no | — |
| 208 | C2×C4 | — | 424 | 1 | no | — |
| 209 | C2×C4 | — | 424 | 2 | no | — |
| 210 | C2×C4 | — | 424 | 2 | no | — |
| 211 | C2×C4 | — | 424 | 2 | no | — |
| 212 | C2×C4 | — | 424 | 2 | no | — |
| 213 | C2×C4 | — | 212 | 2 | no | — |
| 214 | C8 | — | 16 | 1 | no | — |
| 215 | C22 | — | 2 | 2 | no | — |
| 216 | C22 | — | 2 | 1 | no | — |
| 217 | C22 | — | 2 | 1 | no | — |
| 218 | C22 | — | 4 | 1 | no | — |
| 219 | C22 | — | 4 | 1 | no | — |
| 220 | C22 | — | 4 | 1 | no | — |
| 221 | C22 | — | 4 | 2 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 4 | 1 | no | — |
| 224 | C22 | — | 4 | 1 | no | — |
| 225 | C22 | — | 4 | 2 | no | — |
| 226 | C22 | — | 4 | 1 | no | — |
| 227 | C22 | — | 4 | 2 | no | — |
| 228 | C22 | — | 4 | 1 | no | — |
| 229 | C22 | — | 8 | 1 | no | — |
| 230 | C22 | — | 8 | 2 | no | — |
| 231 | C22 | — | 8 | 2 | no | — |
| 232 | C22 | — | 8 | 1 | no | — |
| 233 | C22 | — | 8 | 4 | no | — |
| 234 | C22 | — | 8 | 1 | no | — |
| 235 | C22 | #16810361 | 1 | 1 | yes | characteristic |
| 236 | C22 | #16810411 | 1 | 1 | yes | characteristic |
| 237 | C22 | #16810417 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 8 | 1 | no | — |
| 239 | C4 | — | 8 | 1 | no | — |
| 240 | C4 | — | 848 | 2 | no | — |
| 241 | C4 | — | 848 | 1 | no | — |
| 242 | C4 | — | 424 | 2 | no | — |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 2 | 1 | no | — |
| 246 | C2 | — | 4 | 2 | no | — |
| 247 | C2 | — | 4 | 1 | no | — |
| 248 | C2 | — | 4 | 1 | no | — |
| 249 | C2 | — | 8 | 1 | no | — |
| 250 | C2 | #20272737 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #16811249 | C4 | semidirect |
| 7. #16810390 | C4 | semidirect |
| 136. C53 | #7426 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20275879 | 2 |
| #20272619 | 2 |
| C53×#1316 | 2 |
| #7426 | 53 |