Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
3706 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 315 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 | C2×#16810813 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #20278046 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #20274663 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #20274633 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×C53×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#13133198 | C22 | 1 | 6 | yes | complemented |
| 7 | C22×#11719919 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | #16810813 | — | 2 | 16 | no | — |
| 9 | C2×#13133132 | C22 | 1 | 12 | yes | complemented |
| 10 | #16810542 | — | 2 | 3 | no | — |
| 11 | #16810537 | — | 2 | 3 | no | — |
| 12 | #16810503 | C22 | 1 | 6 | yes | complemented |
| 13 | #16810496 | — | 2 | 6 | no | — |
| 14 | #16810486 | C22 | 1 | 3 | yes | complemented |
| 15 | C53×#585 | — | 2 | 16 | no | — |
| 16 | C2×2−1+4×C53 | — | 2 | 3 | no | — |
| 17 | C2×H(5,𝔽2)×C53 | — | 2 | 3 | no | — |
| 18 | C2×H(5,𝔽2)×C53 | C22 | 1 | 4 | yes | — |
| 19 | C22×#41×C53 | — | 2 | 6 | no | — |
| 20 | C22×#41×C53 | C22 | 1 | 3 | yes | — |
| 21 | C23×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 22 | C23×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 23 | #13133198 | — | 2 | 24 | no | — |
| 24 | C2×#11719919 | — | 2 | 3 | no | — |
| 25 | C2×#11719919 | — | 2 | 6 | no | — |
| 26 | C2×#11719919 | — | 2 | 3 | no | — |
| 27 | C2×C22⋊Dic53 | — | 2 | 6 | no | — |
| 28 | C2×C22⋊Dic53 | — | 2 | 6 | no | — |
| 29 | C2×C22⋊Dic53 | C23 | 1 | 6 | yes | — |
| 30 | #13133132 | — | 2 | 24 | no | — |
| 31 | #13133124 | — | 2 | 3 | no | — |
| 32 | #13133124 | — | 2 | 6 | no | — |
| 33 | #13133124 | — | 2 | 3 | no | — |
| 34 | 2−1+4×C53 | — | 2 | 48 | no | — |
| 35 | H(5,𝔽2)×C53 | — | 2 | 48 | no | — |
| 36 | H(5,𝔽2)×C53 | — | 2 | 24 | no | — |
| 37 | H(5,𝔽2)×C53 | D4 | 1 | 16 | yes | — |
| 38 | C2×#41×C53 | — | 2 | 4 | no | — |
| 39 | C2×#41×C53 | — | 2 | 12 | no | — |
| 40 | C2×#41×C53 | — | 2 | 48 | no | — |
| 41 | C2×#41×C53 | — | 2 | 48 | no | — |
| 42 | C2×#41×C53 | — | 2 | 24 | no | — |
| 43 | C2×#41×C53 | — | 2 | 6 | no | — |
| 44 | C2×#41×C53 | — | 2 | 12 | no | — |
| 45 | C2×#41×C53 | — | 2 | 6 | no | — |
| 46 | C22×Q8×C53 | — | 2 | 6 | no | — |
| 47 | C22×Q8×C53 | D4 | 1 | 3 | yes | — |
| 48 | C22×D4×C53 | — | 2 | 6 | no | — |
| 49 | C22×D4×C53 | — | 2 | 12 | no | — |
| 50 | C22×D4×C53 | D4 | 1 | 3 | yes | — |
| 51 | C22×D4×C53 | C23 | 1 | 6 | yes | — |
| 52 | C23×C4×C53 | — | 2 | 4 | no | — |
| 53 | C23×C4×C53 | D4 | 1 | 6 | yes | — |
| 54 | C23×C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 55 | C24×C53 | — | 2 | 4 | no | — |
| 56 | C24×C53 | C2×D4 | 1 | 6 | yes | — |
| 57 | C24×C53 | C24 | 1 | 1 | yes | characteristic |
| 58 | #41×C53 | — | 2 | 32 | no | — |
| 59 | #41×C53 | — | 2 | 48 | no | — |
| 60 | #41×C53 | — | 2 | 96 | no | — |
| 61 | #41×C53 | — | 2 | 48 | no | — |
| 62 | #41×C53 | — | 2 | 48 | no | — |
| 63 | #41×C53 | — | 2 | 48 | no | — |
| 64 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 65 | C2×Q8×C53 | — | 2 | 48 | no | — |
| 66 | C2×Q8×C53 | — | 2 | 6 | no | — |
| 67 | C2×Q8×C53 | — | 2 | 12 | no | — |
| 68 | C2×D4×C53 | — | 2 | 12 | no | — |
| 69 | C2×D4×C53 | — | 2 | 48 | no | — |
| 70 | C2×D4×C53 | — | 2 | 48 | no | — |
| 71 | C2×D4×C53 | — | 2 | 48 | no | — |
| 72 | C2×D4×C53 | — | 2 | 12 | no | — |
| 73 | C2×D4×C53 | — | 2 | 12 | no | — |
| 74 | C2×D4×C53 | — | 2 | 12 | no | — |
| 75 | C2×D4×C53 | — | 2 | 6 | no | — |
| 76 | C2×D4×C53 | C2×D4 | 1 | 24 | yes | — |
| 77 | C22×C4×C53 | — | 2 | 12 | no | — |
| 78 | C22×C4×C53 | — | 2 | 32 | no | — |
| 79 | C22×C4×C53 | — | 2 | 12 | no | — |
| 80 | C22×C4×C53 | — | 2 | 4 | no | — |
| 81 | C22×C4×C53 | — | 2 | 6 | no | — |
| 82 | C22×C4×C53 | — | 2 | 6 | no | — |
| 83 | C22×C4×C53 | — | 2 | 12 | no | — |
| 84 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 85 | C22×C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 86 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 87 | C22×Dic53 | — | 2 | 3 | no | — |
| 88 | C22×Dic53 | — | 2 | 6 | no | — |
| 89 | C22×Dic53 | — | 2 | 3 | no | — |
| 90 | C22×Dic53 | — | 2 | 4 | no | — |
| 91 | C22×Dic53 | — | 4 | 2 | no | — |
| 92 | C22⋊Dic53 | — | 2 | 12 | no | — |
| 93 | C22⋊Dic53 | — | 4 | 12 | no | — |
| 94 | C22⋊Dic53 | — | 4 | 12 | no | — |
| 95 | #11719919 | — | 4 | 1 | no | — |
| 96 | #11719919 | — | 4 | 3 | no | — |
| 97 | #11719919 | — | 4 | 12 | no | — |
| 98 | C23×C53 | — | 2 | 16 | no | — |
| 99 | C23×C53 | — | 2 | 6 | no | — |
| 100 | C23×C53 | — | 2 | 6 | no | — |
| 101 | C23×C53 | — | 4 | 1 | no | — |
| 102 | C23×C53 | — | 4 | 1 | no | — |
| 103 | C23×C53 | — | 4 | 6 | no | — |
| 104 | C23×C53 | — | 4 | 6 | no | — |
| 105 | C23×C53 | — | 8 | 4 | no | — |
| 106 | C23×C53 | — | 8 | 4 | no | — |
| 107 | C23×C53 | #120 | 1 | 12 | yes | — |
| 108 | C23×C53 | C22×D4 | 1 | 3 | yes | — |
| 109 | C23×C53 | C22×D4 | 1 | 4 | yes | — |
| 110 | Q8×C53 | — | 2 | 16 | no | — |
| 111 | Q8×C53 | — | 2 | 24 | no | — |
| 112 | D4×C53 | — | 2 | 48 | no | — |
| 113 | D4×C53 | — | 2 | 24 | no | — |
| 114 | D4×C53 | — | 2 | 48 | no | — |
| 115 | C2×C4×C53 | — | 2 | 24 | no | — |
| 116 | C2×C4×C53 | — | 2 | 48 | no | — |
| 117 | C2×C4×C53 | — | 2 | 24 | no | — |
| 118 | C2×C4×C53 | — | 2 | 6 | no | — |
| 119 | C2×C4×C53 | — | 2 | 6 | no | — |
| 120 | C2×C4×C53 | — | 2 | 6 | no | — |
| 121 | C2×C4×C53 | — | 2 | 6 | no | — |
| 122 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C53 | #120 | 1 | 3 | yes | — |
| 124 | C2×C4×C53 | #120 | 1 | 12 | yes | — |
| 125 | C2×Dic53 | — | 2 | 6 | no | — |
| 126 | C2×Dic53 | — | 4 | 3 | no | — |
| 127 | C2×Dic53 | — | 4 | 1 | no | — |
| 128 | C2×Dic53 | — | 4 | 6 | no | — |
| 129 | C2×Dic53 | — | 4 | 6 | no | — |
| 130 | C2×Dic53 | — | 4 | 1 | no | — |
| 131 | C2×Dic53 | — | 4 | 6 | no | — |
| 132 | C2×Dic53 | — | 4 | 6 | no | — |
| 133 | #32901 | — | 53 | 1 | no | maximal, Sylow |
| 134 | C22×C53 | — | 2 | 3 | no | — |
| 135 | C22×C53 | — | 2 | 3 | no | — |
| 136 | C22×C53 | — | 2 | 12 | no | — |
| 137 | C22×C53 | — | 4 | 4 | no | — |
| 138 | C22×C53 | — | 4 | 12 | no | — |
| 139 | C22×C53 | — | 4 | 6 | no | — |
| 140 | C22×C53 | — | 4 | 6 | no | — |
| 141 | C22×C53 | — | 8 | 6 | no | — |
| 142 | C22×C53 | — | 8 | 16 | no | — |
| 143 | C22×C53 | C2×#120 | 1 | 6 | yes | — |
| 144 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 145 | C4×C53 | — | 2 | 1 | no | — |
| 146 | C4×C53 | — | 2 | 12 | no | — |
| 147 | C4×C53 | — | 2 | 3 | no | — |
| 148 | Dic53 | — | 4 | 2 | no | — |
| 149 | Dic53 | — | 8 | 6 | no | — |
| 150 | Dic53 | — | 8 | 1 | no | — |
| 151 | C2×#585 | D53 | 1 | 1 | yes | characteristic |
| 152 | C2×#458 | — | 53 | 4 | no | — |
| 153 | #2406 | — | 53 | 3 | no | — |
| 154 | #1790 | — | 53 | 4 | no | — |
| 155 | #1782 | — | 53 | 3 | no | — |
| 156 | C2×C53 | — | 2 | 6 | no | — |
| 157 | C2×C53 | — | 4 | 12 | no | — |
| 158 | C2×C53 | #2846 | 1 | 1 | yes | characteristic |
| 159 | C2×C53 | #3025 | 1 | 1 | yes | characteristic |
| 160 | C2×C53 | #3028 | 1 | 1 | yes | characteristic |
| 161 | #585 | — | 2 | 16 | no | — |
| 162 | C2×2−1+4 | — | 2 | 3 | no | — |
| 163 | C2×H(5,𝔽2) | — | 2 | 3 | no | — |
| 164 | C2×H(5,𝔽2) | C2×D53 | 1 | 4 | yes | — |
| 165 | C22×#41 | — | 2 | 6 | no | — |
| 166 | C22×#41 | C2×D53 | 1 | 3 | yes | — |
| 167 | C2×#123 | — | 53 | 6 | no | — |
| 168 | C22×C4⋊C4 | — | 53 | 1 | no | — |
| 169 | #458 | — | 106 | 16 | no | — |
| 170 | C2×#99 | — | 53 | 12 | no | — |
| 171 | #395 | — | 106 | 3 | no | — |
| 172 | #394 | — | 106 | 3 | no | — |
| 173 | #386 | — | 53 | 6 | no | — |
| 174 | #385 | — | 106 | 6 | no | — |
| 175 | #381 | — | 53 | 3 | no | — |
| 176 | C53 | #32901 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 177 | 2−1+4 | — | 2 | 48 | no | — |
| 178 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 179 | H(5,𝔽2) | — | 2 | 24 | no | — |
| 180 | H(5,𝔽2) | C22⋊D53 | 1 | 16 | yes | — |
| 181 | C2×#41 | — | 2 | 4 | no | — |
| 182 | C2×#41 | — | 2 | 12 | no | — |
| 183 | C2×#41 | — | 2 | 48 | no | — |
| 184 | C2×#41 | — | 2 | 48 | no | — |
| 185 | C2×#41 | — | 2 | 24 | no | — |
| 186 | C2×#41 | — | 2 | 6 | no | — |
| 187 | C2×#41 | — | 2 | 12 | no | — |
| 188 | C2×#41 | — | 2 | 6 | no | — |
| 189 | C22×Q8 | — | 2 | 6 | no | — |
| 190 | C22×Q8 | C22⋊D53 | 1 | 3 | yes | — |
| 191 | C22×D4 | — | 2 | 6 | no | — |
| 192 | C22×D4 | — | 2 | 12 | no | — |
| 193 | C22×D4 | C22⋊D53 | 1 | 3 | yes | — |
| 194 | C22×D4 | C22×D53 | 1 | 6 | yes | — |
| 195 | C23×C4 | — | 2 | 4 | no | — |
| 196 | C23×C4 | — | 53 | 1 | no | — |
| 197 | C23×C4 | — | 53 | 1 | no | — |
| 198 | C23×C4 | C22⋊D53 | 1 | 6 | yes | — |
| 199 | C23×C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 200 | #123 | — | 106 | 24 | no | — |
| 201 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 202 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 203 | C2×C4⋊C4 | — | 106 | 6 | no | — |
| 204 | C2×C22⋊C4 | — | 53 | 6 | no | — |
| 205 | C2×C22⋊C4 | — | 106 | 6 | no | — |
| 206 | C2×C22⋊C4 | — | 106 | 6 | no | — |
| 207 | #99 | — | 106 | 24 | no | — |
| 208 | #95 | — | 106 | 3 | no | — |
| 209 | #95 | — | 106 | 3 | no | — |
| 210 | #95 | — | 106 | 6 | no | — |
| 211 | C24 | — | 2 | 4 | no | — |
| 212 | C24 | C2×C22⋊D53 | 1 | 6 | yes | — |
| 213 | C24 | C23×D53 | 1 | 1 | yes | characteristic |
| 214 | #41 | — | 2 | 32 | no | — |
| 215 | #41 | — | 2 | 48 | no | — |
| 216 | #41 | — | 2 | 96 | no | — |
| 217 | #41 | — | 2 | 48 | no | — |
| 218 | #41 | — | 2 | 48 | no | — |
| 219 | #41 | — | 2 | 48 | no | — |
| 220 | C2×Q8 | — | 2 | 4 | no | — |
| 221 | C2×Q8 | — | 2 | 48 | no | — |
| 222 | C2×Q8 | — | 2 | 6 | no | — |
| 223 | C2×Q8 | — | 2 | 12 | no | — |
| 224 | C2×D4 | — | 2 | 12 | no | — |
| 225 | C2×D4 | — | 2 | 48 | no | — |
| 226 | C2×D4 | — | 2 | 48 | no | — |
| 227 | C2×D4 | — | 2 | 48 | no | — |
| 228 | C2×D4 | — | 2 | 12 | no | — |
| 229 | C2×D4 | — | 2 | 12 | no | — |
| 230 | C2×D4 | — | 2 | 12 | no | — |
| 231 | C2×D4 | — | 2 | 6 | no | — |
| 232 | C2×D4 | C2×C22⋊D53 | 1 | 24 | yes | — |
| 233 | C22×C4 | — | 2 | 12 | no | — |
| 234 | C22×C4 | — | 2 | 32 | no | — |
| 235 | C22×C4 | — | 2 | 12 | no | — |
| 236 | C22×C4 | — | 2 | 4 | no | — |
| 237 | C22×C4 | — | 2 | 6 | no | — |
| 238 | C22×C4 | — | 2 | 6 | no | — |
| 239 | C22×C4 | — | 2 | 12 | no | — |
| 240 | C22×C4 | — | 106 | 3 | no | — |
| 241 | C22×C4 | — | 106 | 6 | no | — |
| 242 | C22×C4 | — | 106 | 3 | no | — |
| 243 | C22×C4 | — | 106 | 4 | no | — |
| 244 | C22×C4 | — | 212 | 2 | no | — |
| 245 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 246 | C22×C4 | C2×C22⋊D53 | 1 | 6 | yes | — |
| 247 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 248 | C4⋊C4 | — | 212 | 1 | no | — |
| 249 | C4⋊C4 | — | 212 | 3 | no | — |
| 250 | C4⋊C4 | — | 212 | 12 | no | — |
| 251 | C22⋊C4 | — | 106 | 12 | no | — |
| 252 | C22⋊C4 | — | 212 | 12 | no | — |
| 253 | C22⋊C4 | — | 212 | 12 | no | — |
| 254 | C23 | — | 2 | 16 | no | — |
| 255 | C23 | — | 2 | 6 | no | — |
| 256 | C23 | — | 2 | 6 | no | — |
| 257 | C23 | — | 4 | 1 | no | — |
| 258 | C23 | — | 4 | 1 | no | — |
| 259 | C23 | — | 4 | 6 | no | — |
| 260 | C23 | — | 4 | 6 | no | — |
| 261 | C23 | — | 8 | 4 | no | — |
| 262 | C23 | — | 8 | 4 | no | — |
| 263 | C23 | #13133186 | 1 | 12 | yes | — |
| 264 | C23 | C22×C22⋊D53 | 1 | 3 | yes | — |
| 265 | C23 | C22×C22⋊D53 | 1 | 4 | yes | — |
| 266 | Q8 | — | 2 | 16 | no | — |
| 267 | Q8 | — | 2 | 24 | no | — |
| 268 | D4 | — | 2 | 48 | no | — |
| 269 | D4 | — | 2 | 24 | no | — |
| 270 | D4 | — | 2 | 48 | no | — |
| 271 | C2×C4 | — | 2 | 24 | no | — |
| 272 | C2×C4 | — | 2 | 48 | no | — |
| 273 | C2×C4 | — | 2 | 24 | no | — |
| 274 | C2×C4 | — | 2 | 6 | no | — |
| 275 | C2×C4 | — | 2 | 6 | no | — |
| 276 | C2×C4 | — | 2 | 6 | no | — |
| 277 | C2×C4 | — | 2 | 6 | no | — |
| 278 | C2×C4 | — | 106 | 6 | no | — |
| 279 | C2×C4 | — | 212 | 6 | no | — |
| 280 | C2×C4 | — | 212 | 6 | no | — |
| 281 | C2×C4 | — | 212 | 6 | no | — |
| 282 | C2×C4 | — | 212 | 6 | no | — |
| 283 | C2×C4 | — | 212 | 1 | no | — |
| 284 | C2×C4 | — | 212 | 3 | no | — |
| 285 | C2×C4 | — | 212 | 1 | no | — |
| 286 | C2×C4 | #13133186 | 1 | 12 | yes | — |
| 287 | C2×C4 | #13133186 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #13133186 | 1 | 3 | yes | — |
| 289 | C22 | — | 2 | 3 | no | — |
| 290 | C22 | — | 2 | 3 | no | — |
| 291 | C22 | — | 2 | 12 | no | — |
| 292 | C22 | — | 4 | 4 | no | — |
| 293 | C22 | — | 4 | 12 | no | — |
| 294 | C22 | — | 4 | 6 | no | — |
| 295 | C22 | — | 4 | 6 | no | — |
| 296 | C22 | — | 8 | 6 | no | — |
| 297 | C22 | — | 8 | 16 | no | — |
| 298 | C22 | C2×#13133186 | 1 | 6 | yes | — |
| 299 | C22 | C2×#13133186 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 2 | 12 | no | — |
| 302 | C4 | — | 2 | 3 | no | — |
| 303 | C4 | — | 424 | 6 | no | — |
| 304 | C4 | — | 424 | 1 | no | — |
| 305 | C4 | — | 212 | 2 | no | — |
| 306 | C2 | — | 2 | 6 | no | — |
| 307 | C2 | — | 4 | 12 | no | — |
| 308 | C2 | #20282992 | 1 | 1 | yes | characteristic |
| 309 | C2 | #20284107 | 1 | 1 | yes | characteristic |
| 310 | C2 | #20284121 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16810813 | C2 | semidirect |
| 2. #20278046 | C2 | semidirect |
| 3. #20274663 | C2 | semidirect |
| 4. #20274633 | C2 | semidirect |
| 6. C2×#13133198 | C22 | semidirect |
| 7. C22×#11719919 | C22 | semidirect |
| 9. C2×#13133132 | C22 | semidirect |
| 12. #16810503 | C22 | semidirect |
| 14. #16810486 | C22 | semidirect |
| 176. C53 | #32901 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16810813 | 2 |
| #20278046 | 2 |
| #20274663 | 2 |
| #20274633 | 2 |
| C2×C53×#585 | 2 |
| #32901 | 53 |