Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
848 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 848 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16811339 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810419 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20275116 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20275109 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×C53×#356 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16811339 | — | 2 | 4 | no | — |
| 7 | #16811339 | — | 2 | 2 | no | — |
| 8 | #16811339 | C4 | 1 | 4 | yes | complemented |
| 9 | C2×#13133206 | — | 2 | 1 | no | — |
| 10 | C2×#13133206 | C22 | 1 | 2 | yes | — |
| 11 | C2×#13133192 | — | 2 | 2 | no | — |
| 12 | #16811101 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×D212 | — | 2 | 1 | no | — |
| 14 | C2×#13133175 | — | 2 | 2 | no | — |
| 15 | #16810632 | — | 2 | 1 | no | — |
| 16 | #16810629 | — | 2 | 2 | no | — |
| 17 | C2×#13133136 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16810419 | — | 2 | 4 | no | — |
| 19 | #16810384 | — | 2 | 2 | no | — |
| 20 | #16810383 | — | 2 | 1 | no | — |
| 21 | C2×C4⋊Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#101×C53 | C22 | 1 | 2 | yes | — |
| 23 | C53×#356 | — | 2 | 2 | no | — |
| 24 | C22×D212 | — | 2 | 1 | no | — |
| 25 | C22×D212 | — | 2 | 1 | no | — |
| 26 | C22×D212 | D4 | 1 | 2 | yes | — |
| 27 | C22×C4×D53 | — | 2 | 2 | no | — |
| 28 | C2×#11719917 | — | 2 | 1 | no | — |
| 29 | #13133206 | — | 2 | 4 | no | — |
| 30 | #13133206 | — | 2 | 4 | no | — |
| 31 | #13133206 | — | 2 | 2 | no | — |
| 32 | #13133206 | — | 2 | 2 | no | — |
| 33 | #13133206 | C2×C4 | 1 | 4 | yes | — |
| 34 | #13133206 | D4 | 1 | 4 | yes | — |
| 35 | #13133192 | — | 2 | 8 | no | — |
| 36 | #13133192 | — | 2 | 8 | no | — |
| 37 | Q8×Dic53 | — | 4 | 2 | no | — |
| 38 | C4×D212 | — | 2 | 4 | no | — |
| 39 | C4×D212 | — | 2 | 4 | no | — |
| 40 | #13133176 | — | 2 | 2 | no | — |
| 41 | #13133176 | — | 2 | 1 | no | — |
| 42 | #13133175 | — | 2 | 8 | no | — |
| 43 | #13133175 | — | 2 | 8 | no | — |
| 44 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 45 | C2×#11719921 | — | 2 | 2 | no | — |
| 46 | C2×#11719921 | — | 2 | 2 | no | — |
| 47 | C2×#11719921 | — | 2 | 2 | no | — |
| 48 | C2×C4×Dic53 | — | 2 | 1 | no | — |
| 49 | C2×C4×Dic53 | D4 | 1 | 2 | yes | — |
| 50 | #13133141 | — | 4 | 2 | no | — |
| 51 | #13133139 | — | 4 | 4 | no | — |
| 52 | #13133136 | — | 2 | 2 | no | — |
| 53 | C22×Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 55 | C2×M16×C53 | — | 2 | 2 | no | — |
| 56 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 57 | C4⋊Q8×C53 | C2×C4 | 1 | 2 | yes | — |
| 58 | C4⋊Q8×C53 | D4 | 1 | 2 | yes | — |
| 59 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 60 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 61 | #101×C53 | — | 2 | 2 | no | — |
| 62 | #101×C53 | — | 2 | 2 | no | — |
| 63 | C23×D53 | — | 2 | 2 | no | — |
| 64 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 65 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 66 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 67 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 68 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 69 | C2×Q8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 70 | C2×Q8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C2×Q8×C53 | C2×D4 | 1 | 2 | yes | — |
| 72 | C2×Q8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C22×C4×C53 | — | 2 | 1 | no | — |
| 74 | C22×C4×C53 | — | 2 | 1 | no | — |
| 75 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 77 | C22×Dic53 | — | 2 | 2 | no | — |
| 78 | C2×D212 | — | 2 | 4 | no | — |
| 79 | C2×D212 | — | 2 | 4 | no | — |
| 80 | C2×D212 | — | 2 | 2 | no | — |
| 81 | C2×D212 | — | 4 | 4 | no | — |
| 82 | C2×D212 | — | 4 | 2 | no | — |
| 83 | C2×D212 | — | 4 | 2 | no | — |
| 84 | C2×D212 | — | 8 | 1 | no | — |
| 85 | C2×D212 | — | 8 | 1 | no | — |
| 86 | C2×D212 | C22⋊C4 | 1 | 4 | yes | — |
| 87 | C2×C4×D53 | — | 2 | 8 | no | — |
| 88 | C2×C4×D53 | — | 4 | 8 | no | — |
| 89 | M16×C53 | — | 4 | 2 | no | — |
| 90 | M16×C53 | — | 4 | 2 | no | — |
| 91 | C2×C8×C53 | — | 4 | 2 | no | — |
| 92 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 93 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 94 | C42×C53 | — | 2 | 1 | no | — |
| 95 | C42×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 96 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 97 | #11719921 | — | 2 | 8 | no | — |
| 98 | #11719921 | — | 2 | 4 | no | — |
| 99 | #11719921 | — | 2 | 4 | no | — |
| 100 | #11719921 | — | 2 | 4 | no | — |
| 101 | #11719921 | — | 2 | 4 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 4 | no | — |
| 103 | C4⋊Dic53 | — | 2 | 4 | no | — |
| 104 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 105 | C4×Dic53 | — | 2 | 4 | no | — |
| 106 | C4×Dic53 | — | 2 | 2 | no | — |
| 107 | C4×Dic53 | — | 4 | 2 | no | — |
| 108 | C4×Dic53 | — | 4 | 1 | no | — |
| 109 | C4×Dic53 | C22⋊C4 | 1 | 4 | yes | — |
| 110 | #11719917 | — | 4 | 1 | no | — |
| 111 | #11719917 | — | 4 | 1 | no | — |
| 112 | C2×#88322 | — | 4 | 1 | no | — |
| 113 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 114 | C22×D53 | — | 2 | 4 | no | — |
| 115 | C22×D53 | — | 4 | 4 | no | — |
| 116 | C22×D53 | — | 4 | 4 | no | — |
| 117 | C22×D53 | — | 4 | 2 | no | — |
| 118 | C22×D53 | — | 4 | 2 | no | — |
| 119 | Q8×C53 | — | 4 | 1 | no | — |
| 120 | Q8×C53 | — | 4 | 2 | no | — |
| 121 | Q8×C53 | — | 4 | 1 | no | — |
| 122 | Q8×C53 | — | 4 | 2 | no | — |
| 123 | Q8×C53 | — | 8 | 1 | no | — |
| 124 | C2×C4×C53 | — | 2 | 4 | no | — |
| 125 | C2×C4×C53 | — | 2 | 2 | no | — |
| 126 | C2×C4×C53 | — | 2 | 2 | no | — |
| 127 | C2×C4×C53 | — | 2 | 1 | no | — |
| 128 | C2×C4×C53 | — | 2 | 2 | no | — |
| 129 | C2×C4×C53 | — | 2 | 1 | no | — |
| 130 | C2×C4×C53 | — | 2 | 2 | no | — |
| 131 | C2×C4×C53 | — | 4 | 1 | no | — |
| 132 | C2×C4×C53 | — | 4 | 1 | no | — |
| 133 | C2×C4×C53 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 134 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C53 | #120 | 1 | 2 | yes | — |
| 136 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 137 | C2×Dic53 | — | 2 | 4 | no | — |
| 138 | C2×Dic53 | — | 4 | 4 | no | — |
| 139 | C2×Dic53 | — | 4 | 2 | no | — |
| 140 | D212 | — | 4 | 4 | no | — |
| 141 | D212 | — | 4 | 8 | no | — |
| 142 | D212 | — | 8 | 2 | no | — |
| 143 | D212 | — | 8 | 4 | no | — |
| 144 | C4×D53 | — | 4 | 16 | no | — |
| 145 | C8×C53 | — | 8 | 2 | no | — |
| 146 | #88322 | — | 8 | 1 | no | — |
| 147 | #12776 | — | 53 | 1 | no | maximal, Sylow |
| 148 | C22×C53 | — | 2 | 2 | no | — |
| 149 | C22×C53 | #379 | 1 | 1 | yes | characteristic |
| 150 | C22×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 151 | C22×C53 | #458 | 1 | 1 | yes | characteristic |
| 152 | C2×D53 | — | 4 | 4 | no | — |
| 153 | C2×D53 | — | 4 | 4 | no | — |
| 154 | C2×D53 | — | 4 | 4 | no | — |
| 155 | C2×D53 | — | 8 | 4 | no | — |
| 156 | C2×D53 | — | 8 | 4 | no | — |
| 157 | C4×C53 | — | 2 | 4 | no | — |
| 158 | C4×C53 | — | 2 | 1 | no | — |
| 159 | C4×C53 | — | 2 | 2 | no | — |
| 160 | C4×C53 | — | 2 | 2 | no | — |
| 161 | C4×C53 | — | 2 | 1 | no | — |
| 162 | C4×C53 | — | 4 | 2 | no | — |
| 163 | Dic53 | — | 4 | 4 | no | — |
| 164 | Dic53 | — | 8 | 2 | no | — |
| 165 | C2×#555 | — | 53 | 1 | no | — |
| 166 | C2×#356 | D53 | 1 | 1 | yes | characteristic, complemented |
| 167 | C2×#355 | — | 53 | 2 | no | — |
| 168 | #1888 | — | 53 | 2 | no | — |
| 169 | #1886 | — | 53 | 1 | no | — |
| 170 | C2×C53 | — | 2 | 1 | no | — |
| 171 | C2×C53 | — | 2 | 1 | no | — |
| 172 | C2×C53 | #1790 | 1 | 1 | yes | characteristic |
| 173 | C2×C53 | #2204 | 1 | 2 | yes | — |
| 174 | D53 | — | 8 | 4 | no | — |
| 175 | #555 | — | 53 | 4 | no | — |
| 176 | #555 | — | 106 | 4 | no | — |
| 177 | #555 | — | 106 | 2 | no | — |
| 178 | C2×C4⋊Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 179 | C2×#124 | — | 53 | 2 | no | — |
| 180 | C2×#124 | — | 106 | 1 | no | — |
| 181 | C2×#121 | — | 106 | 2 | no | — |
| 182 | #519 | — | 53 | 1 | no | — |
| 183 | C2×C4×D4 | — | 106 | 1 | no | — |
| 184 | C2×#117 | — | 106 | 2 | no | — |
| 185 | #419 | — | 106 | 1 | no | — |
| 186 | #418 | — | 106 | 2 | no | — |
| 187 | C2×#101 | — | 53 | 1 | no | — |
| 188 | C2×#101 | C2×D53 | 1 | 2 | yes | — |
| 189 | #356 | — | 2 | 2 | no | — |
| 190 | #355 | — | 106 | 4 | no | — |
| 191 | #339 | — | 106 | 2 | no | — |
| 192 | #339 | — | 106 | 1 | no | — |
| 193 | C53 | #12776 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | C22×Q8 | D212 | 1 | 1 | yes | characteristic |
| 195 | C22×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 196 | C22×D4 | — | 53 | 2 | no | — |
| 197 | C22×D4 | — | 106 | 1 | no | — |
| 198 | C22×D4 | — | 106 | 1 | no | — |
| 199 | C23×C4 | — | 106 | 2 | no | — |
| 200 | C2×M16 | — | 2 | 2 | no | — |
| 201 | C2×M16 | — | 106 | 1 | no | — |
| 202 | C4⋊Q8 | — | 2 | 2 | no | — |
| 203 | C4⋊Q8 | C4×D53 | 1 | 2 | yes | — |
| 204 | C4⋊Q8 | C22⋊D53 | 1 | 2 | yes | — |
| 205 | #124 | — | 53 | 4 | no | — |
| 206 | #124 | — | 53 | 4 | no | — |
| 207 | #124 | — | 106 | 4 | no | — |
| 208 | #124 | — | 106 | 2 | no | — |
| 209 | #124 | — | 106 | 2 | no | — |
| 210 | #124 | — | 106 | 4 | no | — |
| 211 | #121 | — | 106 | 8 | no | — |
| 212 | #121 | — | 106 | 8 | no | — |
| 213 | C4×Q8 | — | 212 | 2 | no | — |
| 214 | C4×D4 | — | 106 | 4 | no | — |
| 215 | C4×D4 | — | 106 | 4 | no | — |
| 216 | #117 | — | 106 | 8 | no | — |
| 217 | #117 | — | 106 | 8 | no | — |
| 218 | #117 | — | 106 | 1 | no | — |
| 219 | #117 | — | 106 | 2 | no | — |
| 220 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 221 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 222 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 223 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 224 | C2×C22⋊C4 | — | 106 | 2 | no | — |
| 225 | C2×C42 | — | 53 | 2 | no | — |
| 226 | C2×C42 | — | 106 | 1 | no | — |
| 227 | C2×C42 | D212 | 1 | 1 | yes | characteristic |
| 228 | #103 | — | 212 | 2 | no | — |
| 229 | #102 | — | 212 | 4 | no | — |
| 230 | #101 | — | 2 | 2 | no | — |
| 231 | #101 | — | 2 | 2 | no | — |
| 232 | #101 | — | 106 | 2 | no | — |
| 233 | C24 | — | 106 | 2 | no | — |
| 234 | C2×Q8 | — | 2 | 1 | no | — |
| 235 | C2×Q8 | — | 4 | 1 | no | — |
| 236 | C2×Q8 | — | 4 | 1 | no | — |
| 237 | C2×Q8 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | — | 4 | 1 | no | — |
| 239 | C2×Q8 | #11719921 | 1 | 2 | yes | — |
| 240 | C2×Q8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 241 | C2×Q8 | D4×D53 | 1 | 2 | yes | — |
| 242 | C2×Q8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 243 | C2×D4 | — | 53 | 4 | no | — |
| 244 | C2×D4 | — | 106 | 4 | no | — |
| 245 | C2×D4 | — | 106 | 4 | no | — |
| 246 | C2×D4 | — | 106 | 2 | no | — |
| 247 | C2×D4 | — | 424 | 1 | no | — |
| 248 | C2×D4 | — | 424 | 1 | no | — |
| 249 | C2×D4 | — | 212 | 2 | no | — |
| 250 | C2×D4 | — | 212 | 2 | no | — |
| 251 | C2×D4 | — | 212 | 4 | no | — |
| 252 | C22×C4 | — | 2 | 1 | no | — |
| 253 | C22×C4 | — | 2 | 1 | no | — |
| 254 | C22×C4 | — | 106 | 8 | no | — |
| 255 | C22×C4 | — | 106 | 2 | no | — |
| 256 | C22×C4 | — | 212 | 8 | no | — |
| 257 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 258 | C22×C4 | D4×D53 | 1 | 2 | yes | — |
| 259 | M16 | — | 4 | 2 | no | — |
| 260 | M16 | — | 4 | 2 | no | — |
| 261 | M16 | — | 212 | 1 | no | — |
| 262 | M16 | — | 212 | 1 | no | — |
| 263 | C2×C8 | — | 4 | 2 | no | — |
| 264 | C2×C8 | — | 212 | 1 | no | — |
| 265 | C4⋊C4 | — | 2 | 4 | no | — |
| 266 | C4⋊C4 | — | 2 | 4 | no | — |
| 267 | C4⋊C4 | — | 106 | 4 | no | — |
| 268 | C4⋊C4 | — | 106 | 4 | no | — |
| 269 | C4⋊C4 | — | 106 | 2 | no | — |
| 270 | C22⋊C4 | — | 106 | 8 | no | — |
| 271 | C22⋊C4 | — | 106 | 4 | no | — |
| 272 | C22⋊C4 | — | 106 | 4 | no | — |
| 273 | C22⋊C4 | — | 106 | 4 | no | — |
| 274 | C22⋊C4 | — | 106 | 4 | no | — |
| 275 | C22⋊C4 | — | 212 | 1 | no | — |
| 276 | C42 | — | 2 | 1 | no | — |
| 277 | C42 | — | 53 | 4 | no | — |
| 278 | C42 | — | 106 | 4 | no | — |
| 279 | C42 | — | 106 | 2 | no | — |
| 280 | C42 | — | 212 | 2 | no | — |
| 281 | C42 | — | 212 | 1 | no | — |
| 282 | C42 | #11719921 | 1 | 2 | yes | — |
| 283 | C23 | — | 106 | 4 | no | — |
| 284 | C23 | — | 212 | 4 | no | — |
| 285 | C23 | — | 212 | 4 | no | — |
| 286 | C23 | — | 212 | 2 | no | — |
| 287 | C23 | — | 212 | 2 | no | — |
| 288 | C23 | #13133187 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 4 | 1 | no | — |
| 290 | Q8 | — | 4 | 2 | no | — |
| 291 | Q8 | — | 4 | 1 | no | — |
| 292 | Q8 | — | 4 | 2 | no | — |
| 293 | Q8 | — | 8 | 1 | no | — |
| 294 | D4 | — | 424 | 2 | no | — |
| 295 | D4 | — | 424 | 4 | no | — |
| 296 | D4 | — | 212 | 4 | no | — |
| 297 | D4 | — | 212 | 8 | no | — |
| 298 | C2×C4 | — | 2 | 4 | no | — |
| 299 | C2×C4 | — | 2 | 2 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 2 | 2 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 2 | no | — |
| 304 | C2×C4 | — | 2 | 2 | no | — |
| 305 | C2×C4 | — | 4 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 106 | 4 | no | — |
| 308 | C2×C4 | — | 212 | 16 | no | — |
| 309 | C2×C4 | — | 212 | 2 | no | — |
| 310 | C2×C4 | — | 212 | 4 | no | — |
| 311 | C2×C4 | C22⋊C4×D53 | 1 | 2 | yes | — |
| 312 | C2×C4 | C2×#11719921 | 1 | 1 | yes | characteristic |
| 313 | C2×C4 | #13133185 | 1 | 2 | yes | — |
| 314 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 315 | C8 | — | 8 | 2 | no | — |
| 316 | C8 | — | 424 | 1 | no | — |
| 317 | C22 | — | 2 | 2 | no | — |
| 318 | C22 | — | 424 | 4 | no | — |
| 319 | C22 | — | 424 | 4 | no | — |
| 320 | C22 | — | 212 | 4 | no | — |
| 321 | C22 | — | 212 | 4 | no | — |
| 322 | C22 | — | 212 | 4 | no | — |
| 323 | C22 | #16810478 | 1 | 1 | yes | characteristic |
| 324 | C22 | #16810811 | 1 | 1 | yes | characteristic |
| 325 | C22 | #16810814 | 1 | 1 | yes | characteristic |
| 326 | C4 | — | 2 | 4 | no | — |
| 327 | C4 | — | 2 | 2 | no | — |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 2 | 2 | no | — |
| 330 | C4 | — | 2 | 1 | no | — |
| 331 | C4 | — | 4 | 2 | no | — |
| 332 | C4 | — | 424 | 2 | no | — |
| 333 | C4 | — | 212 | 4 | no | — |
| 334 | C2 | — | 2 | 1 | no | — |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 424 | 4 | no | — |
| 337 | C2 | #20274661 | 1 | 1 | yes | characteristic |
| 338 | C2 | #20276743 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#16810419 | C2 | semidirect |
| 3. #20275116 | C2 | semidirect |
| 4. #20275109 | C2 | semidirect |
| 5. C2×C53×#356 | C2 | semidirect |
| 8. #16811339 | C4 | semidirect |
| 166. C2×#356 | D53 | semidirect |
| 193. C53 | #12776 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811339 | 2 |
| C2×#16810419 | 2 |
| #20275116 | 2 |
| #20275109 | 2 |
| C2×C53×#356 | 2 |
| #12776 | 53 |