← C59×#33371 #28445480
Subgroups — C59×#33371 #28445480
Order 15104.
1208 conjugacy classes of subgroups.
The subgroup lattice has 274 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C59×#3500 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C59×#2888 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C59×#2135 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C59×#2132 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×2−1+4×C59 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×H(5,𝔽2)×C59 | C22 | 1 | 1 | yes | characteristic |
| 7 | C59×#568 | — | 2 | 2 | no | — |
| 8 | C59×#548 | — | 2 | 2 | no | — |
| 9 | C59×#546 | — | 2 | 6 | no | — |
| 10 | C59×#536 | C22 | 1 | 3 | yes | — |
| 11 | C59×#535 | C4 | 1 | 3 | yes | — |
| 12 | C59×#532 | C22 | 1 | 3 | yes | — |
| 13 | C2×#124×C59 | C4 | 1 | 3 | yes | — |
| 14 | C4×#41×C59 | C4 | 1 | 1 | yes | characteristic |
| 15 | C59×#413 | C22 | 1 | 12 | yes | — |
| 16 | C2×#100×C59 | — | 2 | 6 | no | — |
| 17 | C59×#356 | C22 | 1 | 12 | yes | complemented |
| 18 | C59×#355 | C22 | 1 | 4 | yes | complemented |
| 19 | 2−1+4×C59 | — | 4 | 4 | no | — |
| 20 | H(5,𝔽2)×C59 | — | 4 | 4 | no | — |
| 21 | C2×#41×C59 | — | 2 | 3 | no | — |
| 22 | C2×#41×C59 | C2×C4 | 1 | 3 | yes | — |
| 23 | C2×#41×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×#41×C59 | D4 | 1 | 2 | yes | — |
| 25 | C2×#41×C59 | C23 | 1 | 3 | yes | — |
| 26 | C22×Q8×C59 | — | 2 | 1 | no | — |
| 27 | C22×Q8×C59 | C2×C4 | 1 | 3 | yes | — |
| 28 | C22×D4×C59 | C2×C4 | 1 | 3 | yes | — |
| 29 | C22×D4×C59 | D4 | 1 | 6 | yes | — |
| 30 | #131×C59 | — | 4 | 8 | no | — |
| 31 | C2×M16×C59 | — | 2 | 12 | no | — |
| 32 | C4⋊Q8×C59 | C23 | 1 | 3 | yes | — |
| 33 | #127×C59 | C2×C4 | 1 | 3 | yes | — |
| 34 | #124×C59 | — | 2 | 6 | no | — |
| 35 | #124×C59 | C2×C4 | 1 | 3 | yes | — |
| 36 | #124×C59 | C2×C4 | 1 | 6 | yes | — |
| 37 | #124×C59 | C23 | 1 | 1 | yes | characteristic |
| 38 | #123×C59 | — | 2 | 6 | no | — |
| 39 | C22⋊Q8×C59 | — | 2 | 6 | no | — |
| 40 | #121×C59 | — | 2 | 6 | no | — |
| 41 | #121×C59 | — | 2 | 6 | no | — |
| 42 | #120×C59 | — | 2 | 6 | no | — |
| 43 | C4×Q8×C59 | — | 2 | 1 | no | — |
| 44 | C4×D4×C59 | — | 2 | 3 | no | — |
| 45 | C4×D4×C59 | — | 2 | 6 | no | — |
| 46 | #117×C59 | C2×C4 | 1 | 3 | yes | — |
| 47 | C2×C22⋊C4×C59 | — | 2 | 6 | no | — |
| 48 | C2×C42×C59 | C2×C4 | 1 | 3 | yes | — |
| 49 | #101×C59 | C23 | 1 | 8 | yes | — |
| 50 | #100×C59 | — | 2 | 12 | no | — |
| 51 | C24×C59 | C22⋊C4 | 1 | 6 | yes | — |
| 52 | #41×C59 | — | 2 | 8 | no | — |
| 53 | #41×C59 | — | 2 | 4 | no | — |
| 54 | #41×C59 | — | 4 | 6 | no | — |
| 55 | #41×C59 | — | 4 | 6 | no | — |
| 56 | #41×C59 | — | 4 | 12 | no | — |
| 57 | C2×Q8×C59 | — | 2 | 3 | no | — |
| 58 | C2×Q8×C59 | — | 4 | 6 | no | — |
| 59 | C2×Q8×C59 | — | 4 | 4 | no | — |
| 60 | C2×Q8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×Q8×C59 | C22×C4 | 1 | 3 | yes | — |
| 62 | C2×Q8×C59 | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×D4×C59 | — | 2 | 3 | no | — |
| 64 | C2×D4×C59 | — | 2 | 6 | no | — |
| 65 | C2×D4×C59 | — | 4 | 6 | no | — |
| 66 | C2×D4×C59 | — | 4 | 12 | no | — |
| 67 | C2×D4×C59 | C22⋊C4 | 1 | 3 | yes | — |
| 68 | C2×D4×C59 | C22⋊C4 | 1 | 6 | yes | — |
| 69 | C2×D4×C59 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C2×D4×C59 | C22×C4 | 1 | 3 | yes | — |
| 71 | C2×D4×C59 | C2×D4 | 1 | 6 | yes | — |
| 72 | C22×C4×C59 | — | 2 | 3 | no | — |
| 73 | C22×C4×C59 | — | 2 | 3 | no | — |
| 74 | C22×C4×C59 | — | 2 | 3 | no | — |
| 75 | C22×C4×C59 | C22×C4 | 1 | 3 | yes | — |
| 76 | C22×C4×C59 | C2×D4 | 1 | 6 | yes | — |
| 77 | M16×C59 | — | 2 | 8 | no | — |
| 78 | M16×C59 | — | 4 | 12 | no | — |
| 79 | C2×C8×C59 | — | 4 | 12 | no | — |
| 80 | C4⋊C4×C59 | — | 2 | 3 | no | — |
| 81 | C4⋊C4×C59 | — | 2 | 6 | no | — |
| 82 | C22⋊C4×C59 | — | 2 | 6 | no | — |
| 83 | C22⋊C4×C59 | — | 2 | 12 | no | — |
| 84 | C22⋊C4×C59 | — | 2 | 3 | no | — |
| 85 | C22⋊C4×C59 | — | 2 | 2 | no | — |
| 86 | C42×C59 | — | 2 | 3 | no | — |
| 87 | C42×C59 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C42×C59 | C22×C4 | 1 | 3 | yes | — |
| 89 | C23×C59 | — | 2 | 1 | no | — |
| 90 | C23×C59 | — | 4 | 6 | no | — |
| 91 | C23×C59 | — | 8 | 6 | no | — |
| 92 | C23×C59 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 93 | C23×C59 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 94 | Q8×C59 | — | 4 | 1 | no | — |
| 95 | Q8×C59 | — | 4 | 3 | no | — |
| 96 | Q8×C59 | — | 4 | 6 | no | — |
| 97 | Q8×C59 | #99 | 1 | 4 | yes | — |
| 98 | D4×C59 | — | 2 | 12 | no | — |
| 99 | D4×C59 | — | 2 | 6 | no | — |
| 100 | D4×C59 | — | 4 | 3 | no | — |
| 101 | D4×C59 | — | 4 | 6 | no | — |
| 102 | D4×C59 | — | 4 | 1 | no | — |
| 103 | D4×C59 | — | 4 | 6 | no | — |
| 104 | D4×C59 | — | 8 | 6 | no | — |
| 105 | C2×C4×C59 | — | 2 | 12 | no | — |
| 106 | C2×C4×C59 | — | 2 | 6 | no | — |
| 107 | C2×C4×C59 | — | 2 | 1 | no | — |
| 108 | C2×C4×C59 | — | 2 | 3 | no | — |
| 109 | C2×C4×C59 | — | 2 | 3 | no | — |
| 110 | C2×C4×C59 | — | 4 | 6 | no | — |
| 111 | C2×C4×C59 | — | 4 | 6 | no | — |
| 112 | C2×C4×C59 | — | 8 | 3 | no | — |
| 113 | C2×C4×C59 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 114 | C2×C4×C59 | C23×C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C59 | C22×D4 | 1 | 2 | yes | — |
| 116 | C8×C59 | — | 4 | 8 | no | — |
| 117 | #33371 | C59 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 118 | C22×C59 | — | 2 | 3 | no | — |
| 119 | C22×C59 | — | 2 | 6 | no | — |
| 120 | C22×C59 | — | 4 | 3 | no | — |
| 121 | C22×C59 | — | 4 | 6 | no | — |
| 122 | C22×C59 | — | 4 | 2 | no | — |
| 123 | C22×C59 | — | 4 | 1 | no | — |
| 124 | C22×C59 | — | 16 | 6 | no | — |
| 125 | C22×C59 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C4×C59 | — | 2 | 2 | no | — |
| 127 | C4×C59 | — | 2 | 1 | no | — |
| 128 | C4×C59 | — | 4 | 1 | no | — |
| 129 | C4×C59 | — | 4 | 3 | no | — |
| 130 | C4×C59 | — | 4 | 3 | no | — |
| 131 | C4×C59 | C2×#99 | 1 | 6 | yes | — |
| 132 | #3500 | C2×C59 | 1 | 1 | yes | characteristic |
| 133 | #2888 | C2×C59 | 1 | 2 | yes | complemented |
| 134 | #2135 | C2×C59 | 1 | 6 | yes | complemented |
| 135 | #2132 | C2×C59 | 1 | 6 | yes | complemented |
| 136 | C2×C59 | — | 2 | 1 | no | — |
| 137 | C2×C59 | — | 4 | 3 | no | — |
| 138 | C2×C59 | — | 4 | 6 | no | — |
| 139 | C2×C59 | — | 8 | 1 | no | — |
| 140 | C2×C59 | C22×#99 | 1 | 1 | yes | characteristic |
| 141 | C2×2−1+4 | C4×C59 | 1 | 1 | yes | characteristic |
| 142 | C2×H(5,𝔽2) | C22×C59 | 1 | 1 | yes | characteristic |
| 143 | #568 | — | 2 | 2 | no | — |
| 144 | #548 | — | 2 | 2 | no | — |
| 145 | #546 | — | 2 | 6 | no | — |
| 146 | #536 | C22×C59 | 1 | 3 | yes | — |
| 147 | #535 | C4×C59 | 1 | 3 | yes | — |
| 148 | #532 | C22×C59 | 1 | 3 | yes | — |
| 149 | C2×#124 | C4×C59 | 1 | 3 | yes | — |
| 150 | C4×#41 | C4×C59 | 1 | 1 | yes | characteristic |
| 151 | #413 | C22×C59 | 1 | 12 | yes | — |
| 152 | C2×#100 | — | 2 | 6 | no | — |
| 153 | #356 | C22×C59 | 1 | 12 | yes | complemented |
| 154 | #355 | C22×C59 | 1 | 4 | yes | complemented |
| 155 | C59 | #33371 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 156 | 2−1+4 | — | 4 | 4 | no | — |
| 157 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 158 | C2×#41 | — | 2 | 3 | no | — |
| 159 | C2×#41 | C2×C4×C59 | 1 | 3 | yes | — |
| 160 | C2×#41 | C2×C4×C59 | 1 | 1 | yes | characteristic |
| 161 | C2×#41 | D4×C59 | 1 | 2 | yes | — |
| 162 | C2×#41 | C23×C59 | 1 | 3 | yes | — |
| 163 | C22×Q8 | — | 2 | 1 | no | — |
| 164 | C22×Q8 | C2×C4×C59 | 1 | 3 | yes | — |
| 165 | C22×D4 | C2×C4×C59 | 1 | 3 | yes | — |
| 166 | C22×D4 | D4×C59 | 1 | 6 | yes | — |
| 167 | #131 | — | 4 | 8 | no | — |
| 168 | C2×M16 | — | 2 | 12 | no | — |
| 169 | C4⋊Q8 | C23×C59 | 1 | 3 | yes | — |
| 170 | #127 | C2×C4×C59 | 1 | 3 | yes | — |
| 171 | #124 | — | 2 | 6 | no | — |
| 172 | #124 | C2×C4×C59 | 1 | 3 | yes | — |
| 173 | #124 | C2×C4×C59 | 1 | 6 | yes | — |
| 174 | #124 | C23×C59 | 1 | 1 | yes | characteristic |
| 175 | #123 | — | 2 | 6 | no | — |
| 176 | C22⋊Q8 | — | 2 | 6 | no | — |
| 177 | #121 | — | 2 | 6 | no | — |
| 178 | #121 | — | 2 | 6 | no | — |
| 179 | #120 | — | 2 | 6 | no | — |
| 180 | C4×Q8 | — | 2 | 1 | no | — |
| 181 | C4×D4 | — | 2 | 3 | no | — |
| 182 | C4×D4 | — | 2 | 6 | no | — |
| 183 | #117 | C2×C4×C59 | 1 | 3 | yes | — |
| 184 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 185 | C2×C42 | C2×C4×C59 | 1 | 3 | yes | — |
| 186 | #101 | C23×C59 | 1 | 8 | yes | — |
| 187 | #100 | — | 2 | 12 | no | — |
| 188 | C24 | C22⋊C4×C59 | 1 | 6 | yes | — |
| 189 | #41 | — | 2 | 8 | no | — |
| 190 | #41 | — | 2 | 4 | no | — |
| 191 | #41 | — | 4 | 6 | no | — |
| 192 | #41 | — | 4 | 6 | no | — |
| 193 | #41 | — | 4 | 12 | no | — |
| 194 | C2×Q8 | — | 2 | 3 | no | — |
| 195 | C2×Q8 | — | 4 | 6 | no | — |
| 196 | C2×Q8 | — | 4 | 4 | no | — |
| 197 | C2×Q8 | C22⋊C4×C59 | 1 | 1 | yes | characteristic |
| 198 | C2×Q8 | C22×C4×C59 | 1 | 3 | yes | — |
| 199 | C2×Q8 | C24×C59 | 1 | 1 | yes | characteristic |
| 200 | C2×D4 | — | 2 | 3 | no | — |
| 201 | C2×D4 | — | 2 | 6 | no | — |
| 202 | C2×D4 | — | 4 | 6 | no | — |
| 203 | C2×D4 | — | 4 | 12 | no | — |
| 204 | C2×D4 | C22⋊C4×C59 | 1 | 3 | yes | — |
| 205 | C2×D4 | C22⋊C4×C59 | 1 | 6 | yes | — |
| 206 | C2×D4 | C22×C4×C59 | 1 | 1 | yes | characteristic |
| 207 | C2×D4 | C22×C4×C59 | 1 | 3 | yes | — |
| 208 | C2×D4 | C2×D4×C59 | 1 | 6 | yes | — |
| 209 | C22×C4 | — | 2 | 3 | no | — |
| 210 | C22×C4 | — | 2 | 3 | no | — |
| 211 | C22×C4 | — | 2 | 3 | no | — |
| 212 | C22×C4 | C22×C4×C59 | 1 | 3 | yes | — |
| 213 | C22×C4 | C2×D4×C59 | 1 | 6 | yes | — |
| 214 | M16 | — | 2 | 8 | no | — |
| 215 | M16 | — | 4 | 12 | no | — |
| 216 | C2×C8 | — | 4 | 12 | no | — |
| 217 | C4⋊C4 | — | 2 | 3 | no | — |
| 218 | C4⋊C4 | — | 2 | 6 | no | — |
| 219 | C22⋊C4 | — | 2 | 6 | no | — |
| 220 | C22⋊C4 | — | 2 | 12 | no | — |
| 221 | C22⋊C4 | — | 2 | 3 | no | — |
| 222 | C22⋊C4 | — | 2 | 2 | no | — |
| 223 | C42 | — | 2 | 3 | no | — |
| 224 | C42 | C22×C4×C59 | 1 | 1 | yes | characteristic |
| 225 | C42 | C22×C4×C59 | 1 | 3 | yes | — |
| 226 | C23 | — | 2 | 1 | no | — |
| 227 | C23 | — | 4 | 6 | no | — |
| 228 | C23 | — | 8 | 6 | no | — |
| 229 | C23 | C2×C22⋊C4×C59 | 1 | 6 | yes | — |
| 230 | C23 | C2×C22⋊C4×C59 | 1 | 3 | yes | — |
| 231 | Q8 | — | 4 | 1 | no | — |
| 232 | Q8 | — | 4 | 3 | no | — |
| 233 | Q8 | — | 4 | 6 | no | — |
| 234 | Q8 | #99×C59 | 1 | 4 | yes | — |
| 235 | D4 | — | 2 | 12 | no | — |
| 236 | D4 | — | 2 | 6 | no | — |
| 237 | D4 | — | 4 | 3 | no | — |
| 238 | D4 | — | 4 | 6 | no | — |
| 239 | D4 | — | 4 | 1 | no | — |
| 240 | D4 | — | 4 | 6 | no | — |
| 241 | D4 | — | 8 | 6 | no | — |
| 242 | C2×C4 | — | 2 | 12 | no | — |
| 243 | C2×C4 | — | 2 | 6 | no | — |
| 244 | C2×C4 | — | 2 | 1 | no | — |
| 245 | C2×C4 | — | 2 | 3 | no | — |
| 246 | C2×C4 | — | 2 | 3 | no | — |
| 247 | C2×C4 | — | 4 | 6 | no | — |
| 248 | C2×C4 | — | 4 | 6 | no | — |
| 249 | C2×C4 | — | 8 | 3 | no | — |
| 250 | C2×C4 | C2×C22⋊C4×C59 | 1 | 3 | yes | — |
| 251 | C2×C4 | C23×C4×C59 | 1 | 1 | yes | characteristic |
| 252 | C2×C4 | C22×D4×C59 | 1 | 2 | yes | — |
| 253 | C8 | — | 4 | 8 | no | — |
| 254 | C22 | — | 2 | 3 | no | — |
| 255 | C22 | — | 2 | 6 | no | — |
| 256 | C22 | — | 4 | 3 | no | — |
| 257 | C22 | — | 4 | 6 | no | — |
| 258 | C22 | — | 4 | 2 | no | — |
| 259 | C22 | — | 4 | 1 | no | — |
| 260 | C22 | — | 16 | 6 | no | — |
| 261 | C22 | C22×C22⋊C4×C59 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 2 | 2 | no | — |
| 263 | C4 | — | 2 | 1 | no | — |
| 264 | C4 | — | 4 | 1 | no | — |
| 265 | C4 | — | 4 | 3 | no | — |
| 266 | C4 | — | 4 | 3 | no | — |
| 267 | C4 | C2×#99×C59 | 1 | 6 | yes | — |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 4 | 3 | no | — |
| 270 | C2 | — | 4 | 6 | no | — |
| 271 | C2 | — | 8 | 1 | no | — |
| 272 | C2 | C22×#99×C59 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta