← C52×#3498 #72221175
Subgroups — C52×#3498 #72221175
Order 3200.
6992 conjugacy classes of subgroups.
The subgroup lattice has 267 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C52×H(5,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C52×#550 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C52×D42 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | C52×#535 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C52×#532 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×C52×#127 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C4×C52×#41 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C52×H(5,𝔽2) | — | 2 | 16 | no | — |
| 9 | C2×C52×#41 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C52×#41 | C22 | 1 | 6 | yes | complemented |
| 11 | C2×C52×#41 | C22 | 1 | 4 | yes | complemented |
| 12 | C22×C52×D4 | C22 | 1 | 6 | yes | complemented |
| 13 | C22×C52×D4 | C22 | 1 | 12 | yes | complemented |
| 14 | C22×C52×D4 | C22 | 1 | 12 | yes | complemented |
| 15 | C52×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C52×#127 | C22 | 1 | 3 | yes | complemented |
| 17 | C52×#127 | C22 | 1 | 6 | yes | complemented |
| 18 | C52×#127 | C22 | 1 | 12 | yes | complemented |
| 19 | C52×#124 | C22 | 1 | 6 | yes | complemented |
| 20 | C52×#121 | C22 | 1 | 12 | yes | complemented |
| 21 | C52×#121 | C22 | 1 | 24 | yes | complemented |
| 22 | C52×#120 | C22 | 1 | 24 | yes | complemented |
| 23 | C4×C52×Q8 | C22 | 1 | 2 | yes | complemented |
| 24 | C4×C52×D4 | C22 | 1 | 6 | yes | complemented |
| 25 | C4×C52×D4 | C22 | 1 | 12 | yes | complemented |
| 26 | C52×#117 | C22 | 1 | 3 | yes | complemented |
| 27 | C2×C42×C52 | C22 | 1 | 3 | yes | complemented |
| 28 | C5×#3498 | C5 | 1 | 6 | yes | maximal, direct product factor |
| 29 | C24×C52 | C23 | 1 | 12 | yes | — |
| 30 | C52×#41 | — | 2 | 16 | no | — |
| 31 | C52×#41 | — | 2 | 24 | no | — |
| 32 | C52×#41 | D4 | 1 | 8 | yes | complemented |
| 33 | C2×C52×Q8 | C23 | 1 | 2 | yes | — |
| 34 | C2×C52×Q8 | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×C52×D4 | — | 2 | 48 | no | — |
| 36 | C2×C52×D4 | — | 2 | 24 | no | — |
| 37 | C2×C52×D4 | — | 4 | 24 | no | — |
| 38 | C2×C52×D4 | C23 | 1 | 12 | yes | — |
| 39 | C2×C52×D4 | C23 | 1 | 12 | yes | — |
| 40 | C2×C52×D4 | C23 | 1 | 6 | yes | — |
| 41 | C2×C52×D4 | C23 | 1 | 12 | yes | — |
| 42 | C2×C52×D4 | C23 | 1 | 24 | yes | — |
| 43 | C2×C52×D4 | C23 | 1 | 3 | yes | — |
| 44 | C22×C4×C52 | C23 | 1 | 6 | yes | — |
| 45 | C22×C4×C52 | C23 | 1 | 12 | yes | — |
| 46 | C22×C4×C52 | C23 | 1 | 3 | yes | — |
| 47 | C52×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 48 | C52×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 49 | C52×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 50 | C52×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 51 | C42×C52 | C23 | 1 | 1 | yes | characteristic |
| 52 | C42×C52 | C23 | 1 | 3 | yes | — |
| 53 | C42×C52 | C23 | 1 | 6 | yes | — |
| 54 | C2×C5×H(5,𝔽2) | C2×C5 | 1 | 12 | yes | complemented |
| 55 | C5×#550 | C2×C5 | 1 | 24 | yes | complemented |
| 56 | C5×D42 | C2×C5 | 1 | 72 | yes | complemented |
| 57 | C5×#535 | C2×C5 | 1 | 36 | yes | complemented |
| 58 | C5×#532 | C2×C5 | 1 | 18 | yes | complemented |
| 59 | C2×C5×#127 | C2×C5 | 1 | 18 | yes | complemented |
| 60 | C4×C5×#41 | C2×C5 | 1 | 6 | yes | complemented |
| 61 | C23×C52 | — | 2 | 24 | no | — |
| 62 | C23×C52 | — | 4 | 24 | no | — |
| 63 | C23×C52 | C24 | 1 | 12 | yes | — |
| 64 | C23×C52 | C24 | 1 | 3 | yes | — |
| 65 | C52×Q8 | — | 2 | 4 | no | — |
| 66 | C52×Q8 | C2×D4 | 1 | 4 | yes | complemented |
| 67 | C52×D4 | — | 2 | 24 | no | — |
| 68 | C52×D4 | — | 2 | 24 | no | — |
| 69 | C52×D4 | — | 2 | 12 | no | — |
| 70 | C52×D4 | — | 2 | 24 | no | — |
| 71 | C52×D4 | — | 4 | 24 | no | — |
| 72 | C52×D4 | C2×D4 | 1 | 12 | yes | complemented |
| 73 | C2×C4×C52 | — | 2 | 24 | no | — |
| 74 | C2×C4×C52 | — | 2 | 12 | no | — |
| 75 | C2×C4×C52 | C2×D4 | 1 | 12 | yes | complemented |
| 76 | C2×C4×C52 | C24 | 1 | 6 | yes | — |
| 77 | C2×C4×C52 | C24 | 1 | 4 | yes | — |
| 78 | C2×C4×C52 | C24 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C52 | C24 | 1 | 3 | yes | — |
| 80 | C2×C4×C52 | C24 | 1 | 2 | yes | — |
| 81 | C5×H(5,𝔽2) | — | 2 | 96 | no | — |
| 82 | C2×C5×#41 | C22×C5 | 1 | 6 | yes | — |
| 83 | C2×C5×#41 | C22×C5 | 1 | 36 | yes | complemented |
| 84 | C2×C5×#41 | C22×C5 | 1 | 24 | yes | complemented |
| 85 | C22×C5×D4 | C22×C5 | 1 | 36 | yes | complemented |
| 86 | C22×C5×D4 | C22×C5 | 1 | 72 | yes | complemented |
| 87 | C22×C5×D4 | C22×C5 | 1 | 72 | yes | complemented |
| 88 | C5×C4⋊Q8 | C22×C5 | 1 | 6 | yes | complemented |
| 89 | C5×#127 | C22×C5 | 1 | 18 | yes | complemented |
| 90 | C5×#127 | C22×C5 | 1 | 36 | yes | complemented |
| 91 | C5×#127 | C22×C5 | 1 | 72 | yes | complemented |
| 92 | C5×#124 | C22×C5 | 1 | 36 | yes | complemented |
| 93 | C5×#121 | C22×C5 | 1 | 72 | yes | complemented |
| 94 | C5×#121 | C22×C5 | 1 | 144 | yes | complemented |
| 95 | C5×#120 | C22×C5 | 1 | 144 | yes | complemented |
| 96 | C4×C5×Q8 | C22×C5 | 1 | 12 | yes | complemented |
| 97 | C4×C5×D4 | C22×C5 | 1 | 36 | yes | complemented |
| 98 | C4×C5×D4 | C22×C5 | 1 | 72 | yes | complemented |
| 99 | C5×#117 | C22×C5 | 1 | 18 | yes | complemented |
| 100 | C2×C42×C5 | C22×C5 | 1 | 18 | yes | complemented |
| 101 | #3498 | C52 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 102 | C22×C52 | — | 2 | 6 | no | — |
| 103 | C22×C52 | — | 2 | 12 | no | — |
| 104 | C22×C52 | — | 2 | 24 | no | — |
| 105 | C22×C52 | — | 8 | 24 | no | — |
| 106 | C22×C52 | C22×D4 | 1 | 6 | yes | — |
| 107 | C22×C52 | C25 | 1 | 1 | yes | characteristic |
| 108 | C4×C52 | — | 2 | 4 | no | — |
| 109 | C4×C52 | — | 2 | 6 | no | — |
| 110 | C4×C52 | C22×D4 | 1 | 6 | yes | — |
| 111 | C4×C52 | C22×D4 | 1 | 2 | yes | — |
| 112 | C4×C52 | H(5,𝔽2) | 1 | 4 | yes | complemented |
| 113 | C24×C5 | C23×C5 | 1 | 72 | yes | — |
| 114 | C5×#41 | — | 2 | 96 | no | — |
| 115 | C5×#41 | — | 2 | 144 | no | — |
| 116 | C5×#41 | C5×D4 | 1 | 48 | yes | complemented |
| 117 | C2×C5×Q8 | C23×C5 | 1 | 12 | yes | — |
| 118 | C2×C5×Q8 | C23×C5 | 1 | 6 | yes | — |
| 119 | C2×C5×D4 | — | 2 | 288 | no | — |
| 120 | C2×C5×D4 | — | 2 | 144 | no | — |
| 121 | C2×C5×D4 | — | 4 | 144 | no | — |
| 122 | C2×C5×D4 | C23×C5 | 1 | 72 | yes | — |
| 123 | C2×C5×D4 | C23×C5 | 1 | 72 | yes | — |
| 124 | C2×C5×D4 | C23×C5 | 1 | 72 | yes | — |
| 125 | C2×C5×D4 | C23×C5 | 1 | 36 | yes | — |
| 126 | C2×C5×D4 | C23×C5 | 1 | 144 | yes | — |
| 127 | C2×C5×D4 | C23×C5 | 1 | 18 | yes | — |
| 128 | C22×C4×C5 | C23×C5 | 1 | 36 | yes | — |
| 129 | C22×C4×C5 | C23×C5 | 1 | 72 | yes | — |
| 130 | C22×C4×C5 | C23×C5 | 1 | 18 | yes | — |
| 131 | C5×C4⋊C4 | C23×C5 | 1 | 36 | yes | — |
| 132 | C5×C4⋊C4 | C23×C5 | 1 | 24 | yes | — |
| 133 | C5×C22⋊C4 | C23×C5 | 1 | 36 | yes | — |
| 134 | C5×C22⋊C4 | C23×C5 | 1 | 144 | yes | — |
| 135 | C42×C5 | C23×C5 | 1 | 6 | yes | — |
| 136 | C42×C5 | C23×C5 | 1 | 18 | yes | — |
| 137 | C42×C5 | C23×C5 | 1 | 36 | yes | — |
| 138 | C2×H(5,𝔽2) | C2×C52 | 1 | 2 | yes | complemented |
| 139 | #550 | C2×C52 | 1 | 4 | yes | complemented |
| 140 | D42 | C2×C52 | 1 | 12 | yes | complemented |
| 141 | #535 | C2×C52 | 1 | 6 | yes | complemented |
| 142 | #532 | C2×C52 | 1 | 3 | yes | complemented |
| 143 | C2×#127 | C2×C52 | 1 | 3 | yes | complemented |
| 144 | C4×#41 | C2×C52 | 1 | 1 | yes | characteristic, complemented |
| 145 | C2×C52 | — | 2 | 6 | no | — |
| 146 | C2×C52 | — | 4 | 12 | no | — |
| 147 | C2×C52 | C23×D4 | 1 | 1 | yes | characteristic |
| 148 | C2×C52 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 149 | C23×C5 | — | 2 | 144 | no | — |
| 150 | C23×C5 | — | 4 | 144 | no | — |
| 151 | C23×C5 | C24×C5 | 1 | 72 | yes | — |
| 152 | C23×C5 | C24×C5 | 1 | 18 | yes | — |
| 153 | C5×Q8 | — | 2 | 24 | no | — |
| 154 | C5×Q8 | C2×C5×D4 | 1 | 24 | yes | complemented |
| 155 | C5×D4 | — | 2 | 144 | no | — |
| 156 | C5×D4 | — | 2 | 144 | no | — |
| 157 | C5×D4 | — | 2 | 72 | no | — |
| 158 | C5×D4 | — | 2 | 144 | no | — |
| 159 | C5×D4 | — | 4 | 144 | no | — |
| 160 | C5×D4 | C2×C5×D4 | 1 | 72 | yes | complemented |
| 161 | C2×C4×C5 | — | 2 | 144 | no | — |
| 162 | C2×C4×C5 | — | 2 | 72 | no | — |
| 163 | C2×C4×C5 | C2×C5×D4 | 1 | 72 | yes | complemented |
| 164 | C2×C4×C5 | C24×C5 | 1 | 36 | yes | — |
| 165 | C2×C4×C5 | C24×C5 | 1 | 24 | yes | — |
| 166 | C2×C4×C5 | C24×C5 | 1 | 6 | yes | — |
| 167 | C2×C4×C5 | C24×C5 | 1 | 18 | yes | — |
| 168 | C2×C4×C5 | C24×C5 | 1 | 12 | yes | — |
| 169 | H(5,𝔽2) | — | 2 | 16 | no | — |
| 170 | C2×#41 | C22×C52 | 1 | 1 | yes | characteristic |
| 171 | C2×#41 | C22×C52 | 1 | 6 | yes | complemented |
| 172 | C2×#41 | C22×C52 | 1 | 4 | yes | complemented |
| 173 | C22×D4 | C22×C52 | 1 | 6 | yes | complemented |
| 174 | C22×D4 | C22×C52 | 1 | 12 | yes | complemented |
| 175 | C22×D4 | C22×C52 | 1 | 12 | yes | complemented |
| 176 | C4⋊Q8 | C22×C52 | 1 | 1 | yes | characteristic, complemented |
| 177 | #127 | C22×C52 | 1 | 3 | yes | complemented |
| 178 | #127 | C22×C52 | 1 | 6 | yes | complemented |
| 179 | #127 | C22×C52 | 1 | 12 | yes | complemented |
| 180 | #124 | C22×C52 | 1 | 6 | yes | complemented |
| 181 | #121 | C22×C52 | 1 | 12 | yes | complemented |
| 182 | #121 | C22×C52 | 1 | 24 | yes | complemented |
| 183 | #120 | C22×C52 | 1 | 24 | yes | complemented |
| 184 | C4×Q8 | C22×C52 | 1 | 2 | yes | complemented |
| 185 | C4×D4 | C22×C52 | 1 | 6 | yes | complemented |
| 186 | C4×D4 | C22×C52 | 1 | 12 | yes | complemented |
| 187 | #117 | C22×C52 | 1 | 3 | yes | complemented |
| 188 | C2×C42 | C22×C52 | 1 | 3 | yes | complemented |
| 189 | C52 | #3498 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 190 | C22×C5 | — | 2 | 36 | no | — |
| 191 | C22×C5 | — | 2 | 72 | no | — |
| 192 | C22×C5 | — | 2 | 144 | no | — |
| 193 | C22×C5 | — | 8 | 144 | no | — |
| 194 | C22×C5 | C22×C5×D4 | 1 | 36 | yes | — |
| 195 | C22×C5 | C25×C5 | 1 | 6 | yes | — |
| 196 | C4×C5 | — | 2 | 24 | no | — |
| 197 | C4×C5 | — | 2 | 36 | no | — |
| 198 | C4×C5 | C22×C5×D4 | 1 | 36 | yes | — |
| 199 | C4×C5 | C22×C5×D4 | 1 | 12 | yes | — |
| 200 | C4×C5 | C5×H(5,𝔽2) | 1 | 24 | yes | complemented |
| 201 | C24 | C23×C52 | 1 | 12 | yes | — |
| 202 | #41 | — | 2 | 16 | no | — |
| 203 | #41 | — | 2 | 24 | no | — |
| 204 | #41 | C52×D4 | 1 | 8 | yes | complemented |
| 205 | C2×Q8 | C23×C52 | 1 | 2 | yes | — |
| 206 | C2×Q8 | C23×C52 | 1 | 1 | yes | characteristic |
| 207 | C2×D4 | — | 2 | 48 | no | — |
| 208 | C2×D4 | — | 2 | 24 | no | — |
| 209 | C2×D4 | — | 4 | 24 | no | — |
| 210 | C2×D4 | C23×C52 | 1 | 12 | yes | — |
| 211 | C2×D4 | C23×C52 | 1 | 12 | yes | — |
| 212 | C2×D4 | C23×C52 | 1 | 12 | yes | — |
| 213 | C2×D4 | C23×C52 | 1 | 6 | yes | — |
| 214 | C2×D4 | C23×C52 | 1 | 24 | yes | — |
| 215 | C2×D4 | C23×C52 | 1 | 3 | yes | — |
| 216 | C22×C4 | C23×C52 | 1 | 6 | yes | — |
| 217 | C22×C4 | C23×C52 | 1 | 12 | yes | — |
| 218 | C22×C4 | C23×C52 | 1 | 3 | yes | — |
| 219 | C4⋊C4 | C23×C52 | 1 | 6 | yes | — |
| 220 | C4⋊C4 | C23×C52 | 1 | 4 | yes | — |
| 221 | C22⋊C4 | C23×C52 | 1 | 6 | yes | — |
| 222 | C22⋊C4 | C23×C52 | 1 | 24 | yes | — |
| 223 | C42 | C23×C52 | 1 | 1 | yes | characteristic |
| 224 | C42 | C23×C52 | 1 | 3 | yes | — |
| 225 | C42 | C23×C52 | 1 | 6 | yes | — |
| 226 | C2×C5 | — | 2 | 36 | no | — |
| 227 | C2×C5 | — | 4 | 72 | no | — |
| 228 | C2×C5 | C23×C5×D4 | 1 | 6 | yes | — |
| 229 | C2×C5 | C2×C5×H(5,𝔽2) | 1 | 12 | yes | — |
| 230 | C23 | — | 2 | 24 | no | — |
| 231 | C23 | — | 4 | 24 | no | — |
| 232 | C23 | C24×C52 | 1 | 12 | yes | — |
| 233 | C23 | C24×C52 | 1 | 3 | yes | — |
| 234 | Q8 | — | 2 | 4 | no | — |
| 235 | Q8 | C2×C52×D4 | 1 | 4 | yes | complemented |
| 236 | D4 | — | 2 | 24 | no | — |
| 237 | D4 | — | 2 | 24 | no | — |
| 238 | D4 | — | 2 | 12 | no | — |
| 239 | D4 | — | 2 | 24 | no | — |
| 240 | D4 | — | 4 | 24 | no | — |
| 241 | D4 | C2×C52×D4 | 1 | 12 | yes | complemented |
| 242 | C2×C4 | — | 2 | 24 | no | — |
| 243 | C2×C4 | — | 2 | 12 | no | — |
| 244 | C2×C4 | C2×C52×D4 | 1 | 12 | yes | complemented |
| 245 | C2×C4 | C24×C52 | 1 | 6 | yes | — |
| 246 | C2×C4 | C24×C52 | 1 | 4 | yes | — |
| 247 | C2×C4 | C24×C52 | 1 | 1 | yes | characteristic |
| 248 | C2×C4 | C24×C52 | 1 | 3 | yes | — |
| 249 | C2×C4 | C24×C52 | 1 | 2 | yes | — |
| 250 | C5 | C5×#3498 | 1 | 6 | yes | direct product factor |
| 251 | C22 | — | 2 | 6 | no | — |
| 252 | C22 | — | 2 | 12 | no | — |
| 253 | C22 | — | 2 | 24 | no | — |
| 254 | C22 | — | 8 | 24 | no | — |
| 255 | C22 | C22×C52×D4 | 1 | 6 | yes | — |
| 256 | C22 | C25×C52 | 1 | 1 | yes | characteristic |
| 257 | C4 | — | 2 | 4 | no | — |
| 258 | C4 | — | 2 | 6 | no | — |
| 259 | C4 | C22×C52×D4 | 1 | 6 | yes | — |
| 260 | C4 | C22×C52×D4 | 1 | 2 | yes | — |
| 261 | C4 | C52×H(5,𝔽2) | 1 | 4 | yes | complemented |
| 262 | C2 | — | 2 | 6 | no | — |
| 263 | C2 | — | 4 | 12 | no | — |
| 264 | C2 | C23×C52×D4 | 1 | 1 | yes | characteristic |
| 265 | C2 | C2×C52×H(5,𝔽2) | 1 | 2 | yes | — |
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