← C2×#13107751 #72253238
Subgroups — C2×#13107751 #72253238
Order 3200.
6016 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#41×C5⋊D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13107751 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#11717441 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#11717439 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#11717435 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C22×#87855 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×C52×#41 | C4 | 1 | 1 | yes | characteristic |
| 8 | #41×C5⋊D5 | C4 | 1 | 8 | yes | — |
| 9 | #41×C5⋊D5 | C22 | 1 | 8 | yes | — |
| 10 | C2×#87904 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×Q8×C5⋊D5 | C4 | 1 | 1 | yes | characteristic |
| 12 | C2×#87902 | C22 | 1 | 3 | yes | — |
| 13 | C2×D4×C5⋊D5 | C4 | 1 | 3 | yes | — |
| 14 | C2×#87900 | C4 | 1 | 3 | yes | — |
| 15 | C22×C4×C5⋊D5 | C22 | 1 | 3 | yes | — |
| 16 | C2×#87860 | C22 | 1 | 6 | yes | complemented |
| 17 | C22×#6033 | C22 | 1 | 6 | yes | complemented |
| 18 | #11717441 | C22 | 1 | 16 | yes | complemented |
| 19 | #11717439 | C22 | 1 | 48 | yes | complemented |
| 20 | #11717435 | C22 | 1 | 24 | yes | complemented |
| 21 | C2×#87856 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×#87855 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#87855 | C22 | 1 | 3 | yes | — |
| 24 | C2×#87855 | C22 | 1 | 24 | yes | complemented |
| 25 | C2×#87855 | C22 | 1 | 6 | yes | complemented |
| 26 | C2×D5⋊#131 | — | 5 | 6 | no | maximal |
| 27 | C23×C5⋊D5 | C2×C4 | 1 | 3 | yes | — |
| 28 | C52×#41 | C2×C4 | 1 | 8 | yes | — |
| 29 | C2×C52×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C2×C52×D4 | C2×C4 | 1 | 3 | yes | — |
| 31 | C22×C4×C52 | C2×C4 | 1 | 3 | yes | — |
| 32 | C2×C22⋊(C5⋊D5) | C2×C4 | 1 | 6 | yes | — |
| 33 | C22×#1039 | C23 | 1 | 3 | yes | — |
| 34 | #87904 | C2×C4 | 1 | 4 | yes | — |
| 35 | #87904 | C23 | 1 | 4 | yes | — |
| 36 | Q8×C5⋊D5 | C2×C4 | 1 | 8 | yes | — |
| 37 | #87902 | C2×C4 | 1 | 12 | yes | — |
| 38 | #87902 | C23 | 1 | 12 | yes | — |
| 39 | D4×C5⋊D5 | C2×C4 | 1 | 24 | yes | — |
| 40 | #87900 | C2×C4 | 1 | 24 | yes | — |
| 41 | C2×C4⋊(C5⋊D5) | C2×C4 | 1 | 3 | yes | — |
| 42 | C2×C4×C5⋊D5 | C2×C4 | 1 | 3 | yes | — |
| 43 | C2×C4×C5⋊D5 | C2×C4 | 1 | 3 | yes | — |
| 44 | C2×C4×C5⋊D5 | C2×C4 | 1 | 12 | yes | — |
| 45 | C2×C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C4×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 47 | C2×C4×C5⋊D5 | C23 | 1 | 12 | yes | — |
| 48 | C2×#6045 | C2×C4 | 1 | 3 | yes | — |
| 49 | #87860 | C23 | 1 | 24 | yes | complemented |
| 50 | C2×#6033 | C23 | 1 | 2 | yes | — |
| 51 | C2×#6033 | C23 | 1 | 6 | yes | — |
| 52 | C2×#6033 | C23 | 1 | 24 | yes | complemented |
| 53 | #87856 | C23 | 1 | 24 | yes | complemented |
| 54 | #87855 | C23 | 1 | 4 | yes | — |
| 55 | #87855 | C23 | 1 | 12 | yes | — |
| 56 | #87855 | C23 | 1 | 24 | yes | complemented |
| 57 | C2×D5×#41 | — | 5 | 6 | no | — |
| 58 | D5⋊#131 | — | 5 | 96 | no | — |
| 59 | C2×#4133 | — | 5 | 12 | no | — |
| 60 | C2×#4131 | — | 5 | 36 | no | — |
| 61 | C2×D5⋊M16 | — | 5 | 18 | no | — |
| 62 | C22×D5⋊C8 | — | 5 | 18 | no | — |
| 63 | C23×C52 | C22×C4 | 1 | 3 | yes | — |
| 64 | C22×C5⋊D5 | — | 2 | 12 | no | — |
| 65 | C22×C5⋊D5 | C22×C4 | 1 | 3 | yes | — |
| 66 | C22×C5⋊D5 | C22×C4 | 1 | 12 | yes | — |
| 67 | C22×C5⋊D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C52×Q8 | C22×C4 | 1 | 4 | yes | — |
| 69 | C52×D4 | C22×C4 | 1 | 12 | yes | — |
| 70 | C2×C4×C52 | C22×C4 | 1 | 3 | yes | — |
| 71 | C2×C4×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×C52 | C22×C4 | 1 | 12 | yes | — |
| 73 | C22⋊(C5⋊D5) | C22×C4 | 1 | 24 | yes | — |
| 74 | C2×#1039 | C22×C4 | 1 | 3 | yes | — |
| 75 | C2×#1039 | C22×C4 | 1 | 6 | yes | — |
| 76 | C2×#1039 | C24 | 1 | 1 | yes | characteristic |
| 77 | C2×#1039 | C24 | 1 | 6 | yes | — |
| 78 | C4⋊(C5⋊D5) | C22×C4 | 1 | 12 | yes | — |
| 79 | C4×C5⋊D5 | C22×C4 | 1 | 2 | yes | — |
| 80 | C4×C5⋊D5 | C22×C4 | 1 | 12 | yes | — |
| 81 | C4×C5⋊D5 | C22×C4 | 1 | 12 | yes | — |
| 82 | C4×C5⋊D5 | C22×C4 | 1 | 6 | yes | — |
| 83 | C4×C5⋊D5 | C24 | 1 | 2 | yes | — |
| 84 | C4×C5⋊D5 | C24 | 1 | 6 | yes | — |
| 85 | #6045 | C22×C4 | 1 | 12 | yes | — |
| 86 | #6033 | C24 | 1 | 4 | yes | — |
| 87 | #6033 | C24 | 1 | 12 | yes | — |
| 88 | C2×C5×#41 | AGL(1,𝔽5) | 1 | 6 | yes | — |
| 89 | D5×#41 | — | 5 | 48 | no | — |
| 90 | D5×#41 | — | 5 | 48 | no | — |
| 91 | C2×#713 | — | 5 | 6 | no | — |
| 92 | C2×Q8×D5 | — | 5 | 6 | no | — |
| 93 | C2×#711 | — | 5 | 18 | no | — |
| 94 | C2×D4×D5 | — | 5 | 18 | no | — |
| 95 | C2×#709 | — | 5 | 18 | no | — |
| 96 | C22×C4×D5 | — | 5 | 18 | no | — |
| 97 | C2×#704 | — | 5 | 36 | no | — |
| 98 | C22×#170 | — | 5 | 36 | no | — |
| 99 | #4133 | — | 5 | 96 | no | — |
| 100 | #4131 | — | 5 | 288 | no | — |
| 101 | D5⋊M16 | — | 5 | 144 | no | — |
| 102 | C2×#700 | — | 5 | 36 | no | — |
| 103 | C2×D5⋊C8 | — | 5 | 6 | no | — |
| 104 | C2×D5⋊C8 | — | 5 | 18 | no | — |
| 105 | C2×D5⋊C8 | — | 5 | 36 | no | — |
| 106 | C2×D5⋊C8 | — | 5 | 144 | no | — |
| 107 | C22×#131 | — | 25 | 1 | no | Sylow |
| 108 | C22×C52 | — | 2 | 6 | no | — |
| 109 | C22×C52 | C23×C4 | 1 | 6 | yes | — |
| 110 | C22×C52 | C23×C4 | 1 | 1 | yes | characteristic |
| 111 | C2×C5⋊D5 | — | 2 | 6 | no | — |
| 112 | C2×C5⋊D5 | — | 2 | 24 | no | — |
| 113 | C2×C5⋊D5 | #131 | 1 | 4 | yes | complemented |
| 114 | C2×C5⋊D5 | C23×C4 | 1 | 6 | yes | — |
| 115 | C2×C5⋊D5 | C23×C4 | 1 | 2 | yes | — |
| 116 | C4×C52 | C23×C4 | 1 | 6 | yes | — |
| 117 | C4×C52 | C23×C4 | 1 | 2 | yes | — |
| 118 | #1039 | C23×C4 | 1 | 6 | yes | — |
| 119 | #1039 | C23×C4 | 1 | 1 | yes | characteristic |
| 120 | #1039 | C25 | 1 | 1 | yes | characteristic |
| 121 | C23×D5 | — | 5 | 18 | no | — |
| 122 | C5×#41 | C2×AGL(1,𝔽5) | 1 | 48 | yes | — |
| 123 | C2×C5×Q8 | C2×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 124 | C2×C5×D4 | C2×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 125 | C22×C4×C5 | C2×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 126 | C2×C22⋊D5 | — | 5 | 36 | no | — |
| 127 | C22×Dic5 | — | 5 | 18 | no | — |
| 128 | #713 | — | 5 | 24 | no | — |
| 129 | #713 | — | 5 | 24 | no | — |
| 130 | Q8×D5 | — | 5 | 48 | no | — |
| 131 | #711 | — | 5 | 72 | no | — |
| 132 | #711 | — | 5 | 72 | no | — |
| 133 | D4×D5 | — | 5 | 144 | no | — |
| 134 | #709 | — | 5 | 144 | no | — |
| 135 | C2×D20 | — | 5 | 18 | no | — |
| 136 | C2×C4×D5 | — | 5 | 6 | no | — |
| 137 | C2×C4×D5 | — | 5 | 18 | no | — |
| 138 | C2×C4×D5 | — | 5 | 72 | no | — |
| 139 | C2×C4×D5 | — | 5 | 18 | no | — |
| 140 | C2×C4×D5 | — | 5 | 18 | no | — |
| 141 | C2×C4×D5 | — | 5 | 72 | no | — |
| 142 | C2×Dic10 | — | 5 | 18 | no | — |
| 143 | #704 | — | 5 | 144 | no | — |
| 144 | C2×#170 | — | 5 | 12 | no | — |
| 145 | C2×#170 | — | 5 | 36 | no | — |
| 146 | C2×#170 | — | 5 | 144 | no | — |
| 147 | #700 | — | 5 | 144 | no | — |
| 148 | D5⋊C8 | — | 5 | 24 | no | — |
| 149 | D5⋊C8 | — | 5 | 72 | no | — |
| 150 | D5⋊C8 | — | 5 | 144 | no | — |
| 151 | C22×#41 | — | 25 | 1 | no | — |
| 152 | C2×#131 | — | 25 | 6 | no | — |
| 153 | C2×#131 | — | 25 | 2 | no | — |
| 154 | C2×#131 | — | 25 | 16 | no | — |
| 155 | C22×M16 | — | 25 | 3 | no | — |
| 156 | C23×C8 | — | 25 | 3 | no | — |
| 157 | C2×C52 | — | 2 | 6 | no | — |
| 158 | C2×C52 | C2×#131 | 1 | 2 | yes | complemented |
| 159 | C2×C52 | C24×C4 | 1 | 1 | yes | characteristic |
| 160 | C5⋊D5 | — | 2 | 6 | no | — |
| 161 | C5⋊D5 | C2×#131 | 1 | 4 | yes | complemented |
| 162 | C23×C5 | C22×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 163 | C22×D5 | — | 5 | 6 | no | — |
| 164 | C22×D5 | — | 5 | 18 | no | — |
| 165 | C22×D5 | — | 5 | 72 | no | — |
| 166 | C22×D5 | — | 10 | 72 | no | — |
| 167 | C5×Q8 | C22×AGL(1,𝔽5) | 1 | 24 | yes | — |
| 168 | C5×D4 | C22×AGL(1,𝔽5) | 1 | 72 | yes | — |
| 169 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 18 | yes | — |
| 170 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 72 | yes | — |
| 171 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 172 | C22⋊D5 | — | 5 | 144 | no | — |
| 173 | C2×Dic5 | — | 5 | 18 | no | — |
| 174 | C2×Dic5 | — | 5 | 36 | no | — |
| 175 | C2×Dic5 | — | 5 | 6 | no | — |
| 176 | C2×Dic5 | — | 5 | 36 | no | — |
| 177 | D20 | — | 5 | 72 | no | — |
| 178 | C4×D5 | — | 5 | 12 | no | — |
| 179 | C4×D5 | — | 5 | 72 | no | — |
| 180 | C4×D5 | — | 5 | 72 | no | — |
| 181 | C4×D5 | — | 5 | 36 | no | — |
| 182 | C4×D5 | — | 5 | 12 | no | — |
| 183 | C4×D5 | — | 5 | 36 | no | — |
| 184 | Dic10 | — | 5 | 72 | no | — |
| 185 | #170 | — | 5 | 24 | no | — |
| 186 | #170 | — | 5 | 72 | no | — |
| 187 | C2×#41 | — | 25 | 8 | no | — |
| 188 | C2×#41 | — | 25 | 8 | no | — |
| 189 | C2×#41 | — | 25 | 1 | no | — |
| 190 | C2×#41 | — | 25 | 3 | no | — |
| 191 | C2×#41 | — | 25 | 3 | no | — |
| 192 | C2×#41 | #1043 | 1 | 1 | yes | characteristic |
| 193 | C22×Q8 | — | 25 | 1 | no | — |
| 194 | C22×D4 | — | 25 | 3 | no | — |
| 195 | C23×C4 | — | 25 | 3 | no | — |
| 196 | #131 | — | 25 | 48 | no | — |
| 197 | #131 | — | 25 | 16 | no | — |
| 198 | C2×M16 | — | 25 | 6 | no | — |
| 199 | C2×M16 | — | 25 | 24 | no | — |
| 200 | C2×M16 | — | 25 | 6 | no | — |
| 201 | C22×C8 | — | 25 | 1 | no | — |
| 202 | C22×C8 | — | 25 | 3 | no | — |
| 203 | C22×C8 | — | 25 | 24 | no | — |
| 204 | C22×C8 | — | 25 | 6 | no | — |
| 205 | C22×C8 | — | 25 | 6 | no | — |
| 206 | C52 | C22×#131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 207 | C22×C5 | — | 2 | 36 | no | — |
| 208 | C22×C5 | C23×AGL(1,𝔽5) | 1 | 36 | yes | — |
| 209 | C22×C5 | C23×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 210 | C2×D5 | — | 5 | 24 | no | — |
| 211 | C2×D5 | — | 5 | 12 | no | — |
| 212 | C2×D5 | — | 5 | 36 | no | — |
| 213 | C2×D5 | — | 10 | 144 | no | — |
| 214 | C2×D5 | — | 10 | 36 | no | — |
| 215 | C4×C5 | C23×AGL(1,𝔽5) | 1 | 36 | yes | — |
| 216 | C4×C5 | C23×AGL(1,𝔽5) | 1 | 12 | yes | — |
| 217 | Dic5 | — | 5 | 36 | no | — |
| 218 | Dic5 | — | 5 | 6 | no | — |
| 219 | Dic5 | — | 5 | 6 | no | — |
| 220 | C24 | — | 25 | 3 | no | — |
| 221 | #41 | — | 25 | 4 | no | — |
| 222 | #41 | — | 25 | 4 | no | — |
| 223 | #41 | — | 25 | 12 | no | — |
| 224 | #41 | — | 25 | 12 | no | — |
| 225 | #41 | — | 25 | 24 | no | — |
| 226 | #41 | C2×#1043 | 1 | 8 | yes | — |
| 227 | C2×Q8 | — | 25 | 8 | no | — |
| 228 | C2×Q8 | — | 25 | 3 | no | — |
| 229 | C2×Q8 | C2×#1043 | 1 | 1 | yes | characteristic |
| 230 | C2×D4 | — | 25 | 6 | no | — |
| 231 | C2×D4 | — | 25 | 24 | no | — |
| 232 | C2×D4 | — | 25 | 3 | no | — |
| 233 | C2×D4 | C2×#1043 | 1 | 3 | yes | — |
| 234 | C22×C4 | — | 25 | 3 | no | — |
| 235 | C22×C4 | — | 25 | 3 | no | — |
| 236 | C22×C4 | — | 25 | 12 | no | — |
| 237 | C22×C4 | — | 25 | 3 | no | — |
| 238 | C22×C4 | — | 25 | 12 | no | — |
| 239 | C22×C4 | — | 25 | 1 | no | — |
| 240 | C22×C4 | — | 25 | 3 | no | — |
| 241 | C22×C4 | C2×#1043 | 1 | 3 | yes | — |
| 242 | M16 | — | 25 | 24 | no | — |
| 243 | M16 | — | 25 | 24 | no | — |
| 244 | C2×C8 | — | 25 | 4 | no | — |
| 245 | C2×C8 | — | 25 | 2 | no | — |
| 246 | C2×C8 | — | 25 | 12 | no | — |
| 247 | C2×C8 | — | 25 | 6 | no | — |
| 248 | C2×C8 | — | 25 | 24 | no | — |
| 249 | C2×C8 | — | 25 | 24 | no | — |
| 250 | C2×C5 | — | 2 | 36 | no | — |
| 251 | C2×C5 | D5⋊#131 | 1 | 12 | yes | complemented |
| 252 | C2×C5 | C24×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 253 | D5 | — | 5 | 24 | no | — |
| 254 | D5 | — | 10 | 36 | no | — |
| 255 | C23 | — | 25 | 1 | no | — |
| 256 | C23 | — | 25 | 12 | no | — |
| 257 | C23 | — | 25 | 3 | no | — |
| 258 | C23 | — | 50 | 12 | no | — |
| 259 | C23 | C22×#1043 | 1 | 3 | yes | — |
| 260 | Q8 | — | 25 | 12 | no | — |
| 261 | Q8 | C22×#1043 | 1 | 4 | yes | — |
| 262 | D4 | — | 25 | 24 | no | — |
| 263 | D4 | — | 25 | 12 | no | — |
| 264 | D4 | C22×#1043 | 1 | 12 | yes | — |
| 265 | C2×C4 | — | 25 | 1 | no | — |
| 266 | C2×C4 | — | 25 | 6 | no | — |
| 267 | C2×C4 | — | 25 | 3 | no | — |
| 268 | C2×C4 | — | 25 | 6 | no | — |
| 269 | C2×C4 | — | 25 | 2 | no | — |
| 270 | C2×C4 | — | 25 | 6 | no | — |
| 271 | C2×C4 | — | 25 | 2 | no | — |
| 272 | C2×C4 | — | 25 | 12 | no | — |
| 273 | C2×C4 | — | 25 | 6 | no | — |
| 274 | C2×C4 | — | 25 | 12 | no | — |
| 275 | C2×C4 | C22×#1043 | 1 | 3 | yes | — |
| 276 | C2×C4 | C22×#1043 | 1 | 12 | yes | — |
| 277 | C2×C4 | C22×#1043 | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 25 | 4 | no | — |
| 279 | C8 | — | 25 | 12 | no | — |
| 280 | C5 | C2×D5⋊#131 | 1 | 6 | yes | complemented |
| 281 | C22 | — | 2 | 6 | no | — |
| 282 | C22 | — | 25 | 4 | no | — |
| 283 | C22 | — | 25 | 2 | no | — |
| 284 | C22 | — | 25 | 6 | no | — |
| 285 | C22 | — | 50 | 24 | no | — |
| 286 | C22 | — | 50 | 6 | no | — |
| 287 | C22 | C23×#1043 | 1 | 6 | yes | — |
| 288 | C22 | C23×#1043 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 25 | 1 | no | — |
| 290 | C4 | — | 25 | 1 | no | — |
| 291 | C4 | — | 25 | 6 | no | — |
| 292 | C4 | C23×#1043 | 1 | 2 | yes | — |
| 293 | C4 | C23×#1043 | 1 | 6 | yes | — |
| 294 | C2 | — | 2 | 6 | no | — |
| 295 | C2 | — | 25 | 4 | no | — |
| 296 | C2 | — | 50 | 6 | no | — |
| 297 | C2 | #13107751 | 1 | 2 | yes | direct product factor |
| 298 | C2 | C24×#1043 | 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