← #72242245
Subgroups — #72242245
Order 6400.
581 conjugacy classes of subgroups.
The subgroup lattice has 319 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72242241 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #72242239 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #72242220 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×#13099848 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#11716882 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×C8×#1043 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×#11716583 | C22 | 1 | 1 | yes | characteristic |
| 8 | #13100161 | C4 | 1 | 2 | yes | complemented |
| 9 | C5×#64877 | C4 | 1 | 2 | yes | complemented |
| 10 | #13099848 | C4 | 1 | 1 | yes | characteristic |
| 11 | #13099848 | C4 | 1 | 1 | yes | characteristic |
| 12 | #13099848 | C22 | 1 | 1 | yes | characteristic |
| 13 | #13099848 | C22 | 1 | 1 | yes | characteristic |
| 14 | #13099745 | C4 | 1 | 4 | yes | complemented |
| 15 | #13099743 | C22 | 1 | 4 | yes | — |
| 16 | AGL(1,𝔽5)×M64 | — | 5 | 2 | no | maximal |
| 17 | C2×C4×#1043 | C8 | 1 | 1 | yes | characteristic |
| 18 | C2×#87855 | C8 | 1 | 1 | yes | characteristic |
| 19 | C2×C8×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | C2×#87765 | C8 | 1 | 2 | yes | — |
| 21 | #11716882 | C8 | 1 | 2 | yes | — |
| 22 | #11716882 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #11716882 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C8×#1043 | C8 | 1 | 1 | yes | characteristic |
| 25 | C8×#1043 | C8 | 1 | 1 | yes | characteristic |
| 26 | C8×#1043 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C8×#1043 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C5×#672 | C2×C4 | 1 | 2 | yes | — |
| 29 | #11716583 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | #11716583 | C2×C4 | 1 | 2 | yes | — |
| 31 | #11716583 | C23 | 1 | 1 | yes | characteristic |
| 32 | #11716579 | C2×C4 | 1 | 4 | yes | complemented |
| 33 | C5×#3928 | C2×C4 | 1 | 4 | yes | complemented |
| 34 | C2×C16×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 35 | D5⋊M64 | — | 5 | 2 | no | — |
| 36 | D5×M64 | — | 5 | 2 | no | — |
| 37 | #10598486 | — | 5 | 4 | no | — |
| 38 | AGL(1,𝔽5)×C32 | — | 5 | 4 | no | — |
| 39 | C22×#1043 | C16 | 1 | 2 | yes | — |
| 40 | C2×C4×C5⋊D5 | C2×C8 | 1 | 1 | yes | characteristic |
| 41 | C2×#6033 | C16 | 1 | 2 | yes | — |
| 42 | C4×#1043 | C16 | 1 | 2 | yes | — |
| 43 | C4×#1043 | C2×C8 | 1 | 1 | yes | characteristic |
| 44 | C4×#1043 | C2×C8 | 1 | 1 | yes | characteristic |
| 45 | #87855 | C16 | 1 | 2 | yes | — |
| 46 | #87855 | C2×C8 | 1 | 1 | yes | characteristic |
| 47 | #87855 | C2×C8 | 1 | 1 | yes | characteristic |
| 48 | C2×C52×C8 | C42 | 1 | 1 | yes | characteristic |
| 49 | C2×#6029 | C2×C8 | 1 | 1 | yes | characteristic |
| 50 | C8×C5⋊D5 | C42 | 1 | 1 | yes | characteristic |
| 51 | C8×C5⋊D5 | C2×C8 | 1 | 1 | yes | characteristic |
| 52 | C8×C5⋊D5 | C2×C8 | 1 | 1 | yes | characteristic |
| 53 | C8×C5⋊D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 54 | #87765 | C2×C8 | 1 | 2 | yes | — |
| 55 | #87765 | C2×C8 | 1 | 2 | yes | — |
| 56 | C5×#672 | C42 | 1 | 2 | yes | — |
| 57 | C5×#672 | C22×C4 | 1 | 2 | yes | — |
| 58 | C2×C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 59 | C2×C8×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 60 | C2×D5⋊C16 | — | 5 | 2 | no | — |
| 61 | C2×D5×C16 | — | 5 | 2 | no | — |
| 62 | #64877 | — | 5 | 2 | no | — |
| 63 | #64877 | AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 64 | C16×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 65 | C16×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 66 | C16×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 67 | C16×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 68 | #64842 | — | 5 | 4 | no | — |
| 69 | D5⋊C32 | — | 5 | 4 | no | — |
| 70 | C5×M64 | — | 5 | 2 | no | — |
| 71 | #64719 | — | 5 | 2 | no | — |
| 72 | #64667 | — | 5 | 4 | no | — |
| 73 | D5×C32 | — | 5 | 4 | no | — |
| 74 | C4×M64 | — | 25 | 1 | no | Sylow |
| 75 | C22×C5⋊D5 | C2×C16 | 1 | 1 | yes | characteristic |
| 76 | C2×#1043 | — | 2 | 4 | no | — |
| 77 | C2×#1043 | C2×C16 | 1 | 2 | yes | — |
| 78 | C2×#1043 | C2×C16 | 1 | 2 | yes | — |
| 79 | C2×C4×C52 | C4×C8 | 1 | 1 | yes | characteristic |
| 80 | C2×#1039 | C2×C16 | 1 | 1 | yes | characteristic |
| 81 | C4×C5⋊D5 | C4×C8 | 1 | 1 | yes | characteristic |
| 82 | C4×C5⋊D5 | C2×C16 | 1 | 1 | yes | characteristic |
| 83 | C4×C5⋊D5 | C2×C16 | 1 | 1 | yes | characteristic |
| 84 | C4×C5⋊D5 | C22×C8 | 1 | 1 | yes | characteristic |
| 85 | #6033 | C2×C16 | 1 | 2 | yes | — |
| 86 | #6033 | C2×C16 | 1 | 2 | yes | — |
| 87 | C52×C8 | C4×C8 | 1 | 1 | yes | characteristic |
| 88 | C52×C8 | C2×C42 | 1 | 1 | yes | characteristic |
| 89 | #6029 | C4×C8 | 1 | 1 | yes | characteristic |
| 90 | #6029 | C22×C8 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 92 | C2×C4×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 93 | C2×D5⋊C8 | — | 5 | 2 | no | — |
| 94 | C2×D5⋊C8 | — | 20 | 1 | no | — |
| 95 | C2×C8×D5 | — | 5 | 2 | no | — |
| 96 | C2×C8×D5 | — | 20 | 1 | no | — |
| 97 | C2×#674 | — | 5 | 4 | no | — |
| 98 | C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 99 | C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 100 | C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 101 | C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 102 | C8×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 103 | C8×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 104 | C8×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 105 | C8×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 106 | D5⋊C16 | — | 5 | 4 | no | — |
| 107 | D5⋊C16 | — | 5 | 2 | no | — |
| 108 | D5⋊C16 | — | 5 | 2 | no | — |
| 109 | C2×C5×C16 | — | 5 | 2 | no | — |
| 110 | C2×#672 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 111 | D5×C16 | — | 5 | 2 | no | — |
| 112 | D5×C16 | — | 5 | 2 | no | — |
| 113 | D5×C16 | — | 5 | 2 | no | — |
| 114 | D5×C16 | — | 5 | 2 | no | — |
| 115 | #3928 | — | 5 | 4 | no | — |
| 116 | #3928 | C2×AGL(1,𝔽5) | 1 | 4 | yes | complemented |
| 117 | C5×C32 | — | 5 | 4 | no | — |
| 118 | #3926 | — | 5 | 4 | no | — |
| 119 | C2×M64 | — | 25 | 2 | no | — |
| 120 | C2×C4×C16 | — | 25 | 1 | no | — |
| 121 | #1397 | — | 25 | 2 | no | — |
| 122 | C4×C32 | — | 25 | 2 | no | — |
| 123 | C22×C52 | C4×C16 | 1 | 1 | yes | characteristic |
| 124 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 125 | C2×C5⋊D5 | C4×C16 | 1 | 1 | yes | characteristic |
| 126 | C2×C5⋊D5 | C22×C16 | 1 | 1 | yes | characteristic |
| 127 | #1043 | — | 2 | 2 | no | — |
| 128 | #1043 | M64 | 1 | 4 | yes | complemented |
| 129 | C4×C52 | C4×C16 | 1 | 1 | yes | characteristic |
| 130 | C4×C52 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 131 | #1039 | C4×C16 | 1 | 1 | yes | characteristic |
| 132 | #1039 | C22×C16 | 1 | 1 | yes | characteristic |
| 133 | C22×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 134 | C22×AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 135 | C2×C4×D5 | — | 5 | 2 | no | — |
| 136 | C2×C4×D5 | — | 20 | 1 | no | — |
| 137 | C2×#170 | — | 5 | 4 | no | — |
| 138 | C2×#170 | — | 20 | 2 | no | — |
| 139 | C4×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 140 | C4×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 141 | C4×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 142 | C4×AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 143 | C4×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 144 | C4×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 145 | D5⋊C8 | — | 5 | 4 | no | — |
| 146 | D5⋊C8 | — | 5 | 2 | no | — |
| 147 | D5⋊C8 | — | 5 | 2 | no | — |
| 148 | D5⋊C8 | — | 20 | 2 | no | — |
| 149 | D5⋊C8 | — | 20 | 1 | no | — |
| 150 | D5⋊C8 | — | 20 | 1 | no | — |
| 151 | C2×C5×C8 | — | 4 | 1 | no | — |
| 152 | C2×C5×C8 | C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 153 | C2×#168 | — | 5 | 2 | no | — |
| 154 | C2×#168 | — | 20 | 1 | no | — |
| 155 | C8×D5 | — | 5 | 2 | no | — |
| 156 | C8×D5 | — | 5 | 2 | no | — |
| 157 | C8×D5 | — | 5 | 2 | no | — |
| 158 | C8×D5 | — | 5 | 2 | no | — |
| 159 | C8×D5 | — | 20 | 1 | no | — |
| 160 | C8×D5 | — | 20 | 1 | no | — |
| 161 | C8×D5 | — | 20 | 1 | no | — |
| 162 | C8×D5 | — | 20 | 1 | no | — |
| 163 | #674 | — | 5 | 4 | no | — |
| 164 | #674 | — | 5 | 4 | no | — |
| 165 | C5×C16 | — | 5 | 2 | no | — |
| 166 | C5×C16 | — | 5 | 2 | no | — |
| 167 | #672 | C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 168 | #672 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 169 | C22×C16 | — | 25 | 1 | no | — |
| 170 | C22×C16 | — | 25 | 1 | no | — |
| 171 | C2×C4×C8 | — | 25 | 1 | no | — |
| 172 | M64 | — | 5 | 2 | no | — |
| 173 | M64 | — | 25 | 4 | no | — |
| 174 | M64 | — | 25 | 2 | no | — |
| 175 | C2×C32 | — | 25 | 4 | no | — |
| 176 | C4×C16 | — | 25 | 1 | no | — |
| 177 | C4×C16 | — | 25 | 1 | no | — |
| 178 | C4×C16 | — | 25 | 1 | no | — |
| 179 | C4×C16 | — | 25 | 1 | no | — |
| 180 | C2×C52 | — | 2 | 1 | no | — |
| 181 | C2×C52 | C2×C4×C16 | 1 | 1 | yes | characteristic |
| 182 | C5⋊D5 | — | 2 | 1 | no | — |
| 183 | C5⋊D5 | C2×M64 | 1 | 2 | yes | — |
| 184 | C22×D5 | — | 5 | 2 | no | — |
| 185 | C22×D5 | — | 20 | 1 | no | — |
| 186 | C2×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 187 | C2×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 188 | C2×AGL(1,𝔽5) | — | 10 | 8 | no | — |
| 189 | C2×AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 190 | C2×AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 191 | C2×AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 192 | C2×AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 193 | C2×C4×C5 | — | 4 | 1 | no | — |
| 194 | C2×C4×C5 | C8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 195 | C2×Dic5 | — | 5 | 2 | no | — |
| 196 | C2×Dic5 | — | 20 | 1 | no | — |
| 197 | C4×D5 | — | 5 | 2 | no | — |
| 198 | C4×D5 | — | 5 | 2 | no | — |
| 199 | C4×D5 | — | 5 | 2 | no | — |
| 200 | C4×D5 | — | 5 | 2 | no | — |
| 201 | C4×D5 | — | 20 | 1 | no | — |
| 202 | C4×D5 | — | 20 | 1 | no | — |
| 203 | C4×D5 | — | 20 | 1 | no | — |
| 204 | C4×D5 | — | 20 | 1 | no | — |
| 205 | #170 | — | 5 | 4 | no | — |
| 206 | #170 | — | 5 | 4 | no | — |
| 207 | #170 | — | 20 | 2 | no | — |
| 208 | #170 | — | 20 | 2 | no | — |
| 209 | C5×C8 | — | 4 | 1 | no | — |
| 210 | C5×C8 | — | 4 | 1 | no | — |
| 211 | C5×C8 | C8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 212 | C5×C8 | C2×C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 213 | #168 | — | 5 | 2 | no | — |
| 214 | #168 | — | 5 | 2 | no | — |
| 215 | #168 | — | 20 | 1 | no | — |
| 216 | #168 | — | 20 | 1 | no | — |
| 217 | C22×C8 | — | 25 | 1 | no | — |
| 218 | C22×C8 | — | 25 | 1 | no | — |
| 219 | C2×C42 | — | 25 | 1 | no | — |
| 220 | C2×C16 | — | 5 | 2 | no | — |
| 221 | C2×C16 | — | 25 | 1 | no | — |
| 222 | C2×C16 | — | 25 | 1 | no | — |
| 223 | C2×C16 | — | 25 | 2 | no | — |
| 224 | C2×C16 | — | 25 | 1 | no | — |
| 225 | C2×C16 | — | 25 | 1 | no | — |
| 226 | C2×C16 | — | 25 | 2 | no | — |
| 227 | C2×C16 | — | 25 | 2 | no | — |
| 228 | C4×C8 | — | 25 | 1 | no | — |
| 229 | C4×C8 | — | 25 | 1 | no | — |
| 230 | C4×C8 | — | 25 | 1 | no | — |
| 231 | C4×C8 | — | 25 | 1 | no | — |
| 232 | C32 | — | 5 | 4 | no | — |
| 233 | C32 | — | 25 | 4 | no | — |
| 234 | C52 | C4×M64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 235 | C22×C5 | — | 4 | 1 | no | — |
| 236 | C22×C5 | C16×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 237 | C2×D5 | — | 5 | 2 | no | — |
| 238 | C2×D5 | — | 5 | 2 | no | — |
| 239 | C2×D5 | — | 10 | 4 | no | — |
| 240 | C2×D5 | — | 20 | 1 | no | — |
| 241 | C2×D5 | — | 20 | 1 | no | — |
| 242 | C2×D5 | — | 20 | 2 | no | — |
| 243 | C2×D5 | — | 20 | 2 | no | — |
| 244 | AGL(1,𝔽5) | — | 5 | 8 | no | — |
| 245 | AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 246 | AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 247 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 248 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 249 | C4×C5 | — | 4 | 1 | no | — |
| 250 | C4×C5 | — | 4 | 1 | no | — |
| 251 | C4×C5 | C16×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 252 | C4×C5 | C2×C8×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 253 | Dic5 | — | 5 | 2 | no | — |
| 254 | Dic5 | — | 5 | 2 | no | — |
| 255 | Dic5 | — | 20 | 1 | no | — |
| 256 | Dic5 | — | 20 | 1 | no | — |
| 257 | C22×C4 | — | 25 | 1 | no | — |
| 258 | C22×C4 | — | 25 | 2 | no | — |
| 259 | C2×C8 | — | 25 | 1 | no | — |
| 260 | C2×C8 | — | 25 | 1 | no | — |
| 261 | C2×C8 | — | 25 | 2 | no | — |
| 262 | C2×C8 | — | 25 | 2 | no | — |
| 263 | C2×C8 | — | 25 | 1 | no | — |
| 264 | C2×C8 | — | 25 | 1 | no | — |
| 265 | C2×C8 | — | 25 | 1 | no | — |
| 266 | C2×C8 | — | 25 | 1 | no | — |
| 267 | C2×C8 | — | 25 | 1 | no | — |
| 268 | C2×C8 | AGL(1,𝔽5)2 | 1 | 1 | yes | characteristic |
| 269 | C42 | — | 25 | 1 | no | — |
| 270 | C42 | — | 25 | 1 | no | — |
| 271 | C42 | — | 25 | 2 | no | — |
| 272 | C16 | — | 5 | 2 | no | — |
| 273 | C16 | — | 5 | 2 | no | — |
| 274 | C16 | — | 25 | 2 | no | — |
| 275 | C16 | — | 25 | 2 | no | — |
| 276 | C2×C5 | — | 2 | 2 | no | — |
| 277 | C2×C5 | — | 4 | 1 | no | — |
| 278 | C2×C5 | — | 4 | 1 | no | — |
| 279 | C2×C5 | — | 4 | 1 | no | — |
| 280 | C2×C5 | C2×C16×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 281 | D5 | — | 5 | 4 | no | — |
| 282 | D5 | — | 10 | 2 | no | — |
| 283 | D5 | — | 20 | 2 | no | — |
| 284 | D5 | — | 20 | 1 | no | — |
| 285 | D5 | — | 20 | 1 | no | — |
| 286 | C23 | — | 25 | 1 | no | — |
| 287 | C2×C4 | — | 25 | 2 | no | — |
| 288 | C2×C4 | — | 25 | 1 | no | — |
| 289 | C2×C4 | — | 25 | 1 | no | — |
| 290 | C2×C4 | — | 25 | 1 | no | — |
| 291 | C2×C4 | — | 25 | 1 | no | — |
| 292 | C2×C4 | — | 25 | 1 | no | — |
| 293 | C2×C4 | — | 25 | 2 | no | — |
| 294 | C2×C4 | — | 50 | 4 | no | — |
| 295 | C2×C4 | #11717300 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 25 | 2 | no | — |
| 297 | C8 | — | 25 | 2 | no | — |
| 298 | C8 | — | 25 | 1 | no | — |
| 299 | C8 | — | 25 | 1 | no | — |
| 300 | C8 | #11717300 | 1 | 1 | yes | characteristic |
| 301 | C8 | C2×AGL(1,𝔽5)2 | 1 | 1 | yes | characteristic |
| 302 | C5 | — | 4 | 1 | no | — |
| 303 | C5 | AGL(1,𝔽5)×M64 | 1 | 2 | yes | complemented |
| 304 | C22 | — | 25 | 1 | no | — |
| 305 | C22 | — | 25 | 1 | no | — |
| 306 | C22 | — | 50 | 2 | no | — |
| 307 | C22 | #13103267 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 25 | 4 | no | — |
| 309 | C4 | — | 25 | 1 | no | — |
| 310 | C4 | — | 25 | 1 | no | — |
| 311 | C4 | — | 50 | 2 | no | — |
| 312 | C4 | #13103267 | 1 | 1 | yes | characteristic |
| 313 | C4 | C2×#11717300 | 1 | 1 | yes | characteristic |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 25 | 2 | no | — |
| 316 | C2 | — | 50 | 1 | no | — |
| 317 | C2 | C2×#13103267 | 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