← #72221297
Subgroups — #72221297
Order 3200.
1454 conjugacy classes of subgroups.
The subgroup lattice has 311 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #13107772 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #41×#1043 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13106533 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #13103906 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #13103905 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #13103903 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#11716761 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #41×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 9 | #11717454 | C4 | 1 | 2 | yes | complemented |
| 10 | #11717452 | C4 | 1 | 6 | yes | complemented |
| 11 | #11717448 | C22 | 1 | 3 | yes | — |
| 12 | C2×#87862 | C22 | 1 | 3 | yes | — |
| 13 | Q8×#1043 | C4 | 1 | 2 | yes | — |
| 14 | D4×#1043 | C4 | 1 | 6 | yes | — |
| 15 | #11717438 | C22 | 1 | 3 | yes | — |
| 16 | C2×C4×#1043 | C22 | 1 | 3 | yes | — |
| 17 | C5×#4094 | C4 | 1 | 2 | yes | complemented |
| 18 | #11716936 | C22 | 1 | 3 | yes | — |
| 19 | #11716935 | C4 | 1 | 6 | yes | complemented |
| 20 | C2×#87767 | C22 | 1 | 3 | yes | — |
| 21 | #11716763 | C22 | 1 | 6 | yes | — |
| 22 | #11716761 | C22 | 1 | 1 | yes | characteristic |
| 23 | #11716761 | C22 | 1 | 3 | yes | — |
| 24 | #11716761 | C22 | 1 | 3 | yes | — |
| 25 | #11716761 | C22 | 1 | 3 | yes | — |
| 26 | AGL(1,𝔽5)×#131 | — | 5 | 2 | no | maximal |
| 27 | C22×#1043 | C2×C4 | 1 | 6 | yes | — |
| 28 | C52×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #87904 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | Q8×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | #87902 | C2×C4 | 1 | 3 | yes | — |
| 32 | D4×C5⋊D5 | C2×C4 | 1 | 3 | yes | — |
| 33 | #87900 | C2×C4 | 1 | 3 | yes | — |
| 34 | C2×C4×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 35 | #87867 | C2×C4 | 1 | 6 | yes | complemented |
| 36 | C2×#6034 | C2×C4 | 1 | 6 | yes | complemented |
| 37 | #87863 | C2×C4 | 1 | 6 | yes | complemented |
| 38 | #87862 | C2×C4 | 1 | 6 | yes | complemented |
| 39 | #87862 | C23 | 1 | 1 | yes | characteristic |
| 40 | #87862 | C23 | 1 | 3 | yes | — |
| 41 | C22⋊#1043 | C2×C4 | 1 | 6 | yes | — |
| 42 | C4⋊#1043 | C2×C4 | 1 | 6 | yes | — |
| 43 | C4×#1043 | C2×C4 | 1 | 6 | yes | — |
| 44 | C4×#1043 | C23 | 1 | 1 | yes | characteristic |
| 45 | C4×#1043 | C23 | 1 | 3 | yes | — |
| 46 | C5×#681 | C2×C4 | 1 | 6 | yes | complemented |
| 47 | C2×C5×#168 | C2×C4 | 1 | 6 | yes | complemented |
| 48 | #87769 | C2×C4 | 1 | 6 | yes | complemented |
| 49 | #87767 | C2×C4 | 1 | 6 | yes | complemented |
| 50 | #87767 | C23 | 1 | 1 | yes | characteristic |
| 51 | #87767 | C23 | 1 | 3 | yes | — |
| 52 | #41×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 53 | #41×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 54 | D5⋊#131 | — | 5 | 2 | no | — |
| 55 | D5×#131 | — | 5 | 2 | no | — |
| 56 | #65728 | — | 5 | 6 | no | — |
| 57 | M16×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 58 | #65718 | — | 5 | 6 | no | — |
| 59 | C2×C8×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 60 | C22×C5⋊D5 | C22×C4 | 1 | 3 | yes | — |
| 61 | C2×#1043 | — | 2 | 12 | no | — |
| 62 | C2×#1043 | C22×C4 | 1 | 6 | yes | — |
| 63 | C2×#1043 | C22×C4 | 1 | 2 | yes | — |
| 64 | C52×Q8 | C42 | 1 | 1 | yes | characteristic |
| 65 | C52×D4 | C42 | 1 | 3 | yes | — |
| 66 | C2×C4×C52 | C22×C4 | 1 | 3 | yes | — |
| 67 | C22⋊(C5⋊D5) | C42 | 1 | 6 | yes | — |
| 68 | C2×#1039 | C22×C4 | 1 | 3 | yes | — |
| 69 | C4⋊(C5⋊D5) | C42 | 1 | 3 | yes | — |
| 70 | C4×C5⋊D5 | C22×C4 | 1 | 3 | yes | — |
| 71 | C4×C5⋊D5 | C22×C4 | 1 | 3 | yes | — |
| 72 | C4×C5⋊D5 | C22×C4 | 1 | 3 | yes | — |
| 73 | C4×C5⋊D5 | C24 | 1 | 1 | yes | characteristic |
| 74 | #6045 | C42 | 1 | 3 | yes | — |
| 75 | #6034 | C22×C4 | 1 | 2 | yes | — |
| 76 | #6034 | C22×C4 | 1 | 6 | yes | — |
| 77 | C5×#168 | C22×C4 | 1 | 2 | yes | — |
| 78 | C5×#168 | C22×C4 | 1 | 6 | yes | — |
| 79 | D5×#41 | — | 5 | 2 | no | — |
| 80 | D5×#41 | — | 10 | 2 | no | — |
| 81 | Q8×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 82 | Q8×AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 83 | #4133 | — | 5 | 4 | no | — |
| 84 | D4×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 85 | D4×AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 86 | #4131 | — | 5 | 12 | no | — |
| 87 | #4130 | — | 5 | 6 | no | — |
| 88 | #4130 | — | 10 | 6 | no | — |
| 89 | C2×C4×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 90 | C2×C4×AGL(1,𝔽5) | — | 10 | 6 | no | — |
| 91 | D5⋊M16 | — | 5 | 6 | no | — |
| 92 | C2×D5⋊C8 | — | 5 | 6 | no | — |
| 93 | C5×#131 | — | 5 | 2 | no | — |
| 94 | #4094 | AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 95 | #4053 | — | 5 | 6 | no | — |
| 96 | D5×M16 | — | 5 | 6 | no | — |
| 97 | #4047 | — | 5 | 6 | no | — |
| 98 | C2×C8×D5 | — | 5 | 6 | no | — |
| 99 | #3992 | — | 5 | 12 | no | — |
| 100 | C8×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 101 | C8×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 102 | C8×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 103 | C8×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 104 | C4×#131 | — | 25 | 1 | no | Sylow |
| 105 | C22×C52 | C2×C42 | 1 | 3 | yes | — |
| 106 | C2×C5⋊D5 | — | 2 | 6 | no | — |
| 107 | C2×C5⋊D5 | C2×C42 | 1 | 3 | yes | — |
| 108 | C2×C5⋊D5 | C23×C4 | 1 | 1 | yes | characteristic |
| 109 | #1043 | — | 2 | 6 | no | — |
| 110 | #1043 | #131 | 1 | 4 | yes | complemented |
| 111 | C4×C52 | C2×C42 | 1 | 3 | yes | — |
| 112 | C4×C52 | C23×C4 | 1 | 1 | yes | characteristic |
| 113 | #1039 | C2×C42 | 1 | 3 | yes | — |
| 114 | #1039 | C23×C4 | 1 | 1 | yes | characteristic |
| 115 | C22×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 116 | C22×AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 117 | C5×#41 | — | 2 | 2 | no | — |
| 118 | C5×#41 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 119 | #713 | — | 5 | 2 | no | — |
| 120 | #713 | — | 10 | 2 | no | — |
| 121 | Q8×D5 | — | 5 | 2 | no | — |
| 122 | Q8×D5 | — | 10 | 2 | no | — |
| 123 | #711 | — | 5 | 6 | no | — |
| 124 | #711 | — | 10 | 6 | no | — |
| 125 | D4×D5 | — | 5 | 6 | no | — |
| 126 | D4×D5 | — | 10 | 6 | no | — |
| 127 | #709 | — | 5 | 6 | no | — |
| 128 | #709 | — | 10 | 6 | no | — |
| 129 | C2×C4×D5 | — | 5 | 6 | no | — |
| 130 | C2×C4×D5 | — | 10 | 6 | no | — |
| 131 | C22⋊AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 132 | C22⋊AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 133 | #704 | — | 5 | 12 | no | — |
| 134 | C2×#170 | — | 5 | 12 | no | — |
| 135 | C4⋊AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 136 | C4⋊AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 137 | C4×AGL(1,𝔽5) | — | 5 | 2 | no | — |
| 138 | C4×AGL(1,𝔽5) | — | 5 | 6 | no | — |
| 139 | C4×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 140 | C4×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 141 | C4×AGL(1,𝔽5) | — | 10 | 6 | no | — |
| 142 | C4×AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 143 | #700 | — | 5 | 12 | no | — |
| 144 | D5⋊C8 | — | 5 | 2 | no | — |
| 145 | D5⋊C8 | — | 5 | 6 | no | — |
| 146 | D5⋊C8 | — | 5 | 12 | no | — |
| 147 | C5×M16 | — | 5 | 6 | no | — |
| 148 | C2×C5×C8 | — | 5 | 6 | no | — |
| 149 | #681 | C2×AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 150 | C2×#168 | C2×AGL(1,𝔽5) | 1 | 6 | yes | complemented |
| 151 | #676 | — | 5 | 12 | no | — |
| 152 | C8×D5 | — | 5 | 2 | no | — |
| 153 | C8×D5 | — | 5 | 6 | no | — |
| 154 | C8×D5 | — | 5 | 6 | no | — |
| 155 | C8×D5 | — | 5 | 6 | no | — |
| 156 | C2×#131 | — | 25 | 1 | no | — |
| 157 | C2×#131 | — | 25 | 1 | no | — |
| 158 | C4×#41 | — | 25 | 1 | no | — |
| 159 | #405 | — | 25 | 3 | no | — |
| 160 | #405 | — | 25 | 3 | no | — |
| 161 | C4×M16 | — | 25 | 3 | no | — |
| 162 | C2×C4×C8 | — | 25 | 3 | no | — |
| 163 | C2×C52 | — | 2 | 3 | no | — |
| 164 | C2×C52 | C22×C42 | 1 | 1 | yes | characteristic |
| 165 | C5⋊D5 | — | 2 | 3 | no | — |
| 166 | C5⋊D5 | C2×#131 | 1 | 2 | yes | — |
| 167 | C22×D5 | — | 5 | 6 | no | — |
| 168 | C22×D5 | — | 10 | 6 | no | — |
| 169 | C2×AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 170 | C2×AGL(1,𝔽5) | — | 5 | 12 | no | — |
| 171 | C2×AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 172 | C2×AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 173 | C2×AGL(1,𝔽5) | — | 10 | 24 | no | — |
| 174 | C2×AGL(1,𝔽5) | — | 20 | 24 | no | — |
| 175 | C5×Q8 | — | 2 | 2 | no | — |
| 176 | C5×Q8 | C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 177 | C5×D4 | — | 2 | 6 | no | — |
| 178 | C5×D4 | C4×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 179 | C2×C4×C5 | — | 2 | 6 | no | — |
| 180 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 181 | C22⋊D5 | — | 5 | 12 | no | — |
| 182 | C22⋊D5 | — | 10 | 12 | no | — |
| 183 | C2×Dic5 | — | 5 | 6 | no | — |
| 184 | C2×Dic5 | — | 10 | 6 | no | — |
| 185 | D20 | — | 5 | 6 | no | — |
| 186 | D20 | — | 10 | 6 | no | — |
| 187 | C4×D5 | — | 5 | 6 | no | — |
| 188 | C4×D5 | — | 5 | 6 | no | — |
| 189 | C4×D5 | — | 5 | 6 | no | — |
| 190 | C4×D5 | — | 5 | 2 | no | — |
| 191 | C4×D5 | — | 10 | 6 | no | — |
| 192 | C4×D5 | — | 10 | 6 | no | — |
| 193 | C4×D5 | — | 10 | 6 | no | — |
| 194 | C4×D5 | — | 10 | 2 | no | — |
| 195 | Dic10 | — | 5 | 6 | no | — |
| 196 | Dic10 | — | 10 | 6 | no | — |
| 197 | #170 | — | 5 | 4 | no | — |
| 198 | #170 | — | 5 | 12 | no | — |
| 199 | C5×C8 | — | 5 | 2 | no | — |
| 200 | C5×C8 | — | 5 | 6 | no | — |
| 201 | #168 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 202 | #168 | C22×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 203 | C2×#41 | — | 25 | 1 | no | — |
| 204 | #131 | — | 5 | 2 | no | — |
| 205 | #131 | — | 25 | 6 | no | — |
| 206 | #131 | — | 25 | 6 | no | — |
| 207 | #131 | — | 25 | 2 | no | — |
| 208 | C2×M16 | — | 25 | 3 | no | — |
| 209 | C2×M16 | — | 25 | 3 | no | — |
| 210 | C22×C8 | — | 25 | 3 | no | — |
| 211 | C22×C8 | — | 25 | 3 | no | — |
| 212 | C4×Q8 | — | 25 | 2 | no | — |
| 213 | C4×D4 | — | 25 | 6 | no | — |
| 214 | #117 | — | 25 | 3 | no | — |
| 215 | C2×C42 | — | 25 | 3 | no | — |
| 216 | #97 | — | 25 | 6 | no | — |
| 217 | C4×C8 | — | 25 | 1 | no | — |
| 218 | C4×C8 | — | 25 | 3 | no | — |
| 219 | C4×C8 | — | 25 | 3 | no | — |
| 220 | C4×C8 | — | 25 | 3 | no | — |
| 221 | C52 | C4×#131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 222 | C22×C5 | — | 2 | 6 | no | — |
| 223 | C22×C5 | C2×C4×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 224 | C2×D5 | — | 5 | 2 | no | — |
| 225 | C2×D5 | — | 5 | 6 | no | — |
| 226 | C2×D5 | — | 10 | 2 | no | — |
| 227 | C2×D5 | — | 10 | 6 | no | — |
| 228 | C2×D5 | — | 10 | 12 | no | — |
| 229 | C2×D5 | — | 20 | 12 | no | — |
| 230 | AGL(1,𝔽5) | — | 5 | 8 | no | — |
| 231 | AGL(1,𝔽5) | — | 10 | 8 | no | — |
| 232 | AGL(1,𝔽5) | — | 10 | 12 | no | — |
| 233 | AGL(1,𝔽5) | — | 20 | 12 | no | — |
| 234 | C4×C5 | — | 2 | 2 | no | — |
| 235 | C4×C5 | — | 2 | 6 | no | — |
| 236 | C4×C5 | C2×C4×AGL(1,𝔽5) | 1 | 6 | yes | — |
| 237 | C4×C5 | C23×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 238 | Dic5 | — | 5 | 6 | no | — |
| 239 | Dic5 | — | 5 | 2 | no | — |
| 240 | Dic5 | — | 10 | 6 | no | — |
| 241 | Dic5 | — | 10 | 2 | no | — |
| 242 | #41 | — | 25 | 1 | no | — |
| 243 | #41 | — | 25 | 3 | no | — |
| 244 | #41 | — | 25 | 3 | no | — |
| 245 | #41 | D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 246 | C2×Q8 | — | 25 | 1 | no | — |
| 247 | C2×D4 | — | 25 | 3 | no | — |
| 248 | C22×C4 | — | 25 | 3 | no | — |
| 249 | C22×C4 | — | 25 | 6 | no | — |
| 250 | M16 | — | 5 | 6 | no | — |
| 251 | M16 | — | 25 | 6 | no | — |
| 252 | M16 | — | 25 | 6 | no | — |
| 253 | M16 | — | 25 | 6 | no | — |
| 254 | C2×C8 | — | 5 | 6 | no | — |
| 255 | C2×C8 | — | 25 | 1 | no | — |
| 256 | C2×C8 | — | 25 | 3 | no | — |
| 257 | C2×C8 | — | 25 | 6 | no | — |
| 258 | C2×C8 | — | 25 | 1 | no | — |
| 259 | C2×C8 | — | 25 | 3 | no | — |
| 260 | C2×C8 | — | 25 | 6 | no | — |
| 261 | C2×C8 | — | 25 | 6 | no | — |
| 262 | C4⋊C4 | — | 25 | 6 | no | — |
| 263 | C22⋊C4 | — | 25 | 6 | no | — |
| 264 | C42 | — | 25 | 1 | no | — |
| 265 | C42 | — | 25 | 3 | no | — |
| 266 | C42 | — | 25 | 6 | no | — |
| 267 | C2×C5 | — | 2 | 6 | no | — |
| 268 | C2×C5 | — | 2 | 2 | no | — |
| 269 | C2×C5 | — | 4 | 6 | no | — |
| 270 | C2×C5 | C22×C4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 271 | D5 | — | 5 | 4 | no | — |
| 272 | D5 | — | 10 | 4 | no | — |
| 273 | D5 | — | 10 | 6 | no | — |
| 274 | D5 | — | 20 | 6 | no | — |
| 275 | C23 | — | 25 | 3 | no | — |
| 276 | Q8 | — | 25 | 3 | no | — |
| 277 | Q8 | #87830 | 1 | 1 | yes | characteristic |
| 278 | D4 | — | 25 | 6 | no | — |
| 279 | D4 | — | 25 | 3 | no | — |
| 280 | D4 | #87830 | 1 | 3 | yes | — |
| 281 | C2×C4 | — | 25 | 2 | no | — |
| 282 | C2×C4 | — | 25 | 3 | no | — |
| 283 | C2×C4 | — | 25 | 1 | no | — |
| 284 | C2×C4 | — | 25 | 3 | no | — |
| 285 | C2×C4 | — | 25 | 3 | no | — |
| 286 | C2×C4 | — | 25 | 3 | no | — |
| 287 | C2×C4 | — | 25 | 6 | no | — |
| 288 | C2×C4 | — | 50 | 12 | no | — |
| 289 | C2×C4 | C2×D5⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 290 | C8 | — | 5 | 2 | no | — |
| 291 | C8 | — | 5 | 6 | no | — |
| 292 | C8 | — | 25 | 2 | no | — |
| 293 | C8 | — | 25 | 6 | no | — |
| 294 | C5 | — | 2 | 2 | no | — |
| 295 | C5 | AGL(1,𝔽5)×#131 | 1 | 2 | yes | complemented |
| 296 | C22 | — | 25 | 1 | no | — |
| 297 | C22 | — | 25 | 3 | no | — |
| 298 | C22 | — | 50 | 6 | no | — |
| 299 | C22 | C2×#87830 | 1 | 3 | yes | — |
| 300 | C4 | — | 25 | 4 | no | — |
| 301 | C4 | — | 25 | 1 | no | — |
| 302 | C4 | — | 25 | 3 | no | — |
| 303 | C4 | — | 50 | 6 | no | — |
| 304 | C4 | C2×#87830 | 1 | 3 | yes | — |
| 305 | C4 | C22×D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 2 | 3 | no | — |
| 307 | C2 | — | 25 | 2 | no | — |
| 308 | C2 | — | 50 | 3 | no | — |
| 309 | C2 | C22×#87830 | 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