Finite Group DatabasebetaFinite Group DatabasebetaOrder 3200.
1322 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #127×C5⋊D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#11716900 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #13104827 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13104814 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×D4×C5⋊D5 | — | 2 | 2 | no | — |
| 6 | C2×D4×C5⋊D5 | C22 | 1 | 2 | yes | — |
| 7 | C2×C4⋊#1044 | C22 | 1 | 2 | yes | — |
| 8 | C2×#87862 | C22 | 1 | 2 | yes | — |
| 9 | C52×#127 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #11717260 | — | 2 | 2 | no | — |
| 11 | #11717260 | C4 | 1 | 2 | yes | complemented |
| 12 | #11717194 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C42×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 14 | #11716900 | — | 2 | 8 | no | — |
| 15 | #11716899 | — | 2 | 2 | no | — |
| 16 | #11716895 | — | 2 | 2 | no | — |
| 17 | #10604617 | — | 5 | 2 | no | maximal |
| 18 | C23×C5⋊D5 | — | 2 | 2 | no | — |
| 19 | C22×#1044 | — | 2 | 2 | no | — |
| 20 | C2×C52×D4 | — | 2 | 2 | no | — |
| 21 | C2×C52×D4 | C2×C4 | 1 | 2 | yes | — |
| 22 | C2×C22⋊(C5⋊D5) | — | 2 | 4 | no | — |
| 23 | C2×C22⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 24 | D4×C5⋊D5 | — | 2 | 4 | no | — |
| 25 | D4×C5⋊D5 | — | 2 | 4 | no | — |
| 26 | D4×C5⋊D5 | — | 4 | 8 | no | — |
| 27 | C2×C4⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 28 | C2×C4⋊(C5⋊D5) | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C4×C5⋊D5 | D4 | 1 | 2 | yes | — |
| 30 | C2×C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×#6034 | — | 2 | 2 | no | — |
| 32 | C4⋊#1044 | — | 2 | 4 | no | — |
| 33 | #87862 | — | 2 | 2 | no | — |
| 34 | #87862 | D4 | 1 | 4 | yes | complemented |
| 35 | C42×C52 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C4×#1039 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×#1039 | D4 | 1 | 2 | yes | — |
| 38 | D5×#127 | — | 5 | 2 | no | — |
| 39 | D5×#127 | — | 10 | 2 | no | — |
| 40 | C2×#4007 | — | 5 | 8 | no | — |
| 41 | #65689 | — | 5 | 2 | no | — |
| 42 | #65676 | — | 5 | 2 | no | — |
| 43 | C23×C52 | — | 2 | 2 | no | — |
| 44 | C22×C5⋊D5 | — | 2 | 2 | no | — |
| 45 | C22×C5⋊D5 | — | 4 | 2 | no | — |
| 46 | C22×C5⋊D5 | — | 4 | 2 | no | — |
| 47 | C22×C5⋊D5 | — | 4 | 8 | no | — |
| 48 | C22×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | C2×#1044 | — | 4 | 2 | no | — |
| 50 | C2×#1044 | — | 4 | 4 | no | — |
| 51 | C52×D4 | — | 2 | 4 | no | — |
| 52 | C52×D4 | — | 4 | 4 | no | — |
| 53 | C2×C4×C52 | C22⋊C4 | 1 | 2 | yes | — |
| 54 | C2×C4×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C22⋊(C5⋊D5) | — | 4 | 8 | no | — |
| 56 | C22⋊(C5⋊D5) | — | 4 | 4 | no | — |
| 57 | C2×#1039 | C22⋊C4 | 1 | 2 | yes | — |
| 58 | C2×#1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C4⋊(C5⋊D5) | — | 2 | 4 | no | — |
| 60 | C4⋊(C5⋊D5) | — | 4 | 4 | no | — |
| 61 | C4×C5⋊D5 | — | 2 | 4 | no | — |
| 62 | C4×C5⋊D5 | C2×D4 | 1 | 2 | yes | — |
| 63 | C4×C5⋊D5 | #41 | 1 | 2 | yes | — |
| 64 | #6034 | — | 2 | 4 | no | — |
| 65 | C2×D4×D5 | — | 5 | 4 | no | — |
| 66 | C2×D4×D5 | — | 10 | 4 | no | — |
| 67 | C2×D4×D5 | — | 10 | 4 | no | — |
| 68 | C2×D4×D5 | — | 10 | 8 | no | — |
| 69 | C2×C4⋊AGL(1,𝔽5) | — | 5 | 4 | no | — |
| 70 | C2×D5⋊C8 | — | 5 | 4 | no | — |
| 71 | C5×#127 | — | 2 | 2 | no | — |
| 72 | C5×#127 | AGL(1,𝔽5) | 1 | 2 | yes | complemented |
| 73 | #4086 | — | 5 | 4 | no | — |
| 74 | #4086 | — | 10 | 4 | no | — |
| 75 | #4086 | — | 10 | 4 | no | — |
| 76 | #4086 | — | 10 | 8 | no | — |
| 77 | #4020 | — | 5 | 2 | no | — |
| 78 | #4020 | — | 10 | 2 | no | — |
| 79 | C42×D5 | — | 5 | 2 | no | — |
| 80 | C42×D5 | — | 10 | 2 | no | — |
| 81 | #4007 | — | 10 | 16 | no | — |
| 82 | (C2×AGL(1,𝔽5))⋊C4 | — | 10 | 4 | no | — |
| 83 | #4002 | — | 10 | 4 | no | — |
| 84 | #1961 | — | 25 | 1 | no | Sylow |
| 85 | C22×C52 | — | 4 | 2 | no | — |
| 86 | C22×C52 | — | 4 | 4 | no | — |
| 87 | C22×C52 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C5⋊D5 | — | 4 | 2 | no | — |
| 89 | C2×C5⋊D5 | — | 4 | 4 | no | — |
| 90 | C2×C5⋊D5 | — | 8 | 4 | no | — |
| 91 | C2×C5⋊D5 | — | 8 | 4 | no | — |
| 92 | C2×C5⋊D5 | #124 | 1 | 1 | yes | characteristic |
| 93 | C2×C5⋊D5 | #127 | 1 | 1 | yes | characteristic |
| 94 | C2×C5⋊D5 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 95 | #1044 | — | 8 | 2 | no | — |
| 96 | C4×C52 | — | 2 | 2 | no | — |
| 97 | C4×C52 | C4×D4 | 1 | 2 | yes | — |
| 98 | #1039 | — | 2 | 2 | no | — |
| 99 | #1039 | #121 | 1 | 2 | yes | — |
| 100 | C23×D5 | — | 10 | 4 | no | — |
| 101 | C23×D5 | — | 10 | 8 | no | — |
| 102 | C22×AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 103 | C2×C5×D4 | — | 2 | 8 | no | — |
| 104 | C2×C5×D4 | — | 2 | 4 | no | — |
| 105 | C2×C5×D4 | — | 2 | 4 | no | — |
| 106 | C2×C5×D4 | C2×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 107 | C2×C22⋊D5 | — | 10 | 4 | no | — |
| 108 | C2×C22⋊D5 | — | 10 | 8 | no | — |
| 109 | C2×C22⋊D5 | — | 10 | 16 | no | — |
| 110 | C2×C22⋊D5 | — | 10 | 8 | no | — |
| 111 | D4×D5 | — | 10 | 8 | no | — |
| 112 | D4×D5 | — | 10 | 8 | no | — |
| 113 | D4×D5 | — | 20 | 8 | no | — |
| 114 | D4×D5 | — | 20 | 8 | no | — |
| 115 | D4×D5 | — | 20 | 16 | no | — |
| 116 | D4×D5 | — | 20 | 32 | no | — |
| 117 | C2×D20 | — | 5 | 4 | no | — |
| 118 | C2×D20 | — | 10 | 4 | no | — |
| 119 | C2×D20 | — | 10 | 8 | no | — |
| 120 | C2×D20 | — | 10 | 4 | no | — |
| 121 | C2×C4×D5 | — | 5 | 2 | no | — |
| 122 | C2×C4×D5 | — | 5 | 4 | no | — |
| 123 | C2×C4×D5 | — | 10 | 4 | no | — |
| 124 | C2×C4×D5 | — | 10 | 2 | no | — |
| 125 | C2×#170 | — | 10 | 4 | no | — |
| 126 | C4⋊AGL(1,𝔽5) | — | 10 | 8 | no | — |
| 127 | D5⋊C8 | — | 5 | 8 | no | — |
| 128 | D5⋊C8 | — | 10 | 4 | no | — |
| 129 | C42×C5 | — | 2 | 2 | no | — |
| 130 | C42×C5 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 131 | C4×Dic5 | — | 5 | 4 | no | — |
| 132 | C4×Dic5 | — | 5 | 2 | no | — |
| 133 | C4×Dic5 | — | 10 | 2 | no | — |
| 134 | C4×Dic5 | — | 10 | 4 | no | — |
| 135 | C2×#127 | — | 25 | 1 | no | — |
| 136 | C2×#102 | — | 25 | 4 | no | — |
| 137 | C2×#97 | — | 25 | 1 | no | — |
| 138 | #382 | — | 25 | 1 | no | — |
| 139 | C2×C52 | — | 8 | 2 | no | — |
| 140 | C2×C52 | #390 | 1 | 1 | yes | characteristic |
| 141 | C2×C52 | #418 | 1 | 2 | yes | — |
| 142 | C5⋊D5 | — | 8 | 2 | no | — |
| 143 | C5⋊D5 | #490 | 1 | 2 | yes | — |
| 144 | C5⋊D5 | #496 | 1 | 2 | yes | — |
| 145 | C23×C5 | — | 2 | 8 | no | — |
| 146 | C23×C5 | — | 2 | 4 | no | — |
| 147 | C22×D5 | — | 5 | 2 | no | — |
| 148 | C22×D5 | — | 10 | 2 | no | — |
| 149 | C22×D5 | — | 10 | 8 | no | — |
| 150 | C22×D5 | — | 10 | 4 | no | — |
| 151 | C22×D5 | — | 20 | 4 | no | — |
| 152 | C22×D5 | — | 20 | 4 | no | — |
| 153 | C22×D5 | — | 20 | 16 | no | — |
| 154 | C22×D5 | — | 20 | 8 | no | — |
| 155 | C22×D5 | — | 20 | 8 | no | — |
| 156 | C22×D5 | — | 20 | 32 | no | — |
| 157 | C2×AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 158 | C2×AGL(1,𝔽5) | — | 20 | 8 | no | — |
| 159 | C5×D4 | — | 2 | 8 | no | — |
| 160 | C5×D4 | — | 4 | 8 | no | — |
| 161 | C5×D4 | — | 4 | 16 | no | — |
| 162 | C5×D4 | — | 4 | 8 | no | — |
| 163 | C2×C4×C5 | — | 2 | 4 | no | — |
| 164 | C2×C4×C5 | — | 2 | 2 | no | — |
| 165 | C2×C4×C5 | C22⋊AGL(1,𝔽5) | 1 | 4 | yes | — |
| 166 | C2×C4×C5 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 167 | C22⋊D5 | — | 20 | 8 | no | — |
| 168 | C22⋊D5 | — | 20 | 16 | no | — |
| 169 | C22⋊D5 | — | 20 | 32 | no | — |
| 170 | C22⋊D5 | — | 20 | 16 | no | — |
| 171 | C2×Dic5 | — | 5 | 2 | no | — |
| 172 | C2×Dic5 | — | 5 | 4 | no | — |
| 173 | C2×Dic5 | — | 10 | 4 | no | — |
| 174 | C2×Dic5 | — | 10 | 2 | no | — |
| 175 | D20 | — | 10 | 8 | no | — |
| 176 | D20 | — | 20 | 8 | no | — |
| 177 | D20 | — | 20 | 16 | no | — |
| 178 | D20 | — | 20 | 8 | no | — |
| 179 | C4×D5 | — | 5 | 4 | no | — |
| 180 | C4×D5 | — | 5 | 4 | no | — |
| 181 | C4×D5 | — | 10 | 4 | no | — |
| 182 | C4×D5 | — | 10 | 4 | no | — |
| 183 | C4×D5 | — | 10 | 8 | no | — |
| 184 | C4×D5 | — | 10 | 16 | no | — |
| 185 | #170 | — | 10 | 8 | no | — |
| 186 | C22×D4 | — | 25 | 2 | no | — |
| 187 | C22×D4 | — | 50 | 2 | no | — |
| 188 | C22×C8 | — | 25 | 2 | no | — |
| 189 | #127 | — | 25 | 2 | no | — |
| 190 | #127 | — | 25 | 1 | no | — |
| 191 | #127 | — | 50 | 2 | no | — |
| 192 | #127 | #1044 | 1 | 1 | yes | characteristic, complemented |
| 193 | C2×C4⋊C4 | — | 25 | 2 | no | — |
| 194 | C2×C42 | — | 25 | 1 | no | — |
| 195 | #102 | — | 50 | 8 | no | — |
| 196 | #97 | — | 50 | 2 | no | — |
| 197 | #95 | — | 50 | 2 | no | — |
| 198 | C52 | #1961 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 199 | C22×C5 | — | 2 | 2 | no | — |
| 200 | C22×C5 | — | 4 | 16 | no | — |
| 201 | C22×C5 | — | 4 | 8 | no | — |
| 202 | C22×C5 | — | 4 | 8 | no | — |
| 203 | C22×C5 | — | 4 | 4 | no | — |
| 204 | C22×C5 | C2×C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 205 | C2×D5 | — | 5 | 8 | no | — |
| 206 | C2×D5 | — | 5 | 2 | no | — |
| 207 | C2×D5 | — | 5 | 2 | no | — |
| 208 | C2×D5 | — | 10 | 2 | no | — |
| 209 | C2×D5 | — | 10 | 2 | no | — |
| 210 | C2×D5 | — | 10 | 8 | no | — |
| 211 | C2×D5 | — | 20 | 16 | no | — |
| 212 | C2×D5 | — | 20 | 8 | no | — |
| 213 | C2×D5 | — | 20 | 8 | no | — |
| 214 | C2×D5 | — | 20 | 4 | no | — |
| 215 | C2×D5 | — | 40 | 8 | no | — |
| 216 | C2×D5 | — | 40 | 8 | no | — |
| 217 | C2×D5 | — | 40 | 16 | no | — |
| 218 | C2×D5 | — | 40 | 16 | no | — |
| 219 | AGL(1,𝔽5) | — | 40 | 4 | no | — |
| 220 | C4×C5 | — | 2 | 8 | no | — |
| 221 | C4×C5 | — | 2 | 4 | no | — |
| 222 | C4×C5 | — | 2 | 4 | no | — |
| 223 | C4×C5 | D4×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 224 | Dic5 | — | 5 | 4 | no | — |
| 225 | Dic5 | — | 10 | 4 | no | — |
| 226 | Dic5 | — | 10 | 8 | no | — |
| 227 | Dic5 | — | 10 | 4 | no | — |
| 228 | C24 | — | 50 | 2 | no | — |
| 229 | C2×D4 | — | 2 | 2 | no | — |
| 230 | C2×D4 | — | 25 | 2 | no | — |
| 231 | C2×D4 | — | 50 | 4 | no | — |
| 232 | C2×D4 | — | 50 | 4 | no | — |
| 233 | C2×D4 | — | 50 | 4 | no | — |
| 234 | C2×D4 | — | 50 | 2 | no | — |
| 235 | C2×D4 | — | 50 | 2 | no | — |
| 236 | C2×D4 | — | 100 | 8 | no | — |
| 237 | C2×D4 | C2×#1044 | 1 | 2 | yes | — |
| 238 | C22×C4 | — | 25 | 1 | no | — |
| 239 | C22×C4 | — | 25 | 2 | no | — |
| 240 | C22×C4 | — | 50 | 2 | no | — |
| 241 | C2×C8 | — | 25 | 4 | no | — |
| 242 | C2×C8 | — | 50 | 2 | no | — |
| 243 | C2×C8 | — | 50 | 2 | no | — |
| 244 | C4⋊C4 | — | 50 | 4 | no | — |
| 245 | C42 | — | 25 | 2 | no | — |
| 246 | C42 | — | 25 | 1 | no | — |
| 247 | C42 | C2×#1044 | 1 | 1 | yes | characteristic |
| 248 | C2×C5 | — | 2 | 4 | no | — |
| 249 | C2×C5 | — | 2 | 2 | no | — |
| 250 | C2×C5 | — | 8 | 8 | no | — |
| 251 | C2×C5 | — | 8 | 4 | no | — |
| 252 | C2×C5 | #65767 | 1 | 4 | yes | — |
| 253 | C2×C5 | #65780 | 1 | 2 | yes | — |
| 254 | D5 | — | 5 | 4 | no | — |
| 255 | D5 | — | 5 | 4 | no | — |
| 256 | D5 | — | 10 | 4 | no | — |
| 257 | D5 | — | 10 | 4 | no | — |
| 258 | D5 | — | 40 | 8 | no | — |
| 259 | D5 | — | 40 | 4 | no | — |
| 260 | C23 | — | 2 | 2 | no | — |
| 261 | C23 | — | 25 | 1 | no | — |
| 262 | C23 | — | 50 | 2 | no | — |
| 263 | C23 | — | 100 | 2 | no | — |
| 264 | C23 | — | 100 | 2 | no | — |
| 265 | C23 | — | 100 | 8 | no | — |
| 266 | D4 | — | 2 | 4 | no | — |
| 267 | D4 | — | 4 | 4 | no | — |
| 268 | D4 | — | 50 | 4 | no | — |
| 269 | D4 | — | 100 | 4 | no | — |
| 270 | D4 | — | 100 | 4 | no | — |
| 271 | D4 | — | 100 | 8 | no | — |
| 272 | C2×C4 | — | 25 | 1 | no | — |
| 273 | C2×C4 | — | 25 | 2 | no | — |
| 274 | C2×C4 | — | 25 | 2 | no | — |
| 275 | C2×C4 | — | 25 | 2 | no | — |
| 276 | C2×C4 | — | 50 | 4 | no | — |
| 277 | C2×C4 | — | 100 | 2 | no | — |
| 278 | C2×C4 | — | 100 | 4 | no | — |
| 279 | C2×C4 | C22⋊#1044 | 1 | 2 | yes | — |
| 280 | C2×C4 | C22×#1044 | 1 | 1 | yes | characteristic |
| 281 | C8 | — | 50 | 4 | no | — |
| 282 | C5 | — | 2 | 2 | no | — |
| 283 | C5 | #10604617 | 1 | 2 | yes | complemented |
| 284 | C22 | — | 4 | 4 | no | — |
| 285 | C22 | — | 4 | 2 | no | — |
| 286 | C22 | — | 25 | 4 | no | — |
| 287 | C22 | — | 25 | 1 | no | — |
| 288 | C22 | — | 25 | 1 | no | — |
| 289 | C22 | — | 100 | 4 | no | — |
| 290 | C22 | — | 100 | 2 | no | — |
| 291 | C22 | — | 200 | 4 | no | — |
| 292 | C22 | — | 200 | 4 | no | — |
| 293 | C22 | C2×C22⋊#1044 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 25 | 2 | no | — |
| 296 | C4 | — | 50 | 2 | no | — |
| 297 | C4 | — | 200 | 2 | no | — |
| 298 | C4 | D4×#1044 | 1 | 2 | yes | — |
| 299 | C2 | — | 8 | 2 | no | — |
| 300 | C2 | — | 25 | 2 | no | — |
| 301 | C2 | — | 25 | 2 | no | — |
| 302 | C2 | — | 200 | 2 | no | — |
| 303 | C2 | #13104909 | 1 | 2 | yes | — |
| 304 | C2 | #13104918 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#11716900 | C2 | semidirect |
| 3. #13104827 | C2 | semidirect |
| 4. #13104814 | C2 | semidirect |
| 9. C52×#127 | C4 | semidirect |
| 11. #11717260 | C4 | semidirect |
| 12. #11717194 | C4 | semidirect |
| 34. #87862 | D4 | semidirect |
| 72. C5×#127 | AGL(1,𝔽5) | semidirect |
| 192. #127 | #1044 | semidirect |
| 198. C52 | #1961 | semidirect |
| 283. C5 | #10604617 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #127×C5⋊D5 | 2 |
| C2×#11716900 | 2 |
| #13104827 | 2 |
| #13104814 | 2 |
| #10604617 | 5 |