Finite Group DatabasebetaFinite Group DatabasebetaOrder 3200.
12578 conjugacy classes of subgroups; only the 484 normal subgroups are shown.
This group has 12578 conjugacy classes of subgroups — too many to draw. Only its 484 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 | C25×C5⋊D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#11717092 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C5×#65536 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13101890 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C25×C52 | C22 | 1 | 1 | yes | characteristic |
| 6 | C24×C5⋊D5 | — | 2 | 24 | no | — |
| 7 | C24×C5⋊D5 | C4 | 1 | 8 | yes | complemented |
| 8 | C24×C5⋊D5 | C22 | 1 | 3 | yes | — |
| 9 | C24×C5⋊D5 | C22 | 1 | 3 | yes | — |
| 10 | C22×#6036 | C22 | 1 | 4 | yes | complemented |
| 11 | C2×C5×C22⋊Dic5 | C22 | 1 | 12 | yes | complemented |
| 12 | #11717092 | — | 2 | 24 | no | — |
| 13 | C2×#87779 | C22 | 1 | 12 | yes | complemented |
| 14 | D5×#379 | — | 5 | 2 | no | maximal |
| 15 | C24×C52 | — | 2 | 12 | no | — |
| 16 | C24×C52 | C2×C4 | 1 | 4 | yes | complemented |
| 17 | C24×C52 | C23 | 1 | 3 | yes | — |
| 18 | C23×C5⋊D5 | — | 2 | 128 | no | — |
| 19 | C23×C5⋊D5 | — | 2 | 12 | no | — |
| 20 | C23×C5⋊D5 | — | 2 | 96 | no | — |
| 21 | C23×C5⋊D5 | — | 2 | 24 | no | — |
| 22 | C23×C5⋊D5 | — | 2 | 24 | no | — |
| 23 | C23×C5⋊D5 | C2×C4 | 1 | 12 | yes | — |
| 24 | C23×C5⋊D5 | C2×C4 | 1 | 12 | yes | complemented |
| 25 | C23×C5⋊D5 | D4 | 1 | 24 | yes | — |
| 26 | C23×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 27 | C23×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 28 | C22×C5×Dic5 | C23 | 1 | 8 | yes | complemented |
| 29 | C2×#6036 | — | 2 | 24 | no | — |
| 30 | C5×C22⋊Dic5 | — | 2 | 24 | no | — |
| 31 | #87779 | — | 2 | 24 | no | — |
| 32 | C25×D5 | — | 5 | 2 | no | — |
| 33 | C25×D5 | — | 10 | 2 | no | — |
| 34 | C2×D5×C22⋊C4 | — | 5 | 12 | no | — |
| 35 | C5×#379 | — | 5 | 2 | no | — |
| 36 | #65536 | D5 | 1 | 2 | yes | complemented |
| 37 | #65244 | — | 5 | 12 | no | — |
| 38 | C23×C52 | — | 2 | 32 | no | — |
| 39 | C23×C52 | — | 2 | 24 | no | — |
| 40 | C23×C52 | — | 2 | 12 | no | — |
| 41 | C23×C52 | C22×C4 | 1 | 6 | yes | — |
| 42 | C23×C52 | C2×D4 | 1 | 12 | yes | — |
| 43 | C23×C52 | C24 | 1 | 1 | yes | characteristic |
| 44 | C22×C5⋊D5 | — | 2 | 96 | no | — |
| 45 | C22×C5⋊D5 | — | 2 | 256 | no | — |
| 46 | C22×C5⋊D5 | — | 2 | 24 | no | — |
| 47 | C22×C5⋊D5 | — | 2 | 12 | no | — |
| 48 | C22×C5⋊D5 | — | 2 | 48 | no | — |
| 49 | C22×C5⋊D5 | — | 2 | 48 | no | — |
| 50 | C22×C5⋊D5 | — | 2 | 96 | no | — |
| 51 | C22×C5⋊D5 | C22⋊C4 | 1 | 48 | yes | complemented |
| 52 | C22×C5⋊D5 | C22×C4 | 1 | 6 | yes | — |
| 53 | C22×C5⋊D5 | C22×C4 | 1 | 2 | yes | — |
| 54 | C22×C5⋊D5 | C2×D4 | 1 | 12 | yes | — |
| 55 | C22×C5⋊D5 | C2×D4 | 1 | 12 | yes | — |
| 56 | C2×C5×Dic5 | — | 2 | 24 | no | — |
| 57 | #6036 | — | 4 | 16 | no | — |
| 58 | C25×C5 | — | 2 | 2 | no | — |
| 59 | C25×C5 | C2×D5 | 1 | 2 | yes | — |
| 60 | C24×D5 | — | 5 | 6 | no | — |
| 61 | C24×D5 | — | 5 | 16 | no | — |
| 62 | C24×D5 | — | 5 | 6 | no | — |
| 63 | C24×D5 | — | 10 | 16 | no | — |
| 64 | C24×D5 | — | 10 | 6 | no | — |
| 65 | C24×D5 | — | 10 | 6 | no | — |
| 66 | C24×D5 | — | 10 | 48 | no | — |
| 67 | C24×D5 | — | 10 | 96 | no | — |
| 68 | C22×C4×D5 | — | 5 | 8 | no | — |
| 69 | C2×C5×C22⋊C4 | — | 5 | 12 | no | — |
| 70 | C2×C22⋊Dic5 | C2×D5 | 1 | 12 | yes | complemented |
| 71 | C2×#685 | — | 5 | 24 | no | — |
| 72 | D5×C22⋊C4 | — | 10 | 48 | no | — |
| 73 | C2×#379 | — | 25 | 1 | no | Sylow |
| 74 | C22×C52 | — | 2 | 32 | no | — |
| 75 | C22×C52 | — | 2 | 12 | no | — |
| 76 | C22×C52 | — | 2 | 24 | no | — |
| 77 | C22×C52 | C2×C22⋊C4 | 1 | 12 | yes | complemented |
| 78 | C22×C52 | C23×C4 | 1 | 1 | yes | characteristic |
| 79 | C22×C52 | C22×D4 | 1 | 6 | yes | — |
| 80 | C2×C5⋊D5 | — | 2 | 96 | no | — |
| 81 | C2×C5⋊D5 | — | 2 | 12 | no | — |
| 82 | C2×C5⋊D5 | — | 2 | 24 | no | — |
| 83 | C2×C5⋊D5 | — | 2 | 96 | no | — |
| 84 | C2×C5⋊D5 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 85 | C2×C5⋊D5 | C2×C22⋊C4 | 1 | 12 | yes | complemented |
| 86 | C2×C5⋊D5 | #120 | 1 | 16 | yes | — |
| 87 | C5×Dic5 | — | 4 | 8 | no | — |
| 88 | C24×C5 | — | 2 | 48 | no | — |
| 89 | C24×C5 | — | 2 | 24 | no | — |
| 90 | C24×C5 | — | 2 | 8 | no | — |
| 91 | C24×C5 | — | 2 | 6 | no | — |
| 92 | C24×C5 | C4×D5 | 1 | 8 | yes | complemented |
| 93 | C24×C5 | C22×D5 | 1 | 6 | yes | — |
| 94 | C23×D5 | — | 5 | 2 | no | — |
| 95 | C23×D5 | — | 5 | 24 | no | — |
| 96 | C23×D5 | — | 5 | 48 | no | — |
| 97 | C23×D5 | — | 5 | 6 | no | — |
| 98 | C23×D5 | — | 5 | 24 | no | — |
| 99 | C23×D5 | — | 10 | 24 | no | — |
| 100 | C23×D5 | — | 10 | 24 | no | — |
| 101 | C23×D5 | — | 10 | 48 | no | — |
| 102 | C23×D5 | — | 10 | 2 | no | — |
| 103 | C23×D5 | — | 10 | 6 | no | — |
| 104 | C23×D5 | — | 10 | 48 | no | — |
| 105 | C23×D5 | — | 10 | 48 | no | — |
| 106 | C23×D5 | — | 10 | 192 | no | — |
| 107 | C23×D5 | — | 10 | 384 | no | — |
| 108 | C23×D5 | — | 10 | 96 | no | — |
| 109 | C23×D5 | — | 10 | 96 | no | — |
| 110 | C23×D5 | — | 10 | 256 | no | — |
| 111 | C23×D5 | — | 10 | 512 | no | — |
| 112 | C23×D5 | — | 10 | 48 | no | — |
| 113 | C23×D5 | — | 10 | 24 | no | — |
| 114 | C22×C4×C5 | — | 5 | 8 | no | — |
| 115 | C22×Dic5 | C22×D5 | 1 | 8 | yes | complemented |
| 116 | C2×C4×D5 | — | 10 | 48 | no | — |
| 117 | C5×C22⋊C4 | — | 10 | 24 | no | — |
| 118 | C22⋊Dic5 | — | 2 | 24 | no | — |
| 119 | #685 | — | 10 | 48 | no | — |
| 120 | C26 | — | 25 | 1 | no | — |
| 121 | C22×C22⋊C4 | — | 25 | 6 | no | — |
| 122 | #379 | — | 5 | 2 | no | — |
| 123 | #379 | — | 25 | 6 | no | — |
| 124 | C2×C52 | — | 2 | 12 | no | — |
| 125 | C2×C52 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 126 | C2×C52 | C2×#120 | 1 | 4 | yes | — |
| 127 | C5⋊D5 | — | 2 | 12 | no | — |
| 128 | C5⋊D5 | #379 | 1 | 8 | yes | complemented |
| 129 | C23×C5 | — | 2 | 128 | no | — |
| 130 | C23×C5 | — | 2 | 64 | no | — |
| 131 | C23×C5 | — | 2 | 48 | no | — |
| 132 | C23×C5 | — | 2 | 96 | no | — |
| 133 | C23×C5 | — | 2 | 24 | no | — |
| 134 | C23×C5 | — | 2 | 48 | no | — |
| 135 | C23×C5 | — | 2 | 12 | no | — |
| 136 | C23×C5 | — | 2 | 24 | no | — |
| 137 | C23×C5 | — | 2 | 2 | no | — |
| 138 | C23×C5 | C2×C4×D5 | 1 | 12 | yes | — |
| 139 | C23×C5 | D4×D5 | 1 | 24 | yes | — |
| 140 | C23×C5 | C23×D5 | 1 | 2 | yes | — |
| 141 | C22×D5 | — | 5 | 4 | no | — |
| 142 | C22×D5 | — | 5 | 24 | no | — |
| 143 | C22×D5 | — | 5 | 12 | no | — |
| 144 | C22×D5 | — | 5 | 24 | no | — |
| 145 | C22×D5 | — | 5 | 96 | no | — |
| 146 | C22×D5 | — | 10 | 96 | no | — |
| 147 | C22×D5 | — | 10 | 4 | no | — |
| 148 | C22×D5 | — | 10 | 12 | no | — |
| 149 | C22×D5 | — | 10 | 24 | no | — |
| 150 | C22×D5 | — | 10 | 24 | no | — |
| 151 | C22×D5 | — | 10 | 96 | no | — |
| 152 | C22×D5 | — | 10 | 192 | no | — |
| 153 | C22×D5 | — | 10 | 96 | no | — |
| 154 | C22×D5 | — | 10 | 192 | no | — |
| 155 | C22×D5 | — | 10 | 192 | no | — |
| 156 | C22×D5 | — | 10 | 384 | no | — |
| 157 | C22×D5 | — | 10 | 512 | no | — |
| 158 | C22×D5 | — | 10 | 1024 | no | — |
| 159 | C22×D5 | — | 10 | 384 | no | — |
| 160 | C22×D5 | — | 10 | 192 | no | — |
| 161 | C22×D5 | — | 10 | 48 | no | — |
| 162 | C22×D5 | — | 10 | 96 | no | — |
| 163 | C22×D5 | — | 10 | 24 | no | — |
| 164 | C22×D5 | — | 10 | 48 | no | — |
| 165 | C2×C4×C5 | — | 10 | 24 | no | — |
| 166 | C2×Dic5 | — | 2 | 24 | no | — |
| 167 | C4×D5 | — | 20 | 32 | no | — |
| 168 | C25 | — | 25 | 3 | no | — |
| 169 | C25 | — | 25 | 8 | no | — |
| 170 | C25 | — | 25 | 3 | no | — |
| 171 | C25 | — | 50 | 24 | no | — |
| 172 | C25 | D52 | 1 | 1 | yes | characteristic |
| 173 | C23×C4 | — | 25 | 4 | no | — |
| 174 | C2×C22⋊C4 | — | 5 | 12 | no | — |
| 175 | C2×C22⋊C4 | — | 25 | 12 | no | — |
| 176 | C2×C22⋊C4 | — | 50 | 24 | no | — |
| 177 | C52 | C2×#379 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 178 | C22×C5 | — | 2 | 128 | no | — |
| 179 | C22×C5 | — | 2 | 64 | no | — |
| 180 | C22×C5 | — | 2 | 48 | no | — |
| 181 | C22×C5 | — | 2 | 96 | no | — |
| 182 | C22×C5 | — | 2 | 24 | no | — |
| 183 | C22×C5 | — | 2 | 48 | no | — |
| 184 | C22×C5 | — | 2 | 2 | no | — |
| 185 | C22×C5 | — | 2 | 12 | no | — |
| 186 | C22×C5 | — | 2 | 24 | no | — |
| 187 | C22×C5 | D5×C22⋊C4 | 1 | 24 | yes | complemented |
| 188 | C22×C5 | C22×C4×D5 | 1 | 2 | yes | — |
| 189 | C22×C5 | C2×D4×D5 | 1 | 12 | yes | — |
| 190 | C2×D5 | — | 5 | 24 | no | — |
| 191 | C2×D5 | — | 5 | 32 | no | — |
| 192 | C2×D5 | — | 5 | 24 | no | — |
| 193 | C2×D5 | — | 10 | 24 | no | — |
| 194 | C2×D5 | — | 10 | 24 | no | — |
| 195 | C2×D5 | — | 10 | 32 | no | — |
| 196 | C2×D5 | — | 10 | 192 | no | — |
| 197 | C2×D5 | — | 10 | 384 | no | — |
| 198 | C2×D5 | — | 10 | 384 | no | — |
| 199 | C2×D5 | — | 10 | 192 | no | — |
| 200 | C2×D5 | — | 10 | 48 | no | — |
| 201 | C2×D5 | — | 10 | 96 | no | — |
| 202 | C2×D5 | — | 10 | 24 | no | — |
| 203 | C2×D5 | — | 10 | 48 | no | — |
| 204 | C4×C5 | — | 20 | 8 | no | — |
| 205 | Dic5 | — | 4 | 8 | no | — |
| 206 | C24 | — | 2 | 12 | no | — |
| 207 | C24 | — | 25 | 1 | no | — |
| 208 | C24 | — | 25 | 12 | no | — |
| 209 | C24 | — | 25 | 24 | no | — |
| 210 | C24 | — | 25 | 3 | no | — |
| 211 | C24 | — | 25 | 12 | no | — |
| 212 | C24 | — | 50 | 24 | no | — |
| 213 | C24 | — | 50 | 24 | no | — |
| 214 | C24 | — | 50 | 96 | no | — |
| 215 | C24 | — | 50 | 128 | no | — |
| 216 | C24 | — | 50 | 12 | no | — |
| 217 | C24 | #6036 | 1 | 4 | yes | complemented |
| 218 | C24 | C2×D52 | 1 | 3 | yes | — |
| 219 | C22×C4 | — | 5 | 8 | no | — |
| 220 | C22×C4 | — | 50 | 24 | no | — |
| 221 | C22⋊C4 | — | 10 | 24 | no | — |
| 222 | C22⋊C4 | — | 50 | 24 | no | — |
| 223 | C2×C5 | — | 2 | 48 | no | — |
| 224 | C2×C5 | — | 2 | 24 | no | — |
| 225 | C2×C5 | — | 2 | 6 | no | — |
| 226 | C2×C5 | — | 2 | 8 | no | — |
| 227 | C2×C5 | C2×D5×C22⋊C4 | 1 | 6 | yes | — |
| 228 | C2×C5 | D5×#120 | 1 | 8 | yes | — |
| 229 | D5 | — | 5 | 16 | no | — |
| 230 | D5 | — | 10 | 16 | no | — |
| 231 | D5 | — | 10 | 48 | no | — |
| 232 | D5 | — | 10 | 24 | no | — |
| 233 | C23 | — | 2 | 32 | no | — |
| 234 | C23 | — | 2 | 12 | no | — |
| 235 | C23 | — | 2 | 24 | no | — |
| 236 | C23 | — | 25 | 2 | no | — |
| 237 | C23 | — | 25 | 12 | no | — |
| 238 | C23 | — | 25 | 6 | no | — |
| 239 | C23 | — | 25 | 12 | no | — |
| 240 | C23 | — | 25 | 48 | no | — |
| 241 | C23 | — | 50 | 48 | no | — |
| 242 | C23 | — | 50 | 48 | no | — |
| 243 | C23 | — | 50 | 96 | no | — |
| 244 | C23 | — | 50 | 256 | no | — |
| 245 | C23 | — | 50 | 96 | no | — |
| 246 | C23 | — | 50 | 12 | no | — |
| 247 | C23 | — | 50 | 24 | no | — |
| 248 | C23 | C2×#6036 | 1 | 6 | yes | — |
| 249 | C23 | #87886 | 1 | 12 | yes | — |
| 250 | C23 | C22×D52 | 1 | 1 | yes | characteristic |
| 251 | C2×C4 | — | 10 | 24 | no | — |
| 252 | C2×C4 | — | 100 | 16 | no | — |
| 253 | C5 | — | 2 | 2 | no | — |
| 254 | C5 | D5×#379 | 1 | 2 | yes | complemented |
| 255 | C22 | — | 2 | 32 | no | — |
| 256 | C22 | — | 2 | 12 | no | — |
| 257 | C22 | — | 2 | 24 | no | — |
| 258 | C22 | — | 25 | 12 | no | — |
| 259 | C22 | — | 25 | 16 | no | — |
| 260 | C22 | — | 25 | 12 | no | — |
| 261 | C22 | — | 50 | 96 | no | — |
| 262 | C22 | — | 50 | 96 | no | — |
| 263 | C22 | — | 50 | 12 | no | — |
| 264 | C22 | — | 50 | 24 | no | — |
| 265 | C22 | #11717092 | 1 | 12 | yes | complemented |
| 266 | C22 | C22×#6036 | 1 | 1 | yes | characteristic |
| 267 | C22 | C2×#87886 | 1 | 6 | yes | — |
| 268 | C4 | — | 20 | 8 | no | — |
| 269 | C2 | — | 2 | 12 | no | — |
| 270 | C2 | — | 25 | 8 | no | — |
| 271 | C2 | — | 50 | 12 | no | — |
| 272 | C2 | C2×#11717092 | 1 | 3 | yes | — |
| 273 | C2 | #13106618 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D52 |
| commutator | C22×C52 | C23×C4 |
| Fitting | C25×C52 | C22 |
| socle, cosocle | C23×C52 | C24 |
| prime radical | C25×C5⋊D5 | C2 |
| hypercenter | C25 | D52 |
| hypocenter | C52 | C2×#379 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C23 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#11717092 | C2 | semidirect |
| 3. C5×#65536 | C2 | semidirect |
| 4. #13101890 | C2 | semidirect |
| 7. C24×C5⋊D5 | C4 | semidirect |
| 10. C22×#6036 | C22 | semidirect |
| 11. C2×C5×C22⋊Dic5 | C22 | semidirect |
| 13. C2×#87779 | C22 | semidirect |
| 16. C24×C52 | C2×C4 | semidirect |
| 24. C23×C5⋊D5 | C2×C4 | semidirect |
| 28. C22×C5×Dic5 | C23 | semidirect |
| 36. #65536 | D5 | semidirect |
| 51. C22×C5⋊D5 | C22⋊C4 | semidirect |
| 70. C2×C22⋊Dic5 | C2×D5 | semidirect |
| 77. C22×C52 | C2×C22⋊C4 | semidirect |
| 85. C2×C5⋊D5 | C2×C22⋊C4 | semidirect |
| 92. C24×C5 | C4×D5 | semidirect |
| 115. C22×Dic5 | C22×D5 | semidirect |
| 128. C5⋊D5 | #379 | semidirect |
| 177. C52 | C2×#379 | semidirect |
| 187. C22×C5 | D5×C22⋊C4 | semidirect |
| 217. C24 | #6036 | semidirect |
| 254. C5 | D5×#379 | semidirect |
| 265. C22 | #11717092 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C5⋊D5 | 2 |
| C2×#11717092 | 2 |
| C5×#65536 | 2 |
| #13101890 | 2 |
| D5×#379 | 5 |