Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
634 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #11717273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#4099 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11717097 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11717093 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11717089 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11717088 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11717087 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11717076 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11717074 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C24×C52 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C22⋊(C5⋊D5) | C22 | 1 | 2 | yes | — |
| 12 | C2×C22⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C5×C22⋊D5 | C22 | 1 | 2 | yes | — |
| 14 | C2×C5×C22⋊D5 | C22 | 1 | 2 | yes | — |
| 15 | C2×C5×C22⋊D5 | C22 | 1 | 2 | yes | — |
| 16 | C2×#6038 | C22 | 1 | 2 | yes | complemented |
| 17 | C2×#6037 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×#6036 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×D5×Dic5 | C22 | 1 | 2 | yes | complemented |
| 20 | C22⋊#1039 | C22 | 1 | 2 | yes | — |
| 21 | C22⋊#1039 | C22 | 1 | 1 | yes | characteristic |
| 22 | C5×C22⋊Dic5 | C22 | 1 | 2 | yes | — |
| 23 | C5×C22⋊Dic5 | C22 | 1 | 2 | yes | — |
| 24 | C5×C22⋊Dic5 | C22 | 1 | 2 | yes | — |
| 25 | #87781 | C22 | 1 | 2 | yes | complemented |
| 26 | #87780 | C22 | 1 | 2 | yes | complemented |
| 27 | #87780 | C22 | 1 | 2 | yes | complemented |
| 28 | #87779 | C22 | 1 | 1 | yes | characteristic, complemented |
| 29 | #87778 | C22 | 1 | 2 | yes | complemented |
| 30 | Dic52 | C22 | 1 | 1 | yes | characteristic, complemented |
| 31 | #65929 | — | 5 | 2 | no | maximal |
| 32 | C23×C52 | — | 4 | 2 | no | — |
| 33 | C23×C52 | — | 4 | 1 | no | — |
| 34 | C23×C52 | C23 | 1 | 2 | yes | — |
| 35 | C23×C52 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 37 | C22×C5×D5 | C23 | 1 | 2 | yes | — |
| 38 | C22⋊(C5⋊D5) | — | 4 | 2 | no | — |
| 39 | C22⋊(C5⋊D5) | — | 4 | 1 | no | — |
| 40 | C2×#1039 | C23 | 1 | 2 | yes | — |
| 41 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 42 | C5×C22⋊D5 | — | 4 | 2 | no | — |
| 43 | C5×C22⋊D5 | — | 4 | 2 | no | — |
| 44 | C5×C22⋊D5 | — | 4 | 2 | no | — |
| 45 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 46 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 47 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 48 | #6038 | — | 4 | 2 | no | — |
| 49 | #6037 | — | 4 | 1 | no | — |
| 50 | #6036 | — | 4 | 1 | no | — |
| 51 | D5×Dic5 | — | 4 | 2 | no | — |
| 52 | C5×#120 | — | 5 | 2 | no | — |
| 53 | #4099 | — | 5 | 2 | no | — |
| 54 | #4099 | — | 10 | 2 | no | — |
| 55 | #4099 | D5 | 1 | 2 | yes | complemented |
| 56 | #4086 | — | 5 | 2 | no | — |
| 57 | #4085 | — | 5 | 2 | no | — |
| 58 | #4085 | — | 5 | 2 | no | — |
| 59 | #4084 | — | 5 | 2 | no | — |
| 60 | #4084 | — | 5 | 2 | no | — |
| 61 | #4081 | — | 5 | 2 | no | — |
| 62 | #4030 | — | 5 | 2 | no | — |
| 63 | #4030 | — | 5 | 2 | no | — |
| 64 | #4029 | — | 5 | 2 | no | — |
| 65 | #4029 | — | 5 | 2 | no | — |
| 66 | #4025 | — | 5 | 2 | no | — |
| 67 | #4025 | — | 5 | 2 | no | — |
| 68 | C22×C52 | — | 2 | 2 | no | — |
| 69 | C22×C52 | — | 2 | 1 | no | — |
| 70 | C22×C52 | — | 2 | 2 | no | — |
| 71 | C22×C52 | — | 2 | 2 | no | — |
| 72 | C22×C52 | — | 2 | 2 | no | — |
| 73 | C22×C52 | — | 4 | 4 | no | — |
| 74 | C22×C52 | C24 | 1 | 1 | yes | characteristic |
| 75 | C2×C5⋊D5 | — | 2 | 1 | no | — |
| 76 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 77 | C2×C5×D5 | — | 2 | 2 | no | — |
| 78 | C2×C5×D5 | — | 2 | 2 | no | — |
| 79 | C2×C5×D5 | — | 2 | 2 | no | — |
| 80 | #1039 | — | 2 | 2 | no | — |
| 81 | #1039 | — | 2 | 1 | no | — |
| 82 | C5×Dic5 | — | 2 | 2 | no | — |
| 83 | C5×Dic5 | — | 2 | 2 | no | — |
| 84 | C5×Dic5 | — | 2 | 2 | no | — |
| 85 | C24×C5 | — | 2 | 2 | no | — |
| 86 | C24×C5 | C2×D5 | 1 | 2 | yes | — |
| 87 | C2×C5×D4 | — | 5 | 2 | no | — |
| 88 | C2×C5×D4 | — | 5 | 2 | no | — |
| 89 | C2×C5×D4 | — | 5 | 2 | no | — |
| 90 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 91 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 92 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 93 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 94 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 95 | C2×C22⋊D5 | — | 10 | 4 | no | — |
| 96 | C2×C22⋊D5 | — | 10 | 2 | no | — |
| 97 | C2×C22⋊D5 | C2×D5 | 1 | 2 | yes | — |
| 98 | C2×C22⋊D5 | C2×D5 | 1 | 2 | yes | — |
| 99 | C2×C22⋊D5 | C2×D5 | 1 | 2 | yes | — |
| 100 | C22×Dic5 | — | 5 | 2 | no | — |
| 101 | C2×D20 | — | 5 | 2 | no | — |
| 102 | C2×C4×D5 | — | 5 | 2 | no | — |
| 103 | C2×C4×D5 | — | 5 | 2 | no | — |
| 104 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 105 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 106 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 107 | C22⋊Dic5 | — | 5 | 2 | no | — |
| 108 | C22⋊Dic5 | — | 5 | 2 | no | — |
| 109 | C22⋊Dic5 | — | 5 | 2 | no | — |
| 110 | C22⋊Dic5 | — | 5 | 2 | no | — |
| 111 | C22⋊Dic5 | — | 10 | 4 | no | — |
| 112 | C22⋊Dic5 | — | 10 | 2 | no | — |
| 113 | C22⋊Dic5 | C2×D5 | 1 | 2 | yes | — |
| 114 | C22⋊Dic5 | C2×D5 | 1 | 2 | yes | — |
| 115 | C22⋊Dic5 | C2×D5 | 1 | 2 | yes | — |
| 116 | #685 | — | 5 | 2 | no | — |
| 117 | #685 | — | 5 | 2 | no | — |
| 118 | C4⋊Dic5 | — | 5 | 2 | no | — |
| 119 | C4⋊Dic5 | — | 5 | 2 | no | — |
| 120 | #683 | — | 5 | 2 | no | — |
| 121 | #683 | — | 5 | 2 | no | — |
| 122 | #683 | — | 5 | 2 | no | — |
| 123 | #683 | — | 5 | 2 | no | — |
| 124 | C4×Dic5 | — | 5 | 2 | no | — |
| 125 | #560 | — | 25 | 1 | no | Sylow |
| 126 | C2×C52 | — | 4 | 2 | no | — |
| 127 | C2×C52 | — | 4 | 1 | no | — |
| 128 | C2×C52 | H(5,𝔽2) | 1 | 2 | yes | — |
| 129 | C2×C52 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 130 | C5⋊D5 | — | 4 | 1 | no | — |
| 131 | C5×D5 | — | 4 | 2 | no | — |
| 132 | C23×C5 | — | 2 | 4 | no | — |
| 133 | C23×C5 | — | 2 | 2 | no | — |
| 134 | C23×C5 | — | 4 | 2 | no | — |
| 135 | C23×C5 | — | 4 | 2 | no | — |
| 136 | C23×C5 | — | 4 | 2 | no | — |
| 137 | C23×C5 | — | 4 | 4 | no | — |
| 138 | C23×C5 | — | 4 | 8 | no | — |
| 139 | C23×C5 | — | 5 | 2 | no | — |
| 140 | C23×C5 | C22×D5 | 1 | 2 | yes | — |
| 141 | C23×C5 | C22×D5 | 1 | 2 | yes | — |
| 142 | C23×C5 | C22×D5 | 1 | 2 | yes | — |
| 143 | C22×D5 | — | 5 | 2 | no | — |
| 144 | C22×D5 | — | 10 | 2 | no | — |
| 145 | C22×D5 | C22×D5 | 1 | 2 | yes | — |
| 146 | C5×D4 | — | 20 | 2 | no | — |
| 147 | C5×D4 | — | 20 | 2 | no | — |
| 148 | C5×D4 | — | 20 | 2 | no | — |
| 149 | C2×C4×C5 | — | 5 | 2 | no | — |
| 150 | C2×C4×C5 | — | 5 | 2 | no | — |
| 151 | C2×C4×C5 | — | 5 | 2 | no | — |
| 152 | C22⋊D5 | — | 4 | 2 | no | — |
| 153 | C22⋊D5 | — | 4 | 2 | no | — |
| 154 | C22⋊D5 | — | 4 | 2 | no | — |
| 155 | C22⋊D5 | — | 20 | 2 | no | — |
| 156 | C22⋊D5 | — | 20 | 2 | no | — |
| 157 | C22⋊D5 | — | 20 | 2 | no | — |
| 158 | C22⋊D5 | — | 20 | 2 | no | — |
| 159 | C22⋊D5 | — | 20 | 2 | no | — |
| 160 | C22⋊D5 | — | 20 | 8 | no | — |
| 161 | C22⋊D5 | — | 20 | 4 | no | — |
| 162 | C2×Dic5 | — | 5 | 2 | no | — |
| 163 | C2×Dic5 | — | 5 | 2 | no | — |
| 164 | C2×Dic5 | — | 5 | 2 | no | — |
| 165 | C2×Dic5 | — | 10 | 2 | no | — |
| 166 | C2×Dic5 | — | 10 | 4 | no | — |
| 167 | C2×Dic5 | — | 20 | 2 | no | — |
| 168 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 169 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 170 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 171 | D20 | — | 20 | 2 | no | — |
| 172 | C4×D5 | — | 20 | 2 | no | — |
| 173 | C4×D5 | — | 20 | 2 | no | — |
| 174 | #127 | — | 25 | 1 | no | — |
| 175 | #126 | — | 25 | 2 | no | — |
| 176 | #123 | — | 25 | 1 | no | — |
| 177 | #123 | — | 25 | 2 | no | — |
| 178 | #121 | — | 25 | 2 | no | — |
| 179 | #121 | — | 25 | 2 | no | — |
| 180 | #121 | — | 25 | 2 | no | — |
| 181 | #120 | — | 5 | 2 | no | — |
| 182 | #120 | — | 25 | 1 | no | — |
| 183 | C52 | #560 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C22×C5 | — | 2 | 4 | no | — |
| 185 | C22×C5 | — | 2 | 8 | no | — |
| 186 | C22×C5 | — | 2 | 2 | no | — |
| 187 | C22×C5 | — | 2 | 2 | no | — |
| 188 | C22×C5 | — | 2 | 2 | no | — |
| 189 | C22×C5 | — | 2 | 2 | no | — |
| 190 | C22×C5 | — | 2 | 2 | no | — |
| 191 | C22×C5 | — | 2 | 2 | no | — |
| 192 | C22×C5 | — | 2 | 2 | no | — |
| 193 | C22×C5 | — | 2 | 2 | no | — |
| 194 | C22×C5 | — | 2 | 2 | no | — |
| 195 | C22×C5 | — | 2 | 2 | no | — |
| 196 | C22×C5 | — | 4 | 4 | no | — |
| 197 | C22×C5 | — | 4 | 4 | no | — |
| 198 | C22×C5 | — | 4 | 4 | no | — |
| 199 | C22×C5 | — | 4 | 16 | no | — |
| 200 | C22×C5 | — | 4 | 8 | no | — |
| 201 | C22×C5 | — | 10 | 2 | no | — |
| 202 | C22×C5 | — | 10 | 2 | no | — |
| 203 | C22×C5 | — | 10 | 2 | no | — |
| 204 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 205 | C2×D5 | — | 2 | 2 | no | — |
| 206 | C2×D5 | — | 2 | 2 | no | — |
| 207 | C2×D5 | — | 2 | 2 | no | — |
| 208 | C2×D5 | — | 10 | 2 | no | — |
| 209 | C2×D5 | — | 10 | 2 | no | — |
| 210 | C2×D5 | — | 10 | 2 | no | — |
| 211 | C2×D5 | — | 20 | 2 | no | — |
| 212 | C2×D5 | — | 20 | 4 | no | — |
| 213 | C4×C5 | — | 10 | 2 | no | — |
| 214 | C4×C5 | — | 10 | 2 | no | — |
| 215 | C4×C5 | — | 10 | 2 | no | — |
| 216 | Dic5 | — | 2 | 2 | no | — |
| 217 | Dic5 | — | 2 | 2 | no | — |
| 218 | Dic5 | — | 2 | 2 | no | — |
| 219 | Dic5 | — | 10 | 2 | no | — |
| 220 | Dic5 | — | 10 | 2 | no | — |
| 221 | Dic5 | — | 10 | 2 | no | — |
| 222 | Dic5 | — | 20 | 4 | no | — |
| 223 | Dic5 | — | 20 | 2 | no | — |
| 224 | C24 | D52 | 1 | 1 | yes | characteristic |
| 225 | C2×D4 | — | 5 | 2 | no | — |
| 226 | C2×D4 | — | 5 | 2 | no | — |
| 227 | C2×D4 | — | 5 | 2 | no | — |
| 228 | C2×D4 | — | 25 | 2 | no | — |
| 229 | C2×D4 | — | 25 | 1 | no | — |
| 230 | C2×D4 | — | 25 | 2 | no | — |
| 231 | C2×D4 | — | 25 | 1 | no | — |
| 232 | C22×C4 | — | 25 | 1 | no | — |
| 233 | C22×C4 | — | 25 | 2 | no | — |
| 234 | C4⋊C4 | — | 25 | 2 | no | — |
| 235 | C4⋊C4 | — | 25 | 2 | no | — |
| 236 | C4⋊C4 | — | 25 | 2 | no | — |
| 237 | C22⋊C4 | — | 5 | 2 | no | — |
| 238 | C22⋊C4 | — | 5 | 2 | no | — |
| 239 | C22⋊C4 | — | 5 | 2 | no | — |
| 240 | C22⋊C4 | — | 25 | 1 | no | — |
| 241 | C22⋊C4 | — | 25 | 2 | no | — |
| 242 | C22⋊C4 | — | 25 | 2 | no | — |
| 243 | C22⋊C4 | — | 25 | 1 | no | — |
| 244 | C42 | — | 25 | 1 | no | — |
| 245 | C2×C5 | — | 2 | 2 | no | — |
| 246 | C2×C5 | — | 2 | 4 | no | — |
| 247 | C2×C5 | — | 4 | 8 | no | — |
| 248 | C2×C5 | — | 4 | 4 | no | — |
| 249 | C2×C5 | — | 4 | 2 | no | — |
| 250 | C2×C5 | — | 4 | 2 | no | — |
| 251 | C2×C5 | — | 4 | 2 | no | — |
| 252 | C2×C5 | — | 20 | 2 | no | — |
| 253 | C2×C5 | #4144 | 1 | 2 | yes | — |
| 254 | C2×C5 | #4144 | 1 | 2 | yes | — |
| 255 | C2×C5 | #4144 | 1 | 2 | yes | — |
| 256 | D5 | — | 4 | 2 | no | — |
| 257 | D5 | — | 20 | 2 | no | — |
| 258 | D5 | — | 40 | 2 | no | — |
| 259 | C23 | — | 4 | 1 | no | — |
| 260 | C23 | — | 4 | 2 | no | — |
| 261 | C23 | — | 5 | 2 | no | — |
| 262 | C23 | — | 25 | 1 | no | — |
| 263 | C23 | C2×D52 | 1 | 2 | yes | — |
| 264 | C23 | C2×D52 | 1 | 1 | yes | characteristic |
| 265 | D4 | — | 20 | 2 | no | — |
| 266 | D4 | — | 20 | 2 | no | — |
| 267 | D4 | — | 20 | 2 | no | — |
| 268 | D4 | — | 100 | 2 | no | — |
| 269 | D4 | — | 100 | 2 | no | — |
| 270 | D4 | — | 100 | 1 | no | — |
| 271 | D4 | — | 100 | 1 | no | — |
| 272 | C2×C4 | — | 5 | 2 | no | — |
| 273 | C2×C4 | — | 5 | 2 | no | — |
| 274 | C2×C4 | — | 5 | 2 | no | — |
| 275 | C2×C4 | — | 25 | 1 | no | — |
| 276 | C2×C4 | — | 25 | 2 | no | — |
| 277 | C2×C4 | — | 100 | 1 | no | — |
| 278 | C2×C4 | — | 100 | 2 | no | — |
| 279 | C5 | — | 2 | 2 | no | — |
| 280 | C5 | #65929 | 1 | 2 | yes | complemented |
| 281 | C22 | — | 2 | 2 | no | — |
| 282 | C22 | — | 2 | 2 | no | — |
| 283 | C22 | — | 2 | 2 | no | — |
| 284 | C22 | — | 2 | 1 | no | — |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 4 | 4 | no | — |
| 287 | C22 | — | 10 | 2 | no | — |
| 288 | C22 | — | 10 | 2 | no | — |
| 289 | C22 | — | 10 | 2 | no | — |
| 290 | C22 | — | 50 | 1 | no | — |
| 291 | C22 | — | 50 | 2 | no | — |
| 292 | C22 | C22×D52 | 1 | 1 | yes | characteristic |
| 293 | C4 | — | 10 | 2 | no | — |
| 294 | C4 | — | 10 | 2 | no | — |
| 295 | C4 | — | 10 | 2 | no | — |
| 296 | C4 | — | 50 | 2 | no | — |
| 297 | C4 | — | 50 | 1 | no | — |
| 298 | C2 | — | 4 | 2 | no | — |
| 299 | C2 | — | 4 | 1 | no | — |
| 300 | C2 | — | 20 | 2 | no | — |
| 301 | C2 | — | 100 | 1 | no | — |
| 302 | C2 | #11717495 | 1 | 1 | yes | characteristic |
| 303 | C2 | #11717495 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C24×C52 | C22 |
| hypercenter | C24 | D52 |
| hypocenter | C52 | #560 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717273 | C2 | semidirect |
| 2. C5×#4099 | C2 | semidirect |
| 3. #11717097 | C2 | semidirect |
| 4. #11717093 | C2 | semidirect |
| 5. #11717089 | C2 | semidirect |
| 6. #11717088 | C2 | semidirect |
| 7. #11717087 | C2 | semidirect |
| 8. #11717076 | C2 | semidirect |
| 9. #11717074 | C2 | semidirect |
| 16. C2×#6038 | C22 | semidirect |
| 17. C2×#6037 | C22 | semidirect |
| 18. C2×#6036 | C22 | semidirect |
| 19. C2×D5×Dic5 | C22 | semidirect |
| 25. #87781 | C22 | semidirect |
| 26. #87780 | C22 | semidirect |
| 27. #87780 | C22 | semidirect |
| 28. #87779 | C22 | semidirect |
| 29. #87778 | C22 | semidirect |
| 30. Dic52 | C22 | semidirect |
| 55. #4099 | D5 | semidirect |
| 183. C52 | #560 | semidirect |
| 280. C5 | #65929 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717273 | 2 |
| C5×#4099 | 2 |
| #11717097 | 2 |
| #11717093 | 2 |
| #11717089 | 2 |
| #11717088 | 2 |
| #11717087 | 2 |
| #11717076 | 2 |
| #11717074 | 2 |
| #65929 | 5 |