Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
452 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 87 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 | #11717471 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#87872 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717448 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11717397 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #11717386 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11717385 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11717384 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 9 | C5×#709 | — | 2 | 1 | no | — |
| 10 | C2×C5×Dic10 | — | 2 | 1 | no | — |
| 11 | #87886 | — | 2 | 1 | no | — |
| 12 | C2×#6039 | C22 | 1 | 2 | yes | complemented |
| 13 | C2×#6036 | — | 2 | 1 | no | — |
| 14 | #87880 | C22 | 1 | 2 | yes | — |
| 15 | #87877 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×D52 | — | 2 | 1 | no | — |
| 17 | #87873 | C22 | 1 | 2 | yes | — |
| 18 | #87872 | — | 2 | 1 | no | — |
| 19 | #87872 | — | 2 | 1 | no | — |
| 20 | #87872 | C22 | 1 | 2 | yes | — |
| 21 | #87872 | C22 | 1 | 1 | yes | characteristic |
| 22 | #87872 | C22 | 1 | 2 | yes | complemented |
| 23 | #87871 | — | 2 | 1 | no | — |
| 24 | #87867 | C22 | 1 | 2 | yes | complemented |
| 25 | #87863 | C22 | 1 | 2 | yes | complemented |
| 26 | #87862 | C22 | 1 | 2 | yes | — |
| 27 | #87839 | C22 | 1 | 8 | yes | complemented |
| 28 | #87838 | C22 | 1 | 8 | yes | complemented |
| 29 | C22×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×D52 | — | 2 | 1 | no | — |
| 31 | C2×C4×C52 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 33 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 34 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 35 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 36 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 37 | C5×C22⋊D5 | — | 2 | 2 | no | — |
| 38 | C2×C5×Dic5 | — | 2 | 2 | no | — |
| 39 | C5×D20 | — | 2 | 1 | no | — |
| 40 | C4×C5×D5 | — | 2 | 2 | no | — |
| 41 | C5×Dic10 | — | 2 | 1 | no | — |
| 42 | C5×Dic10 | — | 2 | 2 | no | — |
| 43 | C5×Dic10 | — | 2 | 2 | no | — |
| 44 | #6039 | — | 2 | 2 | no | — |
| 45 | #6039 | C23 | 1 | 4 | yes | — |
| 46 | #6039 | C23 | 1 | 2 | yes | — |
| 47 | #6038 | — | 2 | 2 | no | — |
| 48 | #6037 | C23 | 1 | 2 | yes | — |
| 49 | #6036 | — | 2 | 2 | no | — |
| 50 | #6036 | — | 2 | 2 | no | — |
| 51 | #6036 | — | 2 | 1 | no | — |
| 52 | D5×Dic5 | — | 2 | 2 | no | — |
| 53 | #6034 | C23 | 1 | 4 | yes | — |
| 54 | D5×#41 | — | 10 | 1 | no | — |
| 55 | C2×Q8×D5 | — | 10 | 1 | no | — |
| 56 | D5⋊M16 | — | 10 | 1 | no | — |
| 57 | C22×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 59 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C5×D5 | — | 2 | 2 | no | — |
| 62 | D52 | — | 4 | 2 | no | — |
| 63 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | #1039 | C24 | 1 | 1 | yes | characteristic |
| 67 | C5×Dic5 | — | 2 | 2 | no | — |
| 68 | C5×Dic5 | — | 2 | 4 | no | — |
| 69 | C5×#41 | — | 10 | 1 | no | — |
| 70 | C2×C5×Q8 | — | 10 | 1 | no | — |
| 71 | #713 | — | 10 | 1 | no | — |
| 72 | Q8×D5 | — | 10 | 2 | no | — |
| 73 | Q8×D5 | — | 10 | 2 | no | — |
| 74 | Q8×D5 | — | 10 | 1 | no | — |
| 75 | Q8×D5 | — | 10 | 2 | no | — |
| 76 | Q8×D5 | — | 10 | 2 | no | — |
| 77 | #711 | — | 10 | 2 | no | — |
| 78 | #711 | — | 10 | 1 | no | — |
| 79 | D4×D5 | — | 10 | 1 | no | — |
| 80 | D4×D5 | — | 10 | 2 | no | — |
| 81 | #709 | — | 2 | 1 | no | — |
| 82 | #709 | — | 10 | 2 | no | — |
| 83 | C2×C4×D5 | — | 10 | 1 | no | — |
| 84 | C2×C4×D5 | — | 10 | 1 | no | — |
| 85 | C2×C4×D5 | — | 10 | 1 | no | — |
| 86 | C2×C4×D5 | — | 10 | 2 | no | — |
| 87 | C2×C4×D5 | — | 10 | 2 | no | — |
| 88 | C2×Dic10 | — | 2 | 1 | no | — |
| 89 | C2×Dic10 | — | 10 | 2 | no | — |
| 90 | #704 | — | 10 | 2 | no | — |
| 91 | #700 | — | 10 | 2 | no | — |
| 92 | D5⋊C8 | — | 10 | 2 | no | — |
| 93 | C2×#137 | — | 25 | 1 | no | maximal, Sylow |
| 94 | C2×C52 | — | 2 | 1 | no | — |
| 95 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 96 | C5⋊D5 | — | 2 | 1 | no | — |
| 97 | C5⋊D5 | #137 | 1 | 2 | yes | complemented |
| 98 | C5×D5 | — | 4 | 2 | no | — |
| 99 | C22×D5 | — | 10 | 1 | no | — |
| 100 | C22×D5 | — | 10 | 1 | no | — |
| 101 | C22×D5 | — | 10 | 1 | no | — |
| 102 | C22×D5 | — | 10 | 2 | no | — |
| 103 | C5×Q8 | — | 10 | 2 | no | — |
| 104 | C5×Q8 | — | 10 | 2 | no | — |
| 105 | C5×Q8 | — | 10 | 1 | no | — |
| 106 | C5×D4 | — | 10 | 1 | no | — |
| 107 | C5×D4 | — | 10 | 2 | no | — |
| 108 | C2×C4×C5 | — | 2 | 1 | no | — |
| 109 | C2×C4×C5 | — | 2 | 1 | no | — |
| 110 | C2×C4×C5 | — | 2 | 1 | no | — |
| 111 | C2×C4×C5 | — | 10 | 2 | no | — |
| 112 | C2×C4×C5 | — | 10 | 2 | no | — |
| 113 | C22⋊D5 | — | 2 | 2 | no | — |
| 114 | C22⋊D5 | — | 10 | 2 | no | — |
| 115 | C22⋊D5 | — | 10 | 2 | no | — |
| 116 | C2×Dic5 | — | 2 | 2 | no | — |
| 117 | C2×Dic5 | — | 10 | 1 | no | — |
| 118 | C2×Dic5 | — | 10 | 1 | no | — |
| 119 | C2×Dic5 | — | 10 | 1 | no | — |
| 120 | C2×Dic5 | — | 10 | 2 | no | — |
| 121 | D20 | — | 2 | 1 | no | — |
| 122 | D20 | — | 10 | 2 | no | — |
| 123 | C4×D5 | — | 2 | 2 | no | — |
| 124 | C4×D5 | — | 10 | 1 | no | — |
| 125 | C4×D5 | — | 10 | 1 | no | — |
| 126 | C4×D5 | — | 10 | 1 | no | — |
| 127 | C4×D5 | — | 10 | 1 | no | — |
| 128 | C4×D5 | — | 10 | 1 | no | — |
| 129 | C4×D5 | — | 10 | 1 | no | — |
| 130 | C4×D5 | — | 10 | 1 | no | — |
| 131 | C4×D5 | — | 10 | 1 | no | — |
| 132 | C4×D5 | — | 10 | 1 | no | — |
| 133 | C4×D5 | — | 10 | 1 | no | — |
| 134 | C4×D5 | — | 10 | 1 | no | — |
| 135 | C4×D5 | — | 10 | 1 | no | — |
| 136 | C4×D5 | — | 10 | 2 | no | — |
| 137 | C4×D5 | — | 10 | 4 | no | — |
| 138 | C4×D5 | — | 10 | 4 | no | — |
| 139 | C4×D5 | — | 10 | 2 | no | — |
| 140 | Dic10 | — | 2 | 2 | no | — |
| 141 | Dic10 | — | 2 | 2 | no | — |
| 142 | Dic10 | — | 2 | 1 | no | — |
| 143 | Dic10 | — | 10 | 4 | no | — |
| 144 | Dic10 | — | 10 | 2 | no | — |
| 145 | Dic10 | — | 10 | 4 | no | — |
| 146 | #170 | — | 10 | 4 | no | — |
| 147 | C2×#41 | — | 25 | 1 | no | — |
| 148 | C22×Q8 | — | 25 | 1 | no | — |
| 149 | #137 | — | 25 | 4 | no | — |
| 150 | #137 | — | 25 | 4 | no | — |
| 151 | C2×Q16 | — | 25 | 2 | no | — |
| 152 | C2×SD16 | — | 25 | 2 | no | — |
| 153 | C2×M16 | — | 25 | 1 | no | — |
| 154 | C52 | C2×#137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | C22×C5 | — | 2 | 1 | no | — |
| 156 | C22×C5 | — | 2 | 1 | no | — |
| 157 | C22×C5 | — | 2 | 1 | no | — |
| 158 | C22×C5 | — | 10 | 2 | no | — |
| 159 | C2×D5 | — | 2 | 2 | no | — |
| 160 | C2×D5 | — | 10 | 1 | no | — |
| 161 | C2×D5 | — | 10 | 1 | no | — |
| 162 | C2×D5 | — | 10 | 1 | no | — |
| 163 | C2×D5 | — | 10 | 1 | no | — |
| 164 | C2×D5 | — | 10 | 1 | no | — |
| 165 | C2×D5 | — | 10 | 1 | no | — |
| 166 | C2×D5 | — | 10 | 4 | no | — |
| 167 | C2×D5 | — | 20 | 2 | no | — |
| 168 | C2×D5 | — | 20 | 2 | no | — |
| 169 | C2×D5 | — | 20 | 4 | no | — |
| 170 | C4×C5 | — | 2 | 1 | no | — |
| 171 | C4×C5 | — | 2 | 1 | no | — |
| 172 | C4×C5 | — | 2 | 1 | no | — |
| 173 | C4×C5 | — | 2 | 1 | no | — |
| 174 | C4×C5 | — | 2 | 1 | no | — |
| 175 | C4×C5 | — | 2 | 1 | no | — |
| 176 | C4×C5 | — | 10 | 4 | no | — |
| 177 | C4×C5 | — | 10 | 2 | no | — |
| 178 | Dic5 | — | 2 | 4 | no | — |
| 179 | Dic5 | — | 2 | 2 | no | — |
| 180 | Dic5 | — | 10 | 1 | no | — |
| 181 | Dic5 | — | 10 | 1 | no | — |
| 182 | Dic5 | — | 10 | 1 | no | — |
| 183 | Dic5 | — | 10 | 1 | no | — |
| 184 | Dic5 | — | 10 | 1 | no | — |
| 185 | Dic5 | — | 10 | 1 | no | — |
| 186 | #41 | — | 10 | 1 | no | — |
| 187 | #41 | — | 25 | 2 | no | — |
| 188 | #41 | — | 25 | 2 | no | — |
| 189 | #41 | — | 50 | 1 | no | — |
| 190 | C2×Q8 | — | 10 | 1 | no | — |
| 191 | C2×Q8 | — | 25 | 2 | no | — |
| 192 | C2×Q8 | — | 25 | 2 | no | — |
| 193 | C2×Q8 | — | 25 | 1 | no | — |
| 194 | C2×Q8 | — | 25 | 2 | no | — |
| 195 | C2×Q8 | — | 50 | 1 | no | — |
| 196 | C2×Q8 | — | 50 | 1 | no | — |
| 197 | C2×D4 | — | 25 | 1 | no | — |
| 198 | C2×D4 | — | 50 | 1 | no | — |
| 199 | C22×C4 | — | 25 | 1 | no | — |
| 200 | C22×C4 | — | 50 | 1 | no | — |
| 201 | C22×C4 | — | 50 | 1 | no | — |
| 202 | Q16 | — | 25 | 8 | no | — |
| 203 | SD16 | — | 25 | 8 | no | — |
| 204 | M16 | — | 25 | 2 | no | — |
| 205 | M16 | — | 25 | 2 | no | — |
| 206 | C2×C8 | — | 25 | 2 | no | — |
| 207 | C2×C5 | — | 2 | 2 | no | — |
| 208 | C2×C5 | — | 2 | 1 | no | — |
| 209 | C2×C5 | — | 2 | 1 | no | — |
| 210 | C2×C5 | — | 2 | 1 | no | — |
| 211 | C2×C5 | — | 4 | 1 | no | — |
| 212 | C2×C5 | — | 4 | 1 | no | — |
| 213 | C2×C5 | — | 20 | 2 | no | — |
| 214 | D5 | — | 4 | 2 | no | — |
| 215 | D5 | — | 10 | 2 | no | — |
| 216 | D5 | — | 10 | 2 | no | — |
| 217 | D5 | — | 10 | 2 | no | — |
| 218 | D5 | — | 10 | 2 | no | — |
| 219 | D5 | — | 20 | 1 | no | — |
| 220 | D5 | — | 20 | 1 | no | — |
| 221 | C23 | — | 25 | 1 | no | — |
| 222 | C23 | — | 50 | 1 | no | — |
| 223 | Q8 | — | 10 | 2 | no | — |
| 224 | Q8 | — | 10 | 2 | no | — |
| 225 | Q8 | — | 10 | 1 | no | — |
| 226 | Q8 | — | 25 | 4 | no | — |
| 227 | Q8 | — | 25 | 2 | no | — |
| 228 | Q8 | — | 50 | 2 | no | — |
| 229 | D4 | — | 10 | 2 | no | — |
| 230 | D4 | — | 10 | 1 | no | — |
| 231 | D4 | — | 25 | 2 | no | — |
| 232 | D4 | — | 50 | 2 | no | — |
| 233 | C2×C4 | — | 10 | 2 | no | — |
| 234 | C2×C4 | — | 10 | 2 | no | — |
| 235 | C2×C4 | — | 25 | 1 | no | — |
| 236 | C2×C4 | — | 25 | 1 | no | — |
| 237 | C2×C4 | — | 25 | 1 | no | — |
| 238 | C2×C4 | — | 25 | 1 | no | — |
| 239 | C2×C4 | — | 25 | 1 | no | — |
| 240 | C2×C4 | — | 50 | 1 | no | — |
| 241 | C2×C4 | — | 50 | 2 | no | — |
| 242 | C2×C4 | — | 50 | 2 | no | — |
| 243 | C2×C4 | — | 50 | 2 | no | — |
| 244 | C2×C4 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 245 | C8 | — | 25 | 4 | no | — |
| 246 | C5 | — | 2 | 1 | no | — |
| 247 | C5 | — | 2 | 1 | no | — |
| 248 | C5 | — | 2 | 1 | no | — |
| 249 | C22 | — | 10 | 2 | no | — |
| 250 | C22 | — | 25 | 1 | no | — |
| 251 | C22 | — | 25 | 1 | no | — |
| 252 | C22 | — | 50 | 2 | no | — |
| 253 | C22 | — | 100 | 2 | no | — |
| 254 | C22 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 255 | C4 | — | 10 | 2 | no | — |
| 256 | C4 | — | 10 | 4 | no | — |
| 257 | C4 | — | 25 | 1 | no | — |
| 258 | C4 | — | 25 | 1 | no | — |
| 259 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 260 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 261 | C2 | — | 2 | 1 | no | — |
| 262 | C2 | — | 20 | 2 | no | — |
| 263 | C2 | — | 25 | 2 | no | — |
| 264 | C2 | — | 50 | 1 | no | — |
| 265 | C2 | C22×D52⋊C2 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#87872 | C2 | semidirect |
| 3. #11717448 | C2 | semidirect |
| 4. #11717397 | C2 | semidirect |
| 5. #11717386 | C2 | semidirect |
| 6. #11717385 | C2 | semidirect |
| 7. #11717384 | C2 | semidirect |
| 12. C2×#6039 | C22 | semidirect |
| 22. #87872 | C22 | semidirect |
| 24. #87867 | C22 | semidirect |
| 25. #87863 | C22 | semidirect |
| 27. #87839 | C22 | semidirect |
| 28. #87838 | C22 | semidirect |
| 97. C5⋊D5 | #137 | semidirect |
| 154. C52 | C2×#137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717471 | 2 |
| C2×#87872 | 2 |
| #11717448 | 2 |
| #11717397 | 2 |
| #11717386 | 2 |
| #11717385 | 2 |
| #11717384 | 2 |
| C2×#137 | 25 |