Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
312 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 34 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 | D5×#984 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D5⋊(#131⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11756726 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10604870 | — | 3 | 1 | no | maximal, Hall |
| 5 | D5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | #91954 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C5×#984 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | D5×#226 | — | 2 | 1 | no | — |
| 9 | #91891 | — | 2 | 1 | no | — |
| 10 | #4133⋊C3 | — | 2 | 1 | no | — |
| 11 | #91886 | — | 2 | 1 | no | — |
| 12 | #91154 | — | 4 | 1 | no | maximal |
| 13 | #73124 | — | 5 | 1 | no | maximal, Hall |
| 14 | D5⋊#131 | D3 | 1 | 1 | yes | characteristic |
| 15 | D5×#137 | — | 3 | 1 | no | — |
| 16 | #65793 | — | 3 | 1 | no | — |
| 17 | D5⋊#101 | — | 3 | 1 | no | — |
| 18 | C2×#4009 | — | 3 | 1 | no | — |
| 19 | #64901 | — | 3 | 1 | no | — |
| 20 | #64899 | — | 3 | 1 | no | — |
| 21 | C5×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | D5×Dic6 | — | 4 | 1 | no | — |
| 23 | C4⋊ΓL(1,𝔽16) | — | 4 | 1 | no | — |
| 24 | C3×D5⋊C8 | — | 4 | 1 | no | — |
| 25 | D5×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 26 | C5⋊(#41⋊C3) | D4 | 1 | 1 | yes | characteristic |
| 27 | C5⋊#226 | — | 2 | 1 | no | — |
| 28 | C5×#226 | — | 2 | 1 | no | — |
| 29 | C2×#984 | — | 5 | 1 | no | — |
| 30 | C2×#131⋊C3 | — | 5 | 1 | no | — |
| 31 | #5426 | — | 5 | 1 | no | — |
| 32 | D5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C2×Q8×D5 | — | 3 | 1 | no | — |
| 34 | #4133 | — | 2 | 1 | no | — |
| 35 | #4131 | — | 6 | 1 | no | — |
| 36 | C2×C4⋊AGL(1,𝔽5) | — | 3 | 1 | no | — |
| 37 | D5⋊M16 | — | 3 | 1 | no | — |
| 38 | D5⋊M16 | — | 3 | 1 | no | — |
| 39 | C2×D5⋊C8 | — | 3 | 1 | no | — |
| 40 | C5×#137 | — | 3 | 1 | no | — |
| 41 | #4097 | — | 3 | 1 | no | — |
| 42 | #4088 | — | 3 | 1 | no | — |
| 43 | #4064 | — | 6 | 1 | no | — |
| 44 | D5×Q16 | — | 6 | 1 | no | — |
| 45 | #4061 | — | 6 | 1 | no | — |
| 46 | D5×SD16 | — | 6 | 1 | no | — |
| 47 | #4055 | — | 3 | 1 | no | — |
| 48 | D5×M16 | — | 3 | 1 | no | — |
| 49 | #4009 | — | 6 | 1 | no | — |
| 50 | #4009 | — | 6 | 1 | no | — |
| 51 | #4009 | — | 6 | 1 | no | — |
| 52 | #4007 | — | 6 | 1 | no | — |
| 53 | #4005 | — | 6 | 2 | no | — |
| 54 | #1908 | — | 15 | 1 | no | Sylow |
| 55 | C2×ΓL(1,𝔽16) | — | 8 | 1 | no | — |
| 56 | Dic30 | — | 4 | 1 | no | — |
| 57 | C5×Dic6 | — | 4 | 1 | no | — |
| 58 | C3×C4×D5 | — | 4 | 1 | no | — |
| 59 | C5×SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | #1204 | — | 8 | 1 | no | — |
| 61 | D5×Dic3 | — | 8 | 1 | no | — |
| 62 | C3×#170 | — | 8 | 1 | no | — |
| 63 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 64 | #984 | — | 5 | 1 | no | — |
| 65 | #984 | — | 10 | 1 | no | — |
| 66 | #984 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 67 | C2×#226 | — | 10 | 1 | no | — |
| 68 | #131⋊C3 | — | 10 | 1 | no | — |
| 69 | #859 | — | 10 | 1 | no | — |
| 70 | #816 | — | 20 | 1 | no | — |
| 71 | C22×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 72 | C5×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 73 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 74 | #713 | #67 | 1 | 1 | yes | characteristic |
| 75 | Q8×D5 | — | 6 | 1 | no | — |
| 76 | Q8×D5 | — | 6 | 1 | no | — |
| 77 | Q8×D5 | — | 6 | 1 | no | — |
| 78 | Q8×D5 | — | 6 | 1 | no | — |
| 79 | Q8×D5 | D12 | 1 | 1 | yes | characteristic |
| 80 | #711 | — | 3 | 1 | no | — |
| 81 | D4×D5 | — | 3 | 1 | no | — |
| 82 | #709 | — | 3 | 1 | no | — |
| 83 | C2×C4×D5 | — | 3 | 1 | no | — |
| 84 | C2×C4×D5 | — | 6 | 1 | no | — |
| 85 | C2×Dic10 | — | 3 | 1 | no | — |
| 86 | C2×Dic10 | — | 6 | 1 | no | — |
| 87 | #704 | — | 3 | 2 | no | — |
| 88 | #704 | — | 6 | 1 | no | — |
| 89 | C2×#170 | — | 6 | 1 | no | — |
| 90 | C4⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 91 | C4⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 92 | #700 | — | 6 | 1 | no | — |
| 93 | #700 | — | 6 | 1 | no | — |
| 94 | D5⋊C8 | — | 3 | 1 | no | — |
| 95 | D5⋊C8 | — | 6 | 1 | no | — |
| 96 | D5⋊C8 | — | 6 | 1 | no | — |
| 97 | D5⋊C8 | S4 | 1 | 1 | yes | characteristic |
| 98 | C5×Q16 | — | 6 | 1 | no | — |
| 99 | C5×SD16 | — | 6 | 1 | no | — |
| 100 | C5×M16 | — | 3 | 1 | no | — |
| 101 | #689 | — | 6 | 1 | no | — |
| 102 | #689 | — | 6 | 1 | no | — |
| 103 | #688 | — | 6 | 1 | no | — |
| 104 | #687 | — | 6 | 1 | no | — |
| 105 | #681 | — | 3 | 1 | no | — |
| 106 | Dic20 | — | 6 | 1 | no | — |
| 107 | #677 | — | 6 | 1 | no | — |
| 108 | #676 | — | 6 | 1 | no | — |
| 109 | C8×D5 | — | 6 | 1 | no | — |
| 110 | C2×#137 | — | 15 | 1 | no | — |
| 111 | C2×#131 | — | 5 | 1 | no | — |
| 112 | #417 | — | 15 | 1 | no | — |
| 113 | C2×#103 | — | 15 | 1 | no | — |
| 114 | C2×#101 | — | 15 | 1 | no | — |
| 115 | #343 | — | 15 | 1 | no | — |
| 116 | #341 | — | 15 | 1 | no | — |
| 117 | C2×C3×D5 | — | 4 | 1 | no | — |
| 118 | ΓL(1,𝔽16) | — | 16 | 1 | no | — |
| 119 | C3×C4×C5 | — | 4 | 1 | no | — |
| 120 | Dic15 | — | 8 | 1 | no | — |
| 121 | C3×Dic5 | — | 4 | 1 | no | — |
| 122 | C5×Dic3 | — | 8 | 1 | no | — |
| 123 | C2×Dic6 | — | 20 | 1 | no | — |
| 124 | #41⋊C3 | — | 5 | 1 | no | — |
| 125 | #41⋊C3 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 126 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 127 | #226 | — | 2 | 1 | no | — |
| 128 | #226 | — | 10 | 1 | no | — |
| 129 | C2×C3×C8 | — | 20 | 1 | no | — |
| 130 | C4⋊Dic3 | — | 20 | 1 | no | — |
| 131 | C22×D5 | — | 3 | 1 | no | — |
| 132 | C2×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 133 | C2×AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 134 | C5×Q8 | — | 6 | 1 | no | — |
| 135 | C5×Q8 | — | 6 | 1 | no | — |
| 136 | C5×Q8 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 137 | C5×D4 | — | 3 | 1 | no | — |
| 138 | C2×C4×C5 | — | 3 | 1 | no | — |
| 139 | C2×C4×C5 | — | 6 | 1 | no | — |
| 140 | C22⋊D5 | — | 6 | 1 | no | — |
| 141 | C2×Dic5 | — | 3 | 1 | no | — |
| 142 | C2×Dic5 | — | 6 | 1 | no | — |
| 143 | D20 | — | 3 | 1 | no | — |
| 144 | C4×D5 | — | 3 | 1 | no | — |
| 145 | C4×D5 | — | 3 | 1 | no | — |
| 146 | C4×D5 | — | 3 | 1 | no | — |
| 147 | C4×D5 | — | 12 | 1 | no | — |
| 148 | C4×D5 | — | 12 | 1 | no | — |
| 149 | C4×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 150 | Dic10 | — | 3 | 1 | no | — |
| 151 | Dic10 | — | 6 | 1 | no | — |
| 152 | Dic10 | — | 6 | 1 | no | — |
| 153 | Dic10 | — | 12 | 1 | no | — |
| 154 | Dic10 | — | 12 | 1 | no | — |
| 155 | #170 | — | 2 | 1 | no | — |
| 156 | #170 | — | 6 | 1 | no | — |
| 157 | #170 | — | 12 | 1 | no | — |
| 158 | C5×C8 | — | 6 | 1 | no | — |
| 159 | #168 | — | 6 | 1 | no | — |
| 160 | C2×#41 | — | 5 | 1 | no | — |
| 161 | C22×Q8 | — | 15 | 1 | no | — |
| 162 | #137 | — | 3 | 1 | no | — |
| 163 | #137 | — | 15 | 1 | no | — |
| 164 | #137 | — | 15 | 1 | no | — |
| 165 | #137 | — | 15 | 1 | no | — |
| 166 | #137 | — | 30 | 1 | no | — |
| 167 | #137 | — | 30 | 1 | no | — |
| 168 | C2×Q16 | — | 30 | 1 | no | — |
| 169 | C2×SD16 | — | 30 | 1 | no | — |
| 170 | #131 | — | 10 | 1 | no | — |
| 171 | #131 | — | 30 | 1 | no | — |
| 172 | C2×M16 | — | 15 | 1 | no | — |
| 173 | C2×M16 | — | 15 | 1 | no | — |
| 174 | C2×M16 | — | 15 | 1 | no | — |
| 175 | C22×C8 | — | 15 | 1 | no | — |
| 176 | C2×C4⋊C4 | — | 15 | 1 | no | — |
| 177 | #103 | — | 30 | 1 | no | — |
| 178 | #103 | — | 30 | 1 | no | — |
| 179 | #103 | — | 30 | 1 | no | — |
| 180 | #102 | — | 30 | 1 | no | — |
| 181 | #101 | — | 30 | 2 | no | — |
| 182 | C2×C3×C5 | — | 4 | 1 | no | — |
| 183 | C3×D5 | — | 4 | 1 | no | — |
| 184 | C3×D5 | — | 4 | 1 | no | — |
| 185 | C2×C3×C4 | — | 20 | 1 | no | — |
| 186 | C2×Dic3 | — | 40 | 1 | no | — |
| 187 | C2×Dic3 | — | 40 | 1 | no | — |
| 188 | Dic6 | — | 4 | 1 | no | — |
| 189 | Dic6 | — | 20 | 1 | no | — |
| 190 | Dic6 | — | 40 | 1 | no | — |
| 191 | SL(2,𝔽3) | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 192 | C3×C8 | — | 40 | 1 | no | — |
| 193 | C22×C5 | — | 3 | 1 | no | — |
| 194 | C2×D5 | — | 3 | 1 | no | — |
| 195 | C2×D5 | — | 6 | 1 | no | — |
| 196 | C2×D5 | — | 6 | 1 | no | — |
| 197 | C2×D5 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 198 | AGL(1,𝔽5) | — | 24 | 1 | no | — |
| 199 | C4×C5 | — | 3 | 1 | no | — |
| 200 | C4×C5 | — | 12 | 1 | no | — |
| 201 | C4×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 202 | Dic5 | — | 3 | 1 | no | — |
| 203 | Dic5 | — | 12 | 1 | no | — |
| 204 | Dic5 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 205 | #41 | — | 5 | 1 | no | — |
| 206 | #41 | — | 15 | 1 | no | — |
| 207 | #41 | — | 15 | 1 | no | — |
| 208 | #41 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 209 | C2×Q8 | — | 3 | 1 | no | — |
| 210 | C2×Q8 | — | 5 | 1 | no | — |
| 211 | C2×Q8 | — | 15 | 1 | no | — |
| 212 | C2×Q8 | — | 30 | 1 | no | — |
| 213 | C2×Q8 | — | 30 | 1 | no | — |
| 214 | C2×Q8 | — | 30 | 1 | no | — |
| 215 | C2×Q8 | — | 30 | 1 | no | — |
| 216 | C2×Q8 | — | 30 | 1 | no | — |
| 217 | C2×D4 | — | 15 | 1 | no | — |
| 218 | C22×C4 | — | 15 | 1 | no | — |
| 219 | C22×C4 | — | 30 | 1 | no | — |
| 220 | C22×C4 | — | 30 | 1 | no | — |
| 221 | Q16 | — | 6 | 1 | no | — |
| 222 | Q16 | — | 30 | 1 | no | — |
| 223 | Q16 | — | 30 | 1 | no | — |
| 224 | Q16 | — | 30 | 1 | no | — |
| 225 | SD16 | — | 6 | 1 | no | — |
| 226 | SD16 | — | 30 | 1 | no | — |
| 227 | SD16 | — | 30 | 1 | no | — |
| 228 | SD16 | — | 30 | 1 | no | — |
| 229 | M16 | — | 3 | 1 | no | — |
| 230 | M16 | — | 15 | 1 | no | — |
| 231 | M16 | — | 15 | 2 | no | — |
| 232 | M16 | — | 30 | 1 | no | — |
| 233 | M16 | — | 30 | 1 | no | — |
| 234 | M16 | — | 30 | 1 | no | — |
| 235 | M16 | — | 30 | 1 | no | — |
| 236 | C2×C8 | — | 5 | 1 | no | — |
| 237 | C2×C8 | — | 15 | 1 | no | — |
| 238 | C2×C8 | — | 30 | 1 | no | — |
| 239 | C2×C8 | — | 30 | 1 | no | — |
| 240 | C2×C8 | — | 30 | 1 | no | — |
| 241 | C2×C8 | — | 30 | 1 | no | — |
| 242 | C4⋊C4 | — | 30 | 1 | no | — |
| 243 | C4⋊C4 | — | 30 | 1 | no | — |
| 244 | C3×C5 | — | 4 | 1 | no | Hall |
| 245 | C22×C3 | — | 20 | 1 | no | — |
| 246 | C3×C4 | — | 4 | 1 | no | — |
| 247 | C3×C4 | — | 20 | 1 | no | — |
| 248 | Dic3 | — | 8 | 1 | no | — |
| 249 | Dic3 | — | 40 | 1 | no | — |
| 250 | Dic3 | — | 80 | 1 | no | — |
| 251 | C2×C5 | — | 6 | 1 | no | — |
| 252 | C2×C5 | (C2×S4)⋊C4 | 1 | 1 | yes | characteristic |
| 253 | D5 | — | 6 | 1 | no | — |
| 254 | D5 | #5403 | 1 | 1 | yes | characteristic |
| 255 | D5 | #5406 | 1 | 1 | yes | characteristic |
| 256 | C23 | — | 15 | 1 | no | — |
| 257 | Q8 | — | 6 | 1 | no | — |
| 258 | Q8 | — | 6 | 1 | no | — |
| 259 | Q8 | — | 15 | 1 | no | — |
| 260 | Q8 | — | 30 | 1 | no | — |
| 261 | Q8 | — | 30 | 1 | no | — |
| 262 | Q8 | — | 60 | 1 | no | — |
| 263 | Q8 | — | 60 | 1 | no | — |
| 264 | Q8 | (C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 265 | D4 | — | 3 | 1 | no | — |
| 266 | D4 | — | 15 | 1 | no | — |
| 267 | D4 | — | 30 | 1 | no | — |
| 268 | C2×C4 | — | 3 | 1 | no | — |
| 269 | C2×C4 | — | 5 | 1 | no | — |
| 270 | C2×C4 | — | 6 | 1 | no | — |
| 271 | C2×C4 | — | 15 | 1 | no | — |
| 272 | C2×C4 | — | 15 | 1 | no | — |
| 273 | C2×C4 | — | 15 | 1 | no | — |
| 274 | C2×C4 | — | 15 | 1 | no | — |
| 275 | C2×C4 | — | 30 | 1 | no | — |
| 276 | C2×C4 | — | 60 | 1 | no | — |
| 277 | C2×C4 | — | 60 | 1 | no | — |
| 278 | C2×C4 | — | 60 | 1 | no | — |
| 279 | C2×C4 | — | 120 | 1 | no | — |
| 280 | C8 | — | 6 | 1 | no | — |
| 281 | C8 | — | 10 | 1 | no | — |
| 282 | C8 | — | 30 | 1 | no | — |
| 283 | C8 | — | 30 | 1 | no | — |
| 284 | C8 | — | 60 | 1 | no | — |
| 285 | C2×C3 | — | 4 | 1 | no | — |
| 286 | C2×C3 | — | 20 | 1 | no | — |
| 287 | C2×C3 | — | 20 | 1 | no | — |
| 288 | C5 | #73124 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 289 | C22 | — | 3 | 1 | no | — |
| 290 | C22 | — | 5 | 1 | no | — |
| 291 | C22 | — | 15 | 1 | no | — |
| 292 | C22 | — | 30 | 1 | no | — |
| 293 | C22 | — | 30 | 1 | no | — |
| 294 | C4 | — | 3 | 1 | no | — |
| 295 | C4 | — | 5 | 1 | no | — |
| 296 | C4 | — | 12 | 1 | no | — |
| 297 | C4 | — | 15 | 1 | no | — |
| 298 | C4 | — | 60 | 1 | no | — |
| 299 | C4 | — | 120 | 1 | no | — |
| 300 | C4 | AGL(1,𝔽5)×S4 | 1 | 1 | yes | characteristic |
| 301 | C3 | — | 4 | 1 | no | Sylow |
| 302 | C2 | — | 5 | 1 | no | — |
| 303 | C2 | — | 5 | 1 | no | — |
| 304 | C2 | — | 6 | 1 | no | — |
| 305 | C2 | — | 30 | 1 | no | — |
| 306 | C2 | (C2×AGL(1,𝔽5))⋊S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #91954 | C4 | semidirect |
| 7. C5×#984 | C4 | semidirect |
| 66. #984 | AGL(1,𝔽5) | semidirect |
| 288. C5 | #73124 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#984 | 2 |
| D5⋊(#131⋊C3) | 2 |
| #11756726 | 2 |
| #10604870 | 3 |
| #91154 | 4 |
| #73124 | 5 |