Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
336 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 44 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 | C2×#87840 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87836 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#87835 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#87835 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#87832 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#87831 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #11716776 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D52 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#1044 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×#1044 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×#1044 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×#1043 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6036 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#6036 | C22 | 1 | 1 | yes | characteristic |
| 15 | #87840 | — | 2 | 2 | no | — |
| 16 | #87836 | — | 2 | 2 | no | — |
| 17 | #87835 | — | 2 | 2 | no | — |
| 18 | #87835 | — | 2 | 2 | no | — |
| 19 | #87832 | — | 2 | 2 | no | — |
| 20 | #87831 | — | 2 | 2 | no | — |
| 21 | C22×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×C5×D5 | — | 2 | 1 | no | — |
| 23 | C2×D52 | — | 2 | 1 | no | — |
| 24 | C2×D52 | — | 2 | 1 | no | — |
| 25 | C2×D52 | — | 2 | 1 | no | — |
| 26 | C2×D52 | — | 2 | 1 | no | — |
| 27 | C2×D52 | — | 2 | 1 | no | — |
| 28 | C2×D52 | Q8 | 1 | 2 | yes | — |
| 29 | C2×#1044 | — | 2 | 1 | no | — |
| 30 | C2×#1044 | — | 2 | 1 | no | — |
| 31 | C2×#1044 | — | 2 | 1 | no | — |
| 32 | C2×#1044 | — | 2 | 1 | no | — |
| 33 | C2×#1044 | — | 2 | 1 | no | — |
| 34 | C2×#1044 | — | 2 | 1 | no | — |
| 35 | C2×#1044 | — | 2 | 1 | no | — |
| 36 | C2×#1044 | D4 | 1 | 2 | yes | — |
| 37 | C2×#1044 | D4 | 1 | 2 | yes | — |
| 38 | C2×#1043 | — | 2 | 1 | no | — |
| 39 | C2×#1043 | — | 2 | 1 | no | — |
| 40 | C2×#1043 | — | 2 | 1 | no | — |
| 41 | C2×C5×Dic5 | — | 2 | 1 | no | — |
| 42 | C2×C5×Dic5 | — | 2 | 1 | no | — |
| 43 | #6036 | — | 2 | 1 | no | — |
| 44 | #6036 | — | 2 | 1 | no | — |
| 45 | #6036 | — | 2 | 1 | no | — |
| 46 | #6036 | D4 | 1 | 2 | yes | — |
| 47 | C2×C22⋊AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 48 | C2×C4⋊AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 49 | (C2×AGL(1,𝔽5))⋊C4 | — | 10 | 1 | no | — |
| 50 | C22×C52 | #41 | 1 | 1 | yes | characteristic |
| 51 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 53 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C2×C5⋊D5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 55 | C2×C5⋊D5 | #41 | 1 | 1 | yes | characteristic |
| 56 | C2×C5⋊D5 | #41 | 1 | 1 | yes | characteristic |
| 57 | C2×C5×D5 | — | 2 | 2 | no | — |
| 58 | C2×C5×D5 | — | 4 | 1 | no | — |
| 59 | C2×C5×D5 | — | 4 | 1 | no | — |
| 60 | D52 | — | 2 | 2 | no | — |
| 61 | D52 | — | 2 | 2 | no | — |
| 62 | D52 | — | 4 | 1 | no | — |
| 63 | D52 | — | 4 | 1 | no | — |
| 64 | #1044 | — | 2 | 2 | no | — |
| 65 | #1044 | — | 2 | 2 | no | — |
| 66 | #1044 | — | 4 | 1 | no | — |
| 67 | #1043 | — | 4 | 1 | no | — |
| 68 | C5×Dic5 | — | 2 | 2 | no | — |
| 69 | C5×Dic5 | — | 4 | 1 | no | — |
| 70 | C23×D5 | — | 10 | 1 | no | — |
| 71 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 72 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 73 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 74 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 75 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 76 | C22×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 77 | C2×C4×D5 | — | 10 | 1 | no | — |
| 78 | C2×C4×D5 | — | 10 | 1 | no | — |
| 79 | C22⋊AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 80 | C4⋊AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 81 | #396 | — | 25 | 1 | no | maximal, Sylow |
| 82 | C2×C52 | #121 | 1 | 1 | yes | characteristic |
| 83 | C2×C52 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 84 | C2×C52 | #126 | 1 | 1 | yes | characteristic |
| 85 | C5⋊D5 | #121 | 1 | 1 | yes | characteristic |
| 86 | C5⋊D5 | #121 | 1 | 1 | yes | characteristic |
| 87 | C5⋊D5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 88 | C5⋊D5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 89 | C5×D5 | — | 4 | 2 | no | — |
| 90 | C23×C5 | — | 10 | 1 | no | — |
| 91 | C22×D5 | — | 2 | 1 | no | — |
| 92 | C22×D5 | — | 10 | 2 | no | — |
| 93 | C22×D5 | — | 10 | 1 | no | — |
| 94 | C22×D5 | — | 10 | 1 | no | — |
| 95 | C22×D5 | — | 10 | 1 | no | — |
| 96 | C22×D5 | — | 10 | 2 | no | — |
| 97 | C22×D5 | — | 20 | 1 | no | — |
| 98 | C22×D5 | — | 20 | 1 | no | — |
| 99 | C22×D5 | — | 20 | 1 | no | — |
| 100 | C22×D5 | — | 20 | 1 | no | — |
| 101 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 102 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 103 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 104 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 105 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 106 | C2×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 107 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 108 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 109 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 110 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 111 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 112 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 113 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 114 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 115 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 116 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 117 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 118 | C2×AGL(1,𝔽5) | — | 20 | 1 | no | — |
| 119 | C2×C4×C5 | — | 10 | 1 | no | — |
| 120 | C2×C4×C5 | — | 10 | 1 | no | — |
| 121 | C2×Dic5 | — | 2 | 1 | no | — |
| 122 | C2×Dic5 | — | 2 | 1 | no | — |
| 123 | C4×D5 | — | 10 | 2 | no | — |
| 124 | C4×D5 | — | 10 | 2 | no | — |
| 125 | C4×D5 | — | 20 | 1 | no | — |
| 126 | C4×D5 | — | 20 | 1 | no | — |
| 127 | C2×C4⋊C4 | — | 25 | 1 | no | — |
| 128 | C2×C4⋊C4 | — | 25 | 1 | no | — |
| 129 | C2×C4⋊C4 | — | 25 | 1 | no | — |
| 130 | C2×C22⋊C4 | — | 25 | 1 | no | — |
| 131 | C2×C22⋊C4 | — | 25 | 1 | no | — |
| 132 | C2×C22⋊C4 | — | 25 | 1 | no | — |
| 133 | #95 | — | 25 | 1 | no | — |
| 134 | C52 | #396 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 135 | C22×C5 | — | 2 | 1 | no | — |
| 136 | C22×C5 | — | 2 | 1 | no | — |
| 137 | C22×C5 | — | 2 | 1 | no | — |
| 138 | C22×C5 | — | 10 | 2 | no | — |
| 139 | C22×C5 | — | 20 | 1 | no | — |
| 140 | C22×C5 | — | 20 | 1 | no | — |
| 141 | C2×D5 | — | 2 | 2 | no | — |
| 142 | C2×D5 | — | 4 | 1 | no | — |
| 143 | C2×D5 | — | 4 | 1 | no | — |
| 144 | C2×D5 | — | 10 | 1 | no | — |
| 145 | C2×D5 | — | 10 | 1 | no | — |
| 146 | C2×D5 | — | 10 | 1 | no | — |
| 147 | C2×D5 | — | 10 | 1 | no | — |
| 148 | C2×D5 | — | 10 | 1 | no | — |
| 149 | C2×D5 | — | 10 | 1 | no | — |
| 150 | C2×D5 | — | 10 | 1 | no | — |
| 151 | C2×D5 | — | 10 | 1 | no | — |
| 152 | C2×D5 | — | 10 | 1 | no | — |
| 153 | C2×D5 | — | 10 | 1 | no | — |
| 154 | C2×D5 | — | 10 | 1 | no | — |
| 155 | C2×D5 | — | 10 | 1 | no | — |
| 156 | C2×D5 | — | 10 | 1 | no | — |
| 157 | C2×D5 | — | 10 | 1 | no | — |
| 158 | C2×D5 | — | 10 | 1 | no | — |
| 159 | C2×D5 | — | 10 | 1 | no | — |
| 160 | C2×D5 | — | 10 | 1 | no | — |
| 161 | C2×D5 | — | 10 | 1 | no | — |
| 162 | C2×D5 | — | 20 | 2 | no | — |
| 163 | C2×D5 | — | 20 | 2 | no | — |
| 164 | C2×D5 | — | 20 | 2 | no | — |
| 165 | C2×D5 | — | 20 | 2 | no | — |
| 166 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 167 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 168 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 169 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 170 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 171 | AGL(1,𝔽5) | — | 20 | 2 | no | — |
| 172 | C4×C5 | — | 10 | 2 | no | — |
| 173 | C4×C5 | — | 20 | 1 | no | — |
| 174 | Dic5 | — | 2 | 2 | no | — |
| 175 | Dic5 | — | 4 | 1 | no | — |
| 176 | C24 | — | 25 | 1 | no | — |
| 177 | C22×C4 | — | 25 | 1 | no | — |
| 178 | C22×C4 | — | 25 | 1 | no | — |
| 179 | C22×C4 | — | 25 | 1 | no | — |
| 180 | C22×C4 | — | 25 | 1 | no | — |
| 181 | C22×C4 | — | 25 | 1 | no | — |
| 182 | C22×C4 | — | 25 | 1 | no | — |
| 183 | C4⋊C4 | — | 50 | 2 | no | — |
| 184 | C4⋊C4 | — | 50 | 2 | no | — |
| 185 | C4⋊C4 | — | 50 | 2 | no | — |
| 186 | C22⋊C4 | — | 50 | 2 | no | — |
| 187 | C22⋊C4 | — | 50 | 2 | no | — |
| 188 | C22⋊C4 | — | 50 | 2 | no | — |
| 189 | C2×C5 | — | 2 | 1 | no | — |
| 190 | C2×C5 | — | 2 | 1 | no | — |
| 191 | C2×C5 | — | 2 | 1 | no | — |
| 192 | C2×C5 | — | 2 | 1 | no | — |
| 193 | C2×C5 | — | 2 | 1 | no | — |
| 194 | C2×C5 | — | 2 | 1 | no | — |
| 195 | C2×C5 | — | 2 | 1 | no | — |
| 196 | C2×C5 | — | 2 | 1 | no | — |
| 197 | C2×C5 | — | 2 | 1 | no | — |
| 198 | C2×C5 | — | 20 | 2 | no | — |
| 199 | D5 | — | 4 | 2 | no | — |
| 200 | D5 | — | 10 | 1 | no | — |
| 201 | D5 | — | 10 | 1 | no | — |
| 202 | D5 | — | 10 | 1 | no | — |
| 203 | D5 | — | 10 | 1 | no | — |
| 204 | D5 | — | 10 | 1 | no | — |
| 205 | D5 | — | 10 | 1 | no | — |
| 206 | D5 | — | 10 | 1 | no | — |
| 207 | D5 | — | 10 | 1 | no | — |
| 208 | D5 | — | 10 | 1 | no | — |
| 209 | D5 | — | 10 | 1 | no | — |
| 210 | D5 | — | 10 | 1 | no | — |
| 211 | D5 | — | 10 | 1 | no | — |
| 212 | C23 | — | 10 | 1 | no | — |
| 213 | C23 | — | 25 | 2 | no | — |
| 214 | C23 | — | 25 | 1 | no | — |
| 215 | C23 | — | 50 | 1 | no | — |
| 216 | C23 | — | 50 | 1 | no | — |
| 217 | C23 | — | 50 | 1 | no | — |
| 218 | C23 | — | 50 | 1 | no | — |
| 219 | C23 | — | 50 | 1 | no | — |
| 220 | C2×C4 | — | 10 | 1 | no | — |
| 221 | C2×C4 | — | 10 | 1 | no | — |
| 222 | C2×C4 | — | 25 | 2 | no | — |
| 223 | C2×C4 | — | 25 | 2 | no | — |
| 224 | C2×C4 | — | 25 | 2 | no | — |
| 225 | C2×C4 | — | 50 | 1 | no | — |
| 226 | C2×C4 | — | 50 | 1 | no | — |
| 227 | C2×C4 | — | 50 | 1 | no | — |
| 228 | C2×C4 | — | 50 | 1 | no | — |
| 229 | C2×C4 | — | 50 | 1 | no | — |
| 230 | C2×C4 | — | 50 | 1 | no | — |
| 231 | C2×C4 | — | 50 | 1 | no | — |
| 232 | C2×C4 | — | 50 | 1 | no | — |
| 233 | C2×C4 | — | 50 | 1 | no | — |
| 234 | C2×C4 | — | 50 | 1 | no | — |
| 235 | C2×C4 | — | 50 | 1 | no | — |
| 236 | C2×C4 | — | 50 | 1 | no | — |
| 237 | C2×C4 | — | 50 | 1 | no | — |
| 238 | C5 | — | 2 | 1 | no | — |
| 239 | C5 | — | 2 | 1 | no | — |
| 240 | C5 | — | 2 | 1 | no | — |
| 241 | C22 | — | 10 | 2 | no | — |
| 242 | C22 | — | 20 | 1 | no | — |
| 243 | C22 | — | 20 | 1 | no | — |
| 244 | C22 | — | 25 | 1 | no | — |
| 245 | C22 | — | 25 | 1 | no | — |
| 246 | C22 | — | 25 | 1 | no | — |
| 247 | C22 | — | 25 | 1 | no | — |
| 248 | C22 | — | 25 | 1 | no | — |
| 249 | C22 | — | 25 | 1 | no | — |
| 250 | C22 | — | 50 | 2 | no | — |
| 251 | C22 | — | 50 | 2 | no | — |
| 252 | C22 | — | 100 | 1 | no | — |
| 253 | C22 | — | 100 | 1 | no | — |
| 254 | C22 | (D5⋊AGL(1,𝔽5))⋊C2 | 1 | 1 | yes | characteristic |
| 255 | C4 | — | 10 | 2 | no | — |
| 256 | C4 | — | 20 | 1 | no | — |
| 257 | C4 | — | 50 | 2 | no | — |
| 258 | C4 | — | 50 | 2 | no | — |
| 259 | C4 | — | 100 | 1 | no | — |
| 260 | C4 | — | 100 | 1 | no | — |
| 261 | C2 | — | 20 | 2 | no | — |
| 262 | C2 | — | 25 | 1 | no | — |
| 263 | C2 | — | 25 | 1 | no | — |
| 264 | C2 | — | 25 | 1 | no | — |
| 265 | C2 | — | 25 | 1 | no | — |
| 266 | C2 | #11717309 | 1 | 1 | yes | characteristic |
| 267 | C2 | #11717310 | 1 | 1 | yes | characteristic |
| 268 | C2 | #11717314 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | (D5⋊AGL(1,𝔽5))⋊C2 |
| commutator, cosocle | C22×C5⋊D5 | C23 |
| Fitting, socle | C22×C52 | #41 |
| prime radical | C22×D52 | C22 |
| hypocenter | C52 | #396 |
| LCS | G ▷ C22×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#87840 | C2 | semidirect |
| 2. C2×#87836 | C2 | semidirect |
| 5. C2×#87832 | C2 | semidirect |
| 7. #11716776 | C2 | semidirect |
| 134. C52 | #396 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87840 | 2 |
| C2×#87836 | 2 |
| C2×#87835 | 2 |
| C2×#87832 | 2 |
| C2×#87831 | 2 |
| #11716776 | 2 |
| #396 | 25 |