Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
516 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 516 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 | #11717267 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#4092 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11717033 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11717028 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11717024 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11717009 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11716989 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #11716972 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11716968 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C2×C52×Q8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×C5×D5 | C22 | 1 | 2 | yes | — |
| 13 | C2×#6039 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#6038 | C22 | 1 | 2 | yes | — |
| 15 | C2×#6038 | C22 | 1 | 2 | yes | — |
| 16 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#6036 | C22 | 1 | 2 | yes | — |
| 18 | #87808 | C22 | 1 | 2 | yes | — |
| 19 | #87808 | C22 | 1 | 2 | yes | — |
| 20 | C4×#1039 | C22 | 1 | 1 | yes | characteristic |
| 21 | C5×#685 | C22 | 1 | 2 | yes | — |
| 22 | C5×#685 | C22 | 1 | 2 | yes | — |
| 23 | C5×C4⋊Dic5 | C22 | 1 | 2 | yes | — |
| 24 | C5×#683 | C22 | 1 | 2 | yes | — |
| 25 | C5×#683 | C22 | 1 | 2 | yes | — |
| 26 | #87781 | C22 | 1 | 1 | yes | characteristic |
| 27 | #87781 | C22 | 1 | 1 | yes | characteristic |
| 28 | #87780 | C22 | 1 | 2 | yes | — |
| 29 | #87779 | C22 | 1 | 2 | yes | — |
| 30 | #87778 | C22 | 1 | 2 | yes | — |
| 31 | #65981 | — | 5 | 2 | no | maximal |
| 32 | C22×C5⋊D5 | C23 | 1 | 2 | yes | — |
| 33 | C22×C5×D5 | C23 | 1 | 2 | yes | — |
| 34 | C52×Q8 | — | 4 | 1 | no | — |
| 35 | C2×C4×C52 | C23 | 1 | 2 | yes | — |
| 36 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 39 | C4⋊(C5⋊D5) | — | 4 | 1 | no | — |
| 40 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 41 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 42 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 43 | C4×C5×D5 | — | 4 | 2 | no | — |
| 44 | #6039 | — | 4 | 1 | no | — |
| 45 | #6038 | — | 4 | 2 | no | — |
| 46 | #6038 | — | 4 | 2 | no | — |
| 47 | #6037 | — | 4 | 1 | no | — |
| 48 | #6036 | — | 4 | 2 | no | — |
| 49 | C5×C22⋊Q8 | — | 5 | 2 | no | — |
| 50 | #4093 | — | 5 | 2 | no | — |
| 51 | #4093 | — | 10 | 2 | no | — |
| 52 | #4092 | D5 | 1 | 2 | yes | complemented |
| 53 | #4076 | — | 5 | 2 | no | — |
| 54 | #4075 | — | 5 | 2 | no | — |
| 55 | #4043 | — | 5 | 2 | no | — |
| 56 | #4042 | — | 5 | 2 | no | — |
| 57 | #4041 | — | 5 | 2 | no | — |
| 58 | #4040 | — | 5 | 2 | no | — |
| 59 | #4040 | — | 5 | 2 | no | — |
| 60 | #4035 | — | 5 | 2 | no | — |
| 61 | #4031 | — | 5 | 2 | no | — |
| 62 | #4030 | — | 5 | 2 | no | — |
| 63 | #4030 | — | 5 | 2 | no | — |
| 64 | #4029 | — | 5 | 2 | no | — |
| 65 | C22×C52 | C24 | 1 | 1 | yes | characteristic |
| 66 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 67 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 68 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 69 | C2×C5×D5 | — | 2 | 2 | no | — |
| 70 | C2×C5×D5 | — | 2 | 2 | no | — |
| 71 | C2×C5×D5 | — | 2 | 2 | no | — |
| 72 | C4×C52 | — | 2 | 1 | no | — |
| 73 | C4×C52 | — | 2 | 2 | no | — |
| 74 | #1039 | — | 2 | 1 | no | — |
| 75 | #1039 | — | 2 | 1 | no | — |
| 76 | C5×Dic5 | — | 2 | 2 | no | — |
| 77 | C5×Dic5 | — | 2 | 2 | no | — |
| 78 | C5×Dic5 | — | 2 | 2 | no | — |
| 79 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 80 | C2×C5×Q8 | C2×D5 | 1 | 2 | yes | — |
| 81 | C22×C4×C5 | — | 5 | 2 | no | — |
| 82 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 83 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 84 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 85 | C2×D20 | — | 5 | 2 | no | — |
| 86 | C2×D20 | — | 5 | 2 | no | — |
| 87 | C2×D20 | — | 5 | 2 | no | — |
| 88 | C2×D20 | — | 10 | 2 | no | — |
| 89 | C2×C4×D5 | — | 5 | 2 | no | — |
| 90 | C2×C4×D5 | — | 5 | 2 | no | — |
| 91 | C2×C4×D5 | C2×D5 | 1 | 2 | yes | — |
| 92 | C2×Dic10 | — | 5 | 2 | no | — |
| 93 | C5×C4⋊C4 | — | 5 | 2 | no | — |
| 94 | C5×C4⋊C4 | — | 5 | 2 | no | — |
| 95 | C5×C4⋊C4 | — | 5 | 2 | no | — |
| 96 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 97 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 98 | C22⋊Dic5 | — | 5 | 2 | no | — |
| 99 | #685 | — | 5 | 2 | no | — |
| 100 | #685 | — | 5 | 2 | no | — |
| 101 | #685 | — | 5 | 2 | no | — |
| 102 | #685 | — | 5 | 2 | no | — |
| 103 | #685 | — | 5 | 2 | no | — |
| 104 | #685 | — | 5 | 2 | no | — |
| 105 | #685 | — | 5 | 2 | no | — |
| 106 | #685 | — | 10 | 4 | no | — |
| 107 | #685 | — | 10 | 4 | no | — |
| 108 | #685 | C2×D5 | 1 | 2 | yes | — |
| 109 | #685 | C2×D5 | 1 | 2 | yes | — |
| 110 | C4⋊Dic5 | — | 5 | 2 | no | — |
| 111 | C4⋊Dic5 | C2×D5 | 1 | 2 | yes | — |
| 112 | #683 | — | 5 | 2 | no | — |
| 113 | #683 | — | 5 | 2 | no | — |
| 114 | #683 | — | 5 | 2 | no | — |
| 115 | #683 | C2×D5 | 1 | 2 | yes | — |
| 116 | #683 | C2×D5 | 1 | 2 | yes | — |
| 117 | C4×Dic5 | — | 5 | 2 | no | — |
| 118 | C4×Dic5 | — | 10 | 2 | no | — |
| 119 | #562 | — | 25 | 1 | no | Sylow |
| 120 | C2×C52 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 121 | C2×C52 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 122 | C2×C52 | 2−1+4 | 1 | 1 | yes | characteristic |
| 123 | C5⋊D5 | — | 4 | 2 | no | — |
| 124 | C5×D5 | — | 4 | 2 | no | — |
| 125 | C23×C5 | — | 5 | 2 | no | — |
| 126 | C22×D5 | — | 5 | 2 | no | — |
| 127 | C22×D5 | — | 5 | 2 | no | — |
| 128 | C22×D5 | — | 10 | 4 | no | — |
| 129 | C22×D5 | C22×D5 | 1 | 2 | yes | — |
| 130 | C5×Q8 | — | 4 | 2 | no | — |
| 131 | C5×Q8 | — | 4 | 4 | no | — |
| 132 | C2×C4×C5 | — | 2 | 2 | no | — |
| 133 | C2×C4×C5 | — | 2 | 4 | no | — |
| 134 | C2×C4×C5 | — | 5 | 2 | no | — |
| 135 | C2×C4×C5 | — | 5 | 2 | no | — |
| 136 | C2×C4×C5 | — | 5 | 2 | no | — |
| 137 | C2×C4×C5 | — | 20 | 2 | no | — |
| 138 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 139 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 140 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 141 | C22⋊D5 | — | 20 | 2 | no | — |
| 142 | C22⋊D5 | — | 20 | 2 | no | — |
| 143 | C22⋊D5 | — | 20 | 2 | no | — |
| 144 | C2×Dic5 | — | 5 | 2 | no | — |
| 145 | C2×Dic5 | — | 5 | 2 | no | — |
| 146 | C2×Dic5 | — | 10 | 2 | no | — |
| 147 | C2×Dic5 | — | 10 | 2 | no | — |
| 148 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 149 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 150 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 151 | D20 | — | 20 | 2 | no | — |
| 152 | D20 | — | 20 | 4 | no | — |
| 153 | D20 | — | 20 | 2 | no | — |
| 154 | D20 | — | 20 | 2 | no | — |
| 155 | C4×D5 | — | 4 | 2 | no | — |
| 156 | C4×D5 | — | 20 | 2 | no | — |
| 157 | C4×D5 | — | 20 | 2 | no | — |
| 158 | Dic10 | — | 20 | 2 | no | — |
| 159 | #125 | — | 25 | 1 | no | — |
| 160 | #124 | — | 25 | 1 | no | — |
| 161 | #124 | — | 25 | 1 | no | — |
| 162 | #123 | — | 25 | 2 | no | — |
| 163 | #123 | — | 25 | 2 | no | — |
| 164 | C22⋊Q8 | — | 5 | 2 | no | — |
| 165 | C22⋊Q8 | — | 25 | 2 | no | — |
| 166 | #121 | — | 25 | 2 | no | — |
| 167 | #121 | — | 25 | 2 | no | — |
| 168 | C52 | #562 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 169 | C22×C5 | — | 2 | 2 | no | — |
| 170 | C22×C5 | — | 10 | 2 | no | — |
| 171 | C22×C5 | — | 10 | 2 | no | — |
| 172 | C22×C5 | — | 10 | 2 | no | — |
| 173 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 174 | C2×D5 | — | 2 | 2 | no | — |
| 175 | C2×D5 | — | 2 | 2 | no | — |
| 176 | C2×D5 | — | 2 | 2 | no | — |
| 177 | C2×D5 | — | 10 | 2 | no | — |
| 178 | C2×D5 | — | 10 | 2 | no | — |
| 179 | C2×D5 | — | 10 | 2 | no | — |
| 180 | C2×D5 | — | 10 | 2 | no | — |
| 181 | C2×D5 | — | 10 | 2 | no | — |
| 182 | C2×D5 | — | 10 | 2 | no | — |
| 183 | C2×D5 | — | 20 | 4 | no | — |
| 184 | C2×D5 | — | 20 | 4 | no | — |
| 185 | C2×D5 | — | 20 | 4 | no | — |
| 186 | C4×C5 | — | 2 | 4 | no | — |
| 187 | C4×C5 | — | 2 | 2 | no | — |
| 188 | C4×C5 | — | 2 | 2 | no | — |
| 189 | C4×C5 | — | 2 | 2 | no | — |
| 190 | C4×C5 | — | 4 | 4 | no | — |
| 191 | C4×C5 | — | 10 | 2 | no | — |
| 192 | C4×C5 | — | 10 | 2 | no | — |
| 193 | C4×C5 | — | 10 | 2 | no | — |
| 194 | Dic5 | — | 2 | 2 | no | — |
| 195 | Dic5 | — | 2 | 2 | no | — |
| 196 | Dic5 | — | 2 | 2 | no | — |
| 197 | Dic5 | — | 10 | 2 | no | — |
| 198 | Dic5 | — | 10 | 2 | no | — |
| 199 | Dic5 | — | 20 | 2 | no | — |
| 200 | Dic5 | — | 20 | 2 | no | — |
| 201 | C2×Q8 | — | 25 | 1 | no | — |
| 202 | C2×Q8 | D52 | 1 | 1 | yes | characteristic |
| 203 | C2×D4 | — | 25 | 2 | no | — |
| 204 | C2×D4 | — | 25 | 2 | no | — |
| 205 | C2×D4 | — | 25 | 1 | no | — |
| 206 | C2×D4 | — | 25 | 1 | no | — |
| 207 | C22×C4 | — | 5 | 2 | no | — |
| 208 | C22×C4 | — | 25 | 2 | no | — |
| 209 | C4⋊C4 | — | 5 | 2 | no | — |
| 210 | C4⋊C4 | — | 5 | 2 | no | — |
| 211 | C4⋊C4 | — | 5 | 2 | no | — |
| 212 | C4⋊C4 | — | 25 | 2 | no | — |
| 213 | C4⋊C4 | — | 25 | 1 | no | — |
| 214 | C4⋊C4 | — | 25 | 1 | no | — |
| 215 | C22⋊C4 | — | 5 | 2 | no | — |
| 216 | C22⋊C4 | — | 5 | 2 | no | — |
| 217 | C22⋊C4 | — | 25 | 2 | no | — |
| 218 | C22⋊C4 | — | 25 | 2 | no | — |
| 219 | C22⋊C4 | — | 25 | 2 | no | — |
| 220 | C22⋊C4 | — | 25 | 2 | no | — |
| 221 | C42 | — | 25 | 1 | no | — |
| 222 | C2×C5 | — | 2 | 2 | no | — |
| 223 | C2×C5 | — | 2 | 2 | no | — |
| 224 | C2×C5 | — | 2 | 2 | no | — |
| 225 | C2×C5 | — | 20 | 2 | no | — |
| 226 | C2×C5 | #4144 | 1 | 2 | yes | — |
| 227 | C2×C5 | #4147 | 1 | 2 | yes | — |
| 228 | C2×C5 | #4149 | 1 | 2 | yes | — |
| 229 | D5 | — | 4 | 2 | no | — |
| 230 | D5 | — | 20 | 2 | no | — |
| 231 | D5 | — | 20 | 2 | no | — |
| 232 | D5 | — | 40 | 4 | no | — |
| 233 | C23 | — | 5 | 2 | no | — |
| 234 | C23 | — | 25 | 2 | no | — |
| 235 | Q8 | — | 4 | 1 | no | — |
| 236 | Q8 | — | 100 | 1 | no | — |
| 237 | D4 | — | 100 | 1 | no | — |
| 238 | D4 | — | 100 | 2 | no | — |
| 239 | D4 | — | 100 | 2 | no | — |
| 240 | D4 | — | 100 | 1 | no | — |
| 241 | C2×C4 | — | 5 | 2 | no | — |
| 242 | C2×C4 | — | 5 | 2 | no | — |
| 243 | C2×C4 | — | 5 | 2 | no | — |
| 244 | C2×C4 | — | 20 | 2 | no | — |
| 245 | C2×C4 | — | 25 | 1 | no | — |
| 246 | C2×C4 | — | 25 | 1 | no | — |
| 247 | C2×C4 | — | 100 | 2 | no | — |
| 248 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C2×D52 | 1 | 2 | yes | — |
| 250 | C5 | — | 2 | 2 | no | — |
| 251 | C5 | #65981 | 1 | 2 | yes | complemented |
| 252 | C22 | — | 10 | 2 | no | — |
| 253 | C22 | — | 10 | 2 | no | — |
| 254 | C22 | — | 10 | 2 | no | — |
| 255 | C22 | — | 50 | 2 | no | — |
| 256 | C22 | — | 50 | 2 | no | — |
| 257 | C22 | — | 50 | 2 | no | — |
| 258 | C22 | C22×D52 | 1 | 1 | yes | characteristic |
| 259 | C4 | — | 2 | 2 | no | — |
| 260 | C4 | — | 2 | 1 | no | — |
| 261 | C4 | — | 10 | 2 | no | — |
| 262 | C4 | — | 10 | 2 | no | — |
| 263 | C4 | — | 10 | 2 | no | — |
| 264 | C4 | — | 50 | 1 | no | — |
| 265 | C4 | — | 50 | 1 | no | — |
| 266 | C2 | — | 20 | 2 | no | — |
| 267 | C2 | — | 100 | 2 | no | — |
| 268 | C2 | #11717481 | 1 | 1 | yes | characteristic |
| 269 | C2 | #11717485 | 1 | 1 | yes | characteristic |
| 270 | C2 | #11717495 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C2×C52×Q8 | C22 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | #562 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717267 | C2 | semidirect |
| 2. C5×#4092 | C2 | semidirect |
| 3. #11717033 | C2 | semidirect |
| 4. #11717028 | C2 | semidirect |
| 5. #11717024 | C2 | semidirect |
| 6. #11717009 | C2 | semidirect |
| 7. #11716989 | C2 | semidirect |
| 8. #11716972 | C2 | semidirect |
| 9. #11716968 | C2 | semidirect |
| 52. #4092 | D5 | semidirect |
| 168. C52 | #562 | semidirect |
| 251. C5 | #65981 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717267 | 2 |
| C5×#4092 | 2 |
| #11717033 | 2 |
| #11717028 | 2 |
| #11717024 | 2 |
| #11717009 | 2 |
| #11716989 | 2 |
| #11716972 | 2 |
| #11716968 | 2 |
| #65981 | 5 |