Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
368 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 63 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×#87873 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11717448 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717435 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11716782 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11716781 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C4×C5×D5 | C4 | 1 | 2 | yes | — |
| 8 | C2×#6039 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6038 | C4 | 1 | 2 | yes | — |
| 10 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 11 | #87873 | — | 2 | 1 | no | — |
| 12 | #87873 | — | 2 | 1 | no | — |
| 13 | #87873 | — | 2 | 1 | no | — |
| 14 | #87873 | — | 2 | 1 | no | — |
| 15 | #87867 | C22 | 1 | 2 | yes | — |
| 16 | #87863 | — | 2 | 1 | no | — |
| 17 | #87862 | — | 2 | 1 | no | — |
| 18 | #87860 | C22 | 1 | 2 | yes | — |
| 19 | #87856 | — | 2 | 1 | no | — |
| 20 | #87855 | — | 2 | 1 | no | — |
| 21 | #65761 | — | 5 | 2 | no | maximal |
| 22 | C22×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C22×C5×D5 | C2×C4 | 1 | 2 | yes | — |
| 24 | C2×C4×C52 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 26 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 27 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 28 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 29 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C5×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 31 | C4×C5×D5 | — | 2 | 2 | no | — |
| 32 | C4×C5×D5 | — | 2 | 2 | no | — |
| 33 | #6039 | — | 2 | 1 | no | — |
| 34 | #6039 | — | 2 | 1 | no | — |
| 35 | #6038 | — | 2 | 2 | no | — |
| 36 | #6038 | — | 2 | 2 | no | — |
| 37 | #6037 | — | 2 | 1 | no | — |
| 38 | #6037 | — | 2 | 1 | no | — |
| 39 | #6034 | — | 2 | 2 | no | — |
| 40 | #6033 | — | 2 | 2 | no | — |
| 41 | C2×#709 | — | 5 | 2 | no | — |
| 42 | D5⋊M16 | — | 5 | 2 | no | — |
| 43 | D5⋊M16 | — | 5 | 2 | no | — |
| 44 | D5⋊M16 | — | 10 | 2 | no | — |
| 45 | #4013 | — | 5 | 4 | no | — |
| 46 | #4005 | — | 5 | 4 | no | — |
| 47 | C22×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C2×C5⋊D5 | — | 4 | 1 | no | — |
| 49 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 50 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C5×D5 | — | 2 | 2 | no | — |
| 52 | C2×C5×D5 | — | 4 | 2 | no | — |
| 53 | C4×C52 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C4×C52 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 57 | C5×Dic5 | — | 2 | 2 | no | — |
| 58 | C22×C4×C5 | — | 5 | 2 | no | — |
| 59 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 60 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 61 | #709 | — | 10 | 2 | no | — |
| 62 | #709 | — | 10 | 2 | no | — |
| 63 | #709 | — | 10 | 2 | no | — |
| 64 | #709 | — | 10 | 2 | no | — |
| 65 | C2×D20 | — | 5 | 2 | no | — |
| 66 | C2×C4×D5 | — | 5 | 2 | no | — |
| 67 | C2×C4×D5 | — | 10 | 2 | no | — |
| 68 | C2×C4×D5 | AGL(1,𝔽5) | 1 | 2 | yes | — |
| 69 | C2×Dic10 | — | 5 | 2 | no | — |
| 70 | #704 | — | 5 | 4 | no | — |
| 71 | #704 | — | 5 | 4 | no | — |
| 72 | #704 | — | 10 | 4 | no | — |
| 73 | #700 | — | 10 | 2 | no | — |
| 74 | #700 | — | 10 | 2 | no | — |
| 75 | #700 | — | 10 | 4 | no | — |
| 76 | D5⋊C8 | — | 10 | 2 | no | — |
| 77 | D5⋊C8 | — | 10 | 2 | no | — |
| 78 | D5⋊C8 | — | 10 | 4 | no | — |
| 79 | #413 | — | 25 | 1 | no | Sylow |
| 80 | C2×C52 | — | 2 | 1 | no | — |
| 81 | C2×C52 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C5⋊D5 | — | 2 | 1 | no | — |
| 83 | C5⋊D5 | — | 2 | 1 | no | — |
| 84 | C5×D5 | — | 4 | 2 | no | — |
| 85 | C23×C5 | — | 5 | 2 | no | — |
| 86 | C22×D5 | — | 5 | 2 | no | — |
| 87 | C22×D5 | — | 10 | 2 | no | — |
| 88 | C22×D5 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 89 | C2×C4×C5 | — | 2 | 2 | no | — |
| 90 | C2×C4×C5 | — | 5 | 2 | no | — |
| 91 | C2×C4×C5 | — | 10 | 2 | no | — |
| 92 | C2×C4×C5 | — | 10 | 2 | no | — |
| 93 | C2×C4×C5 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 94 | C22⋊D5 | — | 10 | 2 | no | — |
| 95 | C22⋊D5 | — | 10 | 2 | no | — |
| 96 | C22⋊D5 | — | 10 | 2 | no | — |
| 97 | C22⋊D5 | — | 10 | 2 | no | — |
| 98 | C2×Dic5 | — | 5 | 2 | no | — |
| 99 | C2×Dic5 | — | 10 | 2 | no | — |
| 100 | C2×Dic5 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 101 | D20 | — | 10 | 2 | no | — |
| 102 | D20 | — | 10 | 2 | no | — |
| 103 | C4×D5 | — | 2 | 2 | no | — |
| 104 | C4×D5 | — | 2 | 2 | no | — |
| 105 | C4×D5 | — | 5 | 2 | no | — |
| 106 | C4×D5 | — | 5 | 2 | no | — |
| 107 | C4×D5 | — | 5 | 2 | no | — |
| 108 | C4×D5 | — | 5 | 2 | no | — |
| 109 | C4×D5 | — | 10 | 2 | no | — |
| 110 | C4×D5 | — | 10 | 2 | no | — |
| 111 | C4×D5 | — | 10 | 2 | no | — |
| 112 | C4×D5 | — | 10 | 2 | no | — |
| 113 | Dic10 | — | 10 | 2 | no | — |
| 114 | Dic10 | — | 10 | 2 | no | — |
| 115 | #170 | — | 10 | 4 | no | — |
| 116 | #170 | — | 10 | 4 | no | — |
| 117 | #170 | — | 10 | 8 | no | — |
| 118 | C2×#41 | — | 25 | 1 | no | — |
| 119 | C2×M16 | — | 25 | 1 | no | — |
| 120 | C2×M16 | — | 25 | 1 | no | — |
| 121 | #101 | — | 25 | 2 | no | — |
| 122 | #100 | — | 25 | 2 | no | — |
| 123 | C52 | #413 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 124 | C22×C5 | — | 2 | 2 | no | — |
| 125 | C22×C5 | — | 10 | 2 | no | — |
| 126 | C22×C5 | — | 20 | 2 | no | — |
| 127 | C22×C5 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 128 | C2×D5 | — | 2 | 2 | no | — |
| 129 | C2×D5 | — | 4 | 2 | no | — |
| 130 | C2×D5 | — | 5 | 2 | no | — |
| 131 | C2×D5 | — | 5 | 2 | no | — |
| 132 | C2×D5 | — | 10 | 2 | no | — |
| 133 | C2×D5 | — | 10 | 2 | no | — |
| 134 | C2×D5 | — | 20 | 2 | no | — |
| 135 | C2×D5 | — | 20 | 4 | no | — |
| 136 | C4×C5 | — | 2 | 2 | no | — |
| 137 | C4×C5 | — | 2 | 2 | no | — |
| 138 | C4×C5 | — | 10 | 2 | no | — |
| 139 | C4×C5 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 140 | C4×C5 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 141 | Dic5 | — | 2 | 2 | no | — |
| 142 | Dic5 | — | 5 | 2 | no | — |
| 143 | Dic5 | — | 5 | 2 | no | — |
| 144 | Dic5 | — | 10 | 2 | no | — |
| 145 | Dic5 | — | 10 | 2 | no | — |
| 146 | #41 | — | 50 | 1 | no | — |
| 147 | #41 | — | 50 | 1 | no | — |
| 148 | #41 | — | 50 | 1 | no | — |
| 149 | #41 | — | 50 | 1 | no | — |
| 150 | C2×Q8 | — | 25 | 1 | no | — |
| 151 | C2×D4 | — | 25 | 2 | no | — |
| 152 | C2×D4 | — | 25 | 1 | no | — |
| 153 | C22×C4 | — | 5 | 2 | no | — |
| 154 | C22×C4 | — | 25 | 1 | no | — |
| 155 | M16 | — | 25 | 2 | no | — |
| 156 | M16 | — | 25 | 2 | no | — |
| 157 | M16 | — | 50 | 1 | no | — |
| 158 | M16 | — | 50 | 1 | no | — |
| 159 | C2×C8 | — | 50 | 1 | no | — |
| 160 | C2×C8 | — | 50 | 1 | no | — |
| 161 | C2×C5 | — | 2 | 2 | no | — |
| 162 | C2×C5 | — | 2 | 2 | no | — |
| 163 | C2×C5 | — | 4 | 2 | no | — |
| 164 | C2×C5 | — | 20 | 2 | no | — |
| 165 | C2×C5 | C2×C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 166 | D5 | — | 4 | 2 | no | — |
| 167 | D5 | — | 10 | 4 | no | — |
| 168 | D5 | — | 10 | 2 | no | — |
| 169 | D5 | — | 10 | 2 | no | — |
| 170 | D5 | — | 20 | 2 | no | — |
| 171 | C23 | — | 5 | 2 | no | — |
| 172 | C23 | — | 25 | 1 | no | — |
| 173 | Q8 | — | 50 | 1 | no | — |
| 174 | Q8 | — | 50 | 1 | no | — |
| 175 | D4 | — | 50 | 2 | no | — |
| 176 | D4 | — | 50 | 2 | no | — |
| 177 | D4 | — | 50 | 1 | no | — |
| 178 | D4 | — | 50 | 1 | no | — |
| 179 | C2×C4 | — | 5 | 2 | no | — |
| 180 | C2×C4 | — | 10 | 2 | no | — |
| 181 | C2×C4 | — | 10 | 2 | no | — |
| 182 | C2×C4 | — | 25 | 1 | no | — |
| 183 | C2×C4 | — | 25 | 1 | no | — |
| 184 | C2×C4 | — | 25 | 1 | no | — |
| 185 | C2×C4 | — | 25 | 1 | no | — |
| 186 | C2×C4 | — | 25 | 1 | no | — |
| 187 | C2×C4 | D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 188 | C8 | — | 50 | 2 | no | — |
| 189 | C8 | — | 50 | 2 | no | — |
| 190 | C5 | — | 2 | 2 | no | — |
| 191 | C5 | #65761 | 1 | 2 | yes | complemented |
| 192 | C22 | — | 10 | 2 | no | — |
| 193 | C22 | — | 20 | 2 | no | — |
| 194 | C22 | — | 25 | 1 | no | — |
| 195 | C22 | — | 25 | 1 | no | — |
| 196 | C22 | — | 100 | 1 | no | — |
| 197 | C22 | C2×D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 198 | C4 | — | 10 | 2 | no | — |
| 199 | C4 | — | 25 | 1 | no | — |
| 200 | C4 | — | 25 | 1 | no | — |
| 201 | C4 | #87831 | 1 | 1 | yes | characteristic |
| 202 | C4 | #87831 | 1 | 1 | yes | characteristic |
| 203 | C2 | — | 2 | 1 | no | — |
| 204 | C2 | — | 20 | 2 | no | — |
| 205 | C2 | — | 50 | 1 | no | — |
| 206 | C2 | — | 50 | 1 | no | — |
| 207 | C2 | C2×#87831 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11717448 | C2 | semidirect |
| 3. #11717435 | C2 | semidirect |
| 4. #11716782 | C2 | semidirect |
| 5. #11716781 | C2 | semidirect |
| 123. C52 | #413 | semidirect |
| 191. C5 | #65761 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87873 | 2 |
| #11717448 | 2 |
| #11717435 | 2 |
| #11716782 | 2 |
| #11716781 | 2 |
| #65761 | 5 |