Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
704 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 136 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×#87904 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#87872 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#4090 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11717016 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #11716992 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11716984 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11716982 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×C52×Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | #87904 | — | 2 | 4 | no | — |
| 10 | C2×C4⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4⋊(C5⋊D5) | C22 | 1 | 2 | yes | — |
| 12 | C2×C4×C5⋊D5 | C22 | 1 | 2 | yes | — |
| 13 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C5×Dic10 | C22 | 1 | 2 | yes | — |
| 15 | C2×#6039 | C22 | 1 | 2 | yes | — |
| 16 | C2×#6036 | C22 | 1 | 2 | yes | — |
| 17 | #87872 | — | 2 | 4 | no | — |
| 18 | C5×#683 | C22 | 1 | 8 | yes | — |
| 19 | C4×C5×Dic5 | C22 | 1 | 2 | yes | — |
| 20 | #87781 | C22 | 1 | 2 | yes | — |
| 21 | #87779 | C22 | 1 | 8 | yes | — |
| 22 | #87779 | C22 | 1 | 2 | yes | — |
| 23 | #66058 | — | 5 | 2 | no | maximal |
| 24 | C22×C5⋊D5 | C23 | 1 | 2 | yes | — |
| 25 | C22×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 26 | C52×Q8 | — | 2 | 2 | no | — |
| 27 | C2×C4×C52 | C23 | 1 | 2 | yes | — |
| 28 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 30 | C4⋊(C5⋊D5) | — | 2 | 4 | no | — |
| 31 | C4⋊(C5⋊D5) | — | 2 | 2 | no | — |
| 32 | C4×C5⋊D5 | — | 2 | 4 | no | — |
| 33 | C4×C5⋊D5 | D4 | 1 | 4 | yes | — |
| 34 | C2×C5×Dic5 | C23 | 1 | 4 | yes | — |
| 35 | C2×C5×Dic5 | C23 | 1 | 4 | yes | — |
| 36 | C5×Dic10 | — | 2 | 4 | no | — |
| 37 | #6039 | — | 2 | 4 | no | — |
| 38 | #6036 | — | 2 | 4 | no | — |
| 39 | C2×#713 | — | 5 | 2 | no | — |
| 40 | C2×#713 | — | 10 | 2 | no | — |
| 41 | C2×Q8×D5 | — | 5 | 2 | no | — |
| 42 | C5×C4⋊Q8 | — | 5 | 2 | no | — |
| 43 | #4093 | — | 5 | 2 | no | — |
| 44 | #4090 | D5 | 1 | 2 | yes | complemented |
| 45 | #4042 | — | 5 | 8 | no | — |
| 46 | #4040 | — | 5 | 8 | no | — |
| 47 | #4021 | — | 5 | 2 | no | — |
| 48 | #4018 | — | 5 | 2 | no | — |
| 49 | C22×C52 | C24 | 1 | 1 | yes | characteristic |
| 50 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 51 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 52 | C2×C5⋊D5 | — | 2 | 4 | no | — |
| 53 | C2×C5⋊D5 | C2×D4 | 1 | 2 | yes | — |
| 54 | C4×C52 | — | 2 | 2 | no | — |
| 55 | C4×C52 | C2×D4 | 1 | 2 | yes | — |
| 56 | #1039 | C2×D4 | 1 | 2 | yes | — |
| 57 | C5×Dic5 | — | 2 | 4 | no | — |
| 58 | C5×Dic5 | — | 2 | 4 | no | — |
| 59 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 60 | C2×C5×Q8 | — | 5 | 2 | no | — |
| 61 | C2×C5×Q8 | C2×D5 | 1 | 2 | yes | — |
| 62 | #713 | — | 10 | 8 | no | — |
| 63 | #713 | — | 10 | 16 | no | — |
| 64 | Q8×D5 | — | 10 | 8 | no | — |
| 65 | C2×D20 | — | 5 | 2 | no | — |
| 66 | C2×D20 | — | 5 | 4 | no | — |
| 67 | C2×D20 | — | 10 | 4 | no | — |
| 68 | C2×D20 | — | 10 | 2 | no | — |
| 69 | C2×C4×D5 | — | 5 | 2 | no | — |
| 70 | C2×C4×D5 | — | 5 | 4 | no | — |
| 71 | C2×C4×D5 | — | 5 | 4 | no | — |
| 72 | C2×C4×D5 | — | 10 | 2 | no | — |
| 73 | C2×C4×D5 | — | 10 | 4 | no | — |
| 74 | C2×Dic10 | — | 5 | 4 | no | — |
| 75 | C2×Dic10 | C2×D5 | 1 | 2 | yes | — |
| 76 | C5×C4⋊C4 | — | 5 | 8 | no | — |
| 77 | C42×C5 | — | 5 | 2 | no | — |
| 78 | #685 | — | 5 | 4 | no | — |
| 79 | #685 | — | 5 | 8 | no | — |
| 80 | #685 | — | 5 | 8 | no | — |
| 81 | #683 | — | 5 | 4 | no | — |
| 82 | #683 | C2×D5 | 1 | 8 | yes | — |
| 83 | C4×Dic5 | C2×D5 | 1 | 2 | yes | — |
| 84 | #536 | — | 25 | 1 | no | Sylow |
| 85 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C52 | 2−1+4 | 1 | 2 | yes | — |
| 87 | C5⋊D5 | — | 2 | 2 | no | — |
| 88 | C5⋊D5 | — | 4 | 2 | no | — |
| 89 | C22×D5 | — | 5 | 2 | no | — |
| 90 | C22×D5 | — | 5 | 4 | no | — |
| 91 | C22×D5 | — | 10 | 2 | no | — |
| 92 | C22×D5 | — | 10 | 4 | no | — |
| 93 | C5×Q8 | — | 2 | 8 | no | — |
| 94 | C5×Q8 | — | 2 | 4 | no | — |
| 95 | C5×Q8 | — | 10 | 4 | no | — |
| 96 | C2×C4×C5 | — | 2 | 4 | no | — |
| 97 | C2×C4×C5 | — | 2 | 2 | no | — |
| 98 | C2×C4×C5 | — | 5 | 4 | no | — |
| 99 | C2×C4×C5 | — | 5 | 4 | no | — |
| 100 | C2×C4×C5 | C22×D5 | 1 | 4 | yes | — |
| 101 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 102 | C2×Dic5 | — | 5 | 2 | no | — |
| 103 | C2×Dic5 | — | 10 | 2 | no | — |
| 104 | C2×Dic5 | C22×D5 | 1 | 4 | yes | — |
| 105 | C2×Dic5 | C22×D5 | 1 | 4 | yes | — |
| 106 | D20 | — | 10 | 8 | no | — |
| 107 | D20 | — | 10 | 16 | no | — |
| 108 | D20 | — | 10 | 8 | no | — |
| 109 | D20 | — | 10 | 4 | no | — |
| 110 | C4×D5 | — | 5 | 8 | no | — |
| 111 | C4×D5 | — | 10 | 8 | no | — |
| 112 | C4×D5 | — | 10 | 8 | no | — |
| 113 | C4×D5 | — | 10 | 8 | no | — |
| 114 | C4×D5 | — | 10 | 16 | no | — |
| 115 | Dic10 | — | 2 | 4 | no | — |
| 116 | Dic10 | — | 10 | 8 | no | — |
| 117 | C2×#41 | — | 25 | 1 | no | — |
| 118 | C22×Q8 | — | 25 | 1 | no | — |
| 119 | C4⋊Q8 | — | 5 | 2 | no | — |
| 120 | #124 | — | 25 | 2 | no | — |
| 121 | #123 | — | 25 | 4 | no | — |
| 122 | C22⋊Q8 | — | 25 | 4 | no | — |
| 123 | #117 | — | 25 | 1 | no | — |
| 124 | C52 | #536 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 125 | C22×C5 | — | 2 | 2 | no | — |
| 126 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 127 | C2×D5 | — | 5 | 4 | no | — |
| 128 | C2×D5 | — | 10 | 4 | no | — |
| 129 | C2×D5 | — | 10 | 4 | no | — |
| 130 | C2×D5 | — | 10 | 8 | no | — |
| 131 | C2×D5 | — | 10 | 4 | no | — |
| 132 | C2×D5 | — | 10 | 8 | no | — |
| 133 | C2×D5 | — | 20 | 4 | no | — |
| 134 | C2×D5 | — | 20 | 8 | no | — |
| 135 | C4×C5 | — | 2 | 8 | no | — |
| 136 | C4×C5 | — | 2 | 4 | no | — |
| 137 | C4×C5 | — | 2 | 4 | no | — |
| 138 | C4×C5 | — | 10 | 4 | no | — |
| 139 | C4×C5 | — | 10 | 4 | no | — |
| 140 | C4×C5 | D4×D5 | 1 | 4 | yes | — |
| 141 | Dic5 | — | 2 | 4 | no | — |
| 142 | Dic5 | — | 2 | 4 | no | — |
| 143 | Dic5 | — | 5 | 4 | no | — |
| 144 | Dic5 | — | 10 | 4 | no | — |
| 145 | #41 | — | 50 | 4 | no | — |
| 146 | C2×Q8 | — | 5 | 2 | no | — |
| 147 | C2×Q8 | — | 25 | 2 | no | — |
| 148 | C2×Q8 | — | 50 | 4 | no | — |
| 149 | C2×Q8 | D52 | 1 | 1 | yes | characteristic |
| 150 | C2×D4 | — | 25 | 2 | no | — |
| 151 | C2×D4 | — | 25 | 1 | no | — |
| 152 | C22×C4 | — | 25 | 2 | no | — |
| 153 | C22×C4 | — | 25 | 1 | no | — |
| 154 | C22×C4 | — | 25 | 2 | no | — |
| 155 | C4⋊C4 | — | 5 | 8 | no | — |
| 156 | C4⋊C4 | — | 25 | 2 | no | — |
| 157 | C22⋊C4 | — | 25 | 2 | no | — |
| 158 | C22⋊C4 | — | 25 | 8 | no | — |
| 159 | C42 | — | 5 | 2 | no | — |
| 160 | C2×C5 | — | 2 | 4 | no | — |
| 161 | C2×C5 | — | 2 | 2 | no | — |
| 162 | C2×C5 | C2×D4×D5 | 1 | 2 | yes | — |
| 163 | C2×C5 | #4147 | 1 | 4 | yes | — |
| 164 | D5 | — | 10 | 4 | no | — |
| 165 | D5 | — | 20 | 4 | no | — |
| 166 | D5 | — | 20 | 8 | no | — |
| 167 | D5 | — | 20 | 4 | no | — |
| 168 | C23 | — | 25 | 1 | no | — |
| 169 | C23 | — | 25 | 2 | no | — |
| 170 | Q8 | — | 2 | 2 | no | — |
| 171 | Q8 | — | 10 | 4 | no | — |
| 172 | Q8 | — | 50 | 4 | no | — |
| 173 | D4 | — | 50 | 4 | no | — |
| 174 | D4 | — | 50 | 2 | no | — |
| 175 | C2×C4 | — | 5 | 4 | no | — |
| 176 | C2×C4 | — | 5 | 4 | no | — |
| 177 | C2×C4 | — | 25 | 1 | no | — |
| 178 | C2×C4 | — | 25 | 4 | no | — |
| 179 | C2×C4 | — | 50 | 4 | no | — |
| 180 | C2×C4 | — | 50 | 4 | no | — |
| 181 | C2×C4 | C2×D52 | 1 | 2 | yes | — |
| 182 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 183 | C5 | — | 2 | 2 | no | — |
| 184 | C5 | #66058 | 1 | 2 | yes | complemented |
| 185 | C22 | — | 25 | 2 | no | — |
| 186 | C22 | — | 50 | 2 | no | — |
| 187 | C22 | — | 50 | 4 | no | — |
| 188 | C22 | — | 50 | 2 | no | — |
| 189 | C22 | C22×D52 | 1 | 1 | yes | characteristic |
| 190 | C4 | — | 2 | 2 | no | — |
| 191 | C4 | — | 10 | 4 | no | — |
| 192 | C4 | — | 10 | 4 | no | — |
| 193 | C4 | — | 25 | 2 | no | — |
| 194 | C4 | #87886 | 1 | 2 | yes | — |
| 195 | C2 | — | 50 | 2 | no | — |
| 196 | C2 | — | 100 | 2 | no | — |
| 197 | C2 | #11717481 | 1 | 2 | yes | — |
| 198 | C2 | C2×#87886 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C2×C52×Q8 | C22 |
| prime radical | C2×#87904 | C2 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | #536 |
| 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 |
|---|---|---|
| 2. C2×#87872 | C2 | semidirect |
| 3. C5×#4090 | C2 | semidirect |
| 4. #11717016 | C2 | semidirect |
| 5. #11716992 | C2 | semidirect |
| 6. #11716984 | C2 | semidirect |
| 7. #11716982 | C2 | semidirect |
| 44. #4090 | D5 | semidirect |
| 124. C52 | #536 | semidirect |
| 184. C5 | #66058 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87904 | 2 |
| C2×#87872 | 2 |
| C5×#4090 | 2 |
| #11717016 | 2 |
| #11716992 | 2 |
| #11716984 | 2 |
| #11716982 | 2 |
| #66058 | 5 |