Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
192 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 25 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 | C5×Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10602733 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C5⋊#226 | — | 4 | 1 | no | maximal |
| 4 | #91184 | — | 4 | 2 | no | maximal |
| 5 | #85553 | — | 5 | 1 | no | maximal, Hall |
| 6 | C5×Q82 | D3 | 1 | 1 | yes | characteristic |
| 7 | #65381 | — | 3 | 1 | no | — |
| 8 | #65310 | — | 3 | 2 | no | — |
| 9 | #64766 | — | 3 | 1 | no | — |
| 10 | Q8⋊#168 | — | 3 | 2 | no | — |
| 11 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 12 | C2×C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 13 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 14 | C5⋊#226 | — | 4 | 2 | no | — |
| 15 | Q8⋊SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic |
| 16 | C5×C4⋊Q8 | — | 3 | 1 | no | — |
| 17 | C5×C4⋊Q8 | — | 6 | 1 | no | — |
| 18 | C4×C5×Q8 | — | 3 | 2 | no | — |
| 19 | C2×#689 | — | 6 | 1 | no | — |
| 20 | C2×#689 | — | 6 | 1 | no | — |
| 21 | #4015 | — | 3 | 1 | no | — |
| 22 | #3967 | — | 6 | 2 | no | — |
| 23 | #3939 | — | 6 | 2 | no | — |
| 24 | C4⋊#168 | — | 3 | 2 | no | — |
| 25 | C4×#168 | — | 3 | 1 | no | — |
| 26 | #1633 | — | 15 | 1 | no | Sylow |
| 27 | C2×Dic15 | — | 16 | 1 | no | — |
| 28 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 29 | C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 30 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 31 | C2×#226 | — | 20 | 1 | no | — |
| 32 | #859 | — | 20 | 2 | no | — |
| 33 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 34 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 35 | C2×C5×Q8 | S4 | 1 | 2 | yes | — |
| 36 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 37 | C2×Dic10 | — | 6 | 1 | no | — |
| 38 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 39 | C5×C4⋊C4 | — | 6 | 2 | no | — |
| 40 | C42×C5 | — | 3 | 1 | no | — |
| 41 | C42×C5 | — | 6 | 1 | no | — |
| 42 | #689 | — | 12 | 2 | no | — |
| 43 | #689 | — | 12 | 2 | no | — |
| 44 | C4⋊Dic5 | — | 6 | 2 | no | — |
| 45 | C2×#168 | — | 6 | 1 | no | — |
| 46 | C2×#168 | — | 6 | 2 | no | — |
| 47 | Q82 | D15 | 1 | 1 | yes | characteristic |
| 48 | #494 | — | 15 | 1 | no | — |
| 49 | #477 | — | 15 | 2 | no | — |
| 50 | #333 | — | 15 | 1 | no | — |
| 51 | Q8⋊C8 | — | 15 | 2 | no | — |
| 52 | C22×C3×C5 | — | 16 | 1 | no | — |
| 53 | Dic15 | — | 16 | 2 | no | — |
| 54 | C2×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 55 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 56 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 57 | #226 | — | 20 | 2 | no | — |
| 58 | C5×Q8 | — | 3 | 2 | no | — |
| 59 | C5×Q8 | — | 4 | 1 | no | — |
| 60 | C5×Q8 | — | 8 | 1 | no | — |
| 61 | C5×Q8 | — | 12 | 1 | no | — |
| 62 | C5×Q8 | #226 | 1 | 2 | yes | complemented |
| 63 | C2×C4×C5 | — | 3 | 2 | no | — |
| 64 | C2×C4×C5 | — | 3 | 1 | no | — |
| 65 | C2×C4×C5 | — | 6 | 1 | no | — |
| 66 | C2×Dic5 | — | 12 | 1 | no | — |
| 67 | Dic10 | — | 12 | 1 | no | — |
| 68 | Dic10 | — | 12 | 1 | no | — |
| 69 | #168 | — | 6 | 2 | no | — |
| 70 | #168 | — | 12 | 2 | no | — |
| 71 | C2×Q16 | — | 30 | 1 | no | — |
| 72 | C2×Q16 | — | 30 | 1 | no | — |
| 73 | C4⋊Q8 | — | 3 | 1 | no | — |
| 74 | C4⋊Q8 | — | 6 | 1 | no | — |
| 75 | C4⋊Q8 | — | 15 | 1 | no | — |
| 76 | C4×Q8 | — | 3 | 2 | no | — |
| 77 | #107 | — | 30 | 2 | no | — |
| 78 | C4⋊C8 | — | 15 | 2 | no | — |
| 79 | #103 | — | 30 | 2 | no | — |
| 80 | C4×C8 | — | 15 | 1 | no | — |
| 81 | C2×C3×C5 | — | 16 | 2 | no | — |
| 82 | C2×C3×C5 | — | 16 | 1 | no | — |
| 83 | C2×Dic3 | — | 80 | 1 | no | — |
| 84 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 85 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 86 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 87 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 88 | C4×C5 | — | 3 | 2 | no | — |
| 89 | C4×C5 | — | 3 | 2 | no | — |
| 90 | C4×C5 | — | 6 | 1 | no | — |
| 91 | C4×C5 | — | 12 | 1 | no | — |
| 92 | Dic5 | — | 24 | 1 | no | — |
| 93 | C2×Q8 | — | 2 | 1 | no | — |
| 94 | C2×Q8 | — | 3 | 1 | no | — |
| 95 | C2×Q8 | — | 30 | 1 | no | — |
| 96 | C2×Q8 | C5⋊S4 | 1 | 2 | yes | — |
| 97 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 98 | Q16 | — | 60 | 2 | no | — |
| 99 | Q16 | — | 60 | 2 | no | — |
| 100 | C2×C8 | — | 30 | 1 | no | — |
| 101 | C2×C8 | — | 30 | 2 | no | — |
| 102 | C4⋊C4 | — | 3 | 2 | no | — |
| 103 | C4⋊C4 | — | 6 | 2 | no | — |
| 104 | C4⋊C4 | — | 30 | 2 | no | — |
| 105 | C42 | — | 3 | 1 | no | — |
| 106 | C42 | — | 6 | 1 | no | — |
| 107 | C3×C5 | — | 16 | 1 | no | Hall |
| 108 | C22×C3 | — | 16 | 1 | no | — |
| 109 | Dic3 | — | 80 | 2 | no | — |
| 110 | C2×C5 | #5930 | 1 | 2 | yes | — |
| 111 | C2×C5 | #5935 | 1 | 1 | yes | characteristic |
| 112 | Q8 | — | 3 | 2 | no | — |
| 113 | Q8 | — | 4 | 1 | no | — |
| 114 | Q8 | — | 8 | 1 | no | — |
| 115 | Q8 | — | 12 | 1 | no | — |
| 116 | Q8 | — | 60 | 1 | no | — |
| 117 | Q8 | — | 60 | 1 | no | — |
| 118 | Q8 | C5⋊#226 | 1 | 2 | yes | complemented |
| 119 | C2×C4 | — | 3 | 2 | no | — |
| 120 | C2×C4 | — | 3 | 1 | no | — |
| 121 | C2×C4 | — | 6 | 1 | no | — |
| 122 | C2×C4 | — | 60 | 1 | no | — |
| 123 | C8 | — | 30 | 2 | no | — |
| 124 | C8 | — | 60 | 2 | no | — |
| 125 | C2×C3 | — | 16 | 2 | no | — |
| 126 | C2×C3 | — | 16 | 1 | no | — |
| 127 | C5 | #85553 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 128 | C22 | C5⋊#1020 | 1 | 1 | yes | characteristic |
| 129 | C4 | — | 3 | 2 | no | — |
| 130 | C4 | — | 3 | 2 | no | — |
| 131 | C4 | — | 6 | 1 | no | — |
| 132 | C4 | — | 12 | 1 | no | — |
| 133 | C4 | — | 120 | 1 | no | — |
| 134 | C3 | — | 16 | 1 | no | Sylow |
| 135 | C2 | C5⋊#5930 | 1 | 2 | yes | — |
| 136 | C2 | C5⋊#5935 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C5⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C5×Q8⋊SL(2,𝔽3) | C2 |
| Fitting | C5×Q82 | D3 |
| socle | C22×C5 | #1020 |
| LCS | G ▷ C5×Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 62. C5×Q8 | #226 | semidirect |
| 118. Q8 | C5⋊#226 | semidirect |
| 127. C5 | #85553 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Q8⋊SL(2,𝔽3) | 2 |
| #10602733 | 3 |
| C2×C5⋊#226 | 4 |
| #91184 | 4 |
| #85553 | 5 |