Finite Group Databasebeta

← C3×C5×(C2×D3)⋊C4 #10623860

Subgroups — C3×C5×(C2×D3)⋊C4 #10623860

Order 720.

152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×D3C211yesmaximal, characteristic
2C2×C32×C4×C5C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic3C211yesmaximal, characteristic, complemented
4C3×C5×C22⋊C4—31nomaximal
5C5×(C2×D3)⋊C4C311yesmaximal, characteristic, direct product factor
6C22×C32×C5C2211yescharacteristic
7C2×C3×C5×D3—21no—
8C2×C3×C5×D3—21no—
9C2×C3×C5×D3C412yescomplemented
10C32×C4×C5—21no—
11C3×C5×Dic3—21no—
12C3×(C2×D3)⋊C4C511yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C5—31no—
14C22×C5×D3C2×C311yescharacteristic
15C2×C3×C4×C5—21no—
16C2×C3×C4×C5—31no—
17C2×C3×C4×C5D311yescharacteristic, complemented
18C2×C3×C4×C5C2×C311yescharacteristic, complemented
19C2×C5×Dic3C2×C311yescharacteristic, complemented
20C2×C32×C5C2×C411yescharacteristic
21C2×C32×C5D411yescharacteristic
22C2×C32×C5D411yescharacteristic
23C3×C5×D3—22no—
24C5×C22⋊C4—31noHall
25C22×C3×D3C2×C511yescharacteristic
26C2×C32×C4C2×C511yescharacteristic, complemented
27C2×C3×Dic3C2×C511yescharacteristic, complemented
28C22×C3×C5—21no—
29C22×C3×C5—32no—
30C22×C3×C5—61no—
31C22×C3×C5—61no—
32C22×C3×C5C2×D311yescharacteristic
33C22×C3×C5C22×C311yescharacteristic
34C2×C5×D3—21no—
35C2×C5×D3—21no—
36C2×C5×D3C3×C412yescomplemented
37C3×C4×C5—22no—
38C3×C4×C5—21no—
39C3×C4×C5—21no—
40C3×C4×C5—61no—
41C5×Dic3—21no—
42C3×C22⋊C4—31no—
43(C2×D3)⋊C4C3×C511yescharacteristic, direct product factor
44C32×C5C22⋊C411yescharacteristic, complemented, Hall
45C23×C5—31no—
46C2×C4×C5—31no—
47C2×C4×C5C3×D311yescharacteristic, complemented
48C22×C32C22×C511yescharacteristic
49C2×C3×D3—21no—
50C2×C3×D3—21no—
51C2×C3×D3C4×C512yescomplemented
52C32×C4—21no—
53C3×Dic3—21no—
54C2×C3×C5—21no—
55C2×C3×C5—21no—
56C2×C3×C5—21no—
57C2×C3×C5—62no—
58C2×C3×C5C4×D311yescharacteristic
59C2×C3×C5D1211yescharacteristic
60C2×C3×C5#6711yescharacteristic
61C2×C3×C5C2×C3×C411yescharacteristic
62C2×C3×C5C3×D411yescharacteristic
63C2×C3×C5C3×D411yescharacteristic
64C5×D3—22no—
65C23×C3—31no—
66C22×D3C2×C3×C511yescharacteristic
67C2×C3×C4—21no—
68C2×C3×C4—31no—
69C2×C3×C4C5×D311yescharacteristic, complemented
70C2×C3×C4C2×C3×C511yescharacteristic, complemented
71C2×Dic3C2×C3×C511yescharacteristic, complemented
72C22×C5—32no—
73C22×C5—61no—
74C22×C5—61no—
75C22×C5C2×C3×D311yescharacteristic
76C4×C5—21no—
77C4×C5—61no—
78C2×C32C2×C4×C511yescharacteristic
79C2×C32C5×D411yescharacteristic
80C2×C32C5×D411yescharacteristic
81C3×D3—22no—
82C22⋊C4—31noSylow
83C3×C5—21no—
84C3×C5(C2×D3)⋊C411yescharacteristic, direct product factor
85C3×C5C3×C22⋊C411yescharacteristic, complemented
86C22×C3—21no—
87C22×C3—32no—
88C22×C3—61no—
89C22×C3—61no—
90C22×C3C2×C5×D311yescharacteristic
91C22×C3C22×C3×C511yescharacteristic
92C2×D3—21no—
93C2×D3—21no—
94C2×D3C3×C4×C512yescomplemented
95C3×C4—22no—
96C3×C4—21no—
97C3×C4—21no—
98C3×C4—61no—
99Dic3—21no—
100C2×C5—62no—
101C2×C5C3×C4×D311yescharacteristic
102C2×C5C3×D1211yescharacteristic
103C2×C5C3×#6711yescharacteristic
104C32C5×C22⋊C411yescharacteristic, complemented, Sylow
105C23—31no—
106C2×C4—31no—
107C2×C4C3×C5×D311yescharacteristic, complemented
108C2×C3—21no—
109C2×C3—21no—
110C2×C3—21no—
111C2×C3—62no—
112C2×C3C4×C5×D311yescharacteristic
113C2×C3C5×D1211yescharacteristic
114C2×C3C5×#6711yescharacteristic
115C2×C3C2×C3×C4×C511yescharacteristic
116C2×C3C3×C5×D411yescharacteristic
117C2×C3C3×C5×D411yescharacteristic
118D3—22no—
119C5C3×(C2×D3)⋊C411yescharacteristic, direct product factor, Sylow
120C22—32no—
121C22—61no—
122C22—61no—
123C22C2×C3×C5×D311yescharacteristic
124C4—21no—
125C4—61no—
126C3—21no—
127C3C5×(C2×D3)⋊C411yescharacteristic, direct product factor
128C3C3×C5×C22⋊C411yescharacteristic, complemented
129C2—62no—
130C2C3×C4×C5×D311yescharacteristic
131C2C3×C5×D1211yescharacteristic
132C2C3×C5×#6711yescharacteristic
subgroupG/H
centerC22×C3×C5C2×D3
commutatorC2×C3C2×C3×C4×C5
FrattiniC22C2×C3×C5×D3
FittingC2×C32×C4×C5C2
socleC22×C32×C5C22
prime radicalC22×C3×C5×D3C2
cosocleC22×C3C22×C3×C5
hypercenterC2×C3×C4×C5D3
hypocenterC3C3×C5×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C5 ◁ C2×C3×C4×C5

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C32×C4×C5C2semidirect
3. C2×C3×C5×Dic3C2semidirect
5. C5×(C2×D3)⋊C4C3direct (N × H)
9. C2×C3×C5×D3C4semidirect
12. C3×(C2×D3)⋊C4C5direct (N × H)
17. C2×C3×C4×C5D3semidirect
18. C2×C3×C4×C5C2×C3semidirect
19. C2×C5×Dic3C2×C3semidirect
26. C2×C32×C4C2×C5semidirect
27. C2×C3×Dic3C2×C5semidirect
36. C2×C5×D3C3×C4semidirect
43. (C2×D3)⋊C4C3×C5direct (N × H)
44. C32×C5C22⋊C4semidirect
47. C2×C4×C5C3×D3semidirect
51. C2×C3×D3C4×C5semidirect
69. C2×C3×C4C5×D3semidirect
70. C2×C3×C4C2×C3×C5semidirect
71. C2×Dic3C2×C3×C5semidirect
84. C3×C5(C2×D3)⋊C4direct (N × H)
85. C3×C5C3×C22⋊C4semidirect
94. C2×D3C3×C4×C5semidirect
104. C32C5×C22⋊C4semidirect
107. C2×C4C3×C5×D3semidirect
119. C5C3×(C2×D3)⋊C4direct (N × H)
127. C3C5×(C2×D3)⋊C4direct (N × H)
128. C3C3×C5×C22⋊C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C5×D32
C2×C32×C4×C52
C2×C3×C5×Dic32
C3×C5×C22⋊C43
C5×(C2×D3)⋊C43
C3×(C2×D3)⋊C45