Finite Group Databasebeta

← C2×C3×#1203 #10624142

Subgroups — C2×C3×#1203 #10624142

Order 720.

232 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 80 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×D15C211yesmaximal, characteristic, complemented
2C22×C3×C5×D3C211yesmaximal, characteristic, complemented
3C2×C32×Dic5C211yesmaximal, characteristic, complemented
4C3×#1203C214yesmaximal, direct product factor
5C2×C3×C22⋊D5—31nomaximal
6C2×#1203C311yesmaximal, characteristic, direct product factor
7C22×C32×C5C2211yescharacteristic
8C2×C3×D15—22no—
9C2×C3×D15C2212yescomplemented
10C2×C3×C5×D3—22no—
11C2×C3×C5×D3C2212yescomplemented
12C32×Dic5C2212yescomplemented
13C2×C3×D12—51nomaximal, Hall
14C23×C3×C5—31no—
15C22×D15C2×C311yescharacteristic, complemented
16C22×C5×D3C2×C311yescharacteristic, complemented
17C22×C3×D5—31no—
18C3×C22⋊D5—34no—
19C2×C3×Dic5—21no—
20C2×C3×Dic5D311yescharacteristic, complemented
21C2×C3×Dic5C2×C311yescharacteristic, complemented
22#1203C2×C314yesdirect product factor
23C2×C32×C5D412yescomplemented
24C2×C32×C5C2311yescharacteristic
25C3×D15—22no—
26C3×C5×D3—22no—
27C2×C22⋊D5—31noHall
28C22×C3×D3—51no—
29C22×C3×D3D511yescharacteristic, complemented
30C2×C32×C4—51no—
31C3×D12—54no—
32C22×C3×C5—21no—
33C22×C3×C5—32no—
34C22×C3×C5—62no—
35C22×C3×C5C2×D311yescharacteristic
36C22×C3×C5C22×C311yescharacteristic
37C2×D15—22no—
38C2×D15C22×C312yescomplemented
39C2×C5×D3—22no—
40C2×C5×D3C22×C312yescomplemented
41C2×C3×D5—32no—
42C2×C3×D5—62no—
43C3×Dic5—22no—
44C3×Dic5C2×D312yescomplemented
45C3×Dic5C22×C312yescomplemented
46C2×C3×D4—151no—
47C2×D12—51no—
48C32×C5C2×D411yescharacteristic, complemented, Hall
49C23×C5—31no—
50C22×D5—31no—
51C22⋊D5—34no—
52C2×Dic5C3×D311yescharacteristic, complemented
53C22×C32C2×D511yescharacteristic
54C2×C3×D3—22no—
55C2×C3×D3—52no—
56C2×C3×D3—102no—
57C2×C3×D3C2×D512yescomplemented
58C32×C4—52no—
59C2×C3×C5—22no—
60C2×C3×C5—21no—
61C2×C3×C5—62no—
62C2×C3×C5D1212yescomplemented
63C2×C3×C5C3×D412yescomplemented
64C2×C3×C5C22×D311yescharacteristic
65C2×C3×C5C23×C311yescharacteristic
66D15—22no—
67C3×D5—62no—
68C5×D3—22no—
69C23×C3—31no—
70C23×C3—151no—
71C22×D3—51no—
72C22×D3C3×D511yescharacteristic, complemented
73C3×D4—154no—
74C2×C3×C4—51no—
75C2×C3×C4—51no—
76C2×C3×C4—101no—
77D12—54no—
78C22×C5—32no—
79C22×C5—62no—
80C22×C5C2×C3×D311yescharacteristic
81C2×D5—32no—
82C2×D5—62no—
83Dic5C2×C3×D312yescomplemented
84C2×C32C22⋊D512yescomplemented
85C2×C32C22×D511yescharacteristic
86C3×D3—22no—
87C3×D3—102no—
88C2×D4—151noSylow
89C3×C5—21no—
90C3×C5C2×D1211yescharacteristic, complemented
91C3×C5C2×C3×D411yescharacteristic, complemented
92C22×C3—21no—
93C22×C3—32no—
94C22×C3—62no—
95C22×C3—152no—
96C22×C3—302no—
97C22×C3D3×D511yescharacteristic
98C22×C3C2×C3×D511yescharacteristic
99C2×D3—22no—
100C2×D3—52no—
101C2×D3—102no—
102C2×D3C2×C3×D512yescomplemented
103C3×C4—52no—
104C3×C4—52no—
105C3×C4—102no—
106C2×C5—62no—
107C2×C5C3×D1212yescomplemented
108C2×C5C22×C3×D311yescharacteristic
109D5—62no—
110C32C2×C22⋊D511yescharacteristic, complemented, Sylow
111C23—31no—
112C23—151no—
113D4—154no—
114C2×C4—51no—
115C2×C3—22no—
116C2×C3—21no—
117C2×C3—62no—
118C2×C3—302no—
119C2×C3#120312yesdirect product factor
120C2×C3C3×C22⋊D512yescomplemented
121C2×C3C2×D3×D511yescharacteristic
122C2×C3C22×C3×D511yescharacteristic
123D3—22no—
124D3—102no—
125C5C2×C3×D1211yescharacteristic, complemented, Sylow
126C22—32no—
127C22—62no—
128C22—152no—
129C22—302no—
130C22C3×D3×D511yescharacteristic
131C4—52no—
132C3—21no—
133C3C2×#120311yescharacteristic, direct product factor
134C3C2×C3×C22⋊D511yescharacteristic, complemented
135C2—62no—
136C2—302no—
137C2C3×#120312yesdirect product factor
138C2C2×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×D5
commutator, cosocleC2×C3×C5C23×C3
FrattiniC2C2×C3×D3×D5
Fitting, socleC22×C32×C5C22
hypocenterC3×C5C2×C3×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×D15C2semidirect
2. C22×C3×C5×D3C2semidirect
3. C2×C32×Dic5C2semidirect
4. C3×#1203C2direct (N × H)
6. C2×#1203C3direct (N × H)
9. C2×C3×D15C22semidirect
11. C2×C3×C5×D3C22semidirect
12. C32×Dic5C22semidirect
15. C22×D15C2×C3semidirect
16. C22×C5×D3C2×C3semidirect
20. C2×C3×Dic5D3semidirect
21. C2×C3×Dic5C2×C3semidirect
22. #1203C2×C3direct (N × H)
23. C2×C32×C5D4semidirect
29. C22×C3×D3D5semidirect
38. C2×D15C22×C3semidirect
40. C2×C5×D3C22×C3semidirect
44. C3×Dic5C2×D3semidirect
45. C3×Dic5C22×C3semidirect
48. C32×C5C2×D4semidirect
52. C2×Dic5C3×D3semidirect
57. C2×C3×D3C2×D5semidirect
62. C2×C3×C5D12semidirect
63. C2×C3×C5C3×D4semidirect
72. C22×D3C3×D5semidirect
83. Dic5C2×C3×D3semidirect
84. C2×C32C22⋊D5semidirect
90. C3×C5C2×D12semidirect
91. C3×C5C2×C3×D4semidirect
102. C2×D3C2×C3×D5semidirect
107. C2×C5C3×D12semidirect
110. C32C2×C22⋊D5semidirect
119. C2×C3#1203direct (N × H)
120. C2×C3C3×C22⋊D5semidirect
125. C5C2×C3×D12semidirect
133. C3C2×#1203direct (N × H)
134. C3C2×C3×C22⋊D5semidirect
137. C2C3×#1203direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D152
C22×C3×C5×D32
C2×C32×Dic52
C3×#12032
C2×C3×C22⋊D53
C2×#12033
C2×C3×D125