Finite Group Databasebeta

← C2×#67150 #10624212

Subgroups — C2×#67150 #10624212

Order 720.

232 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 66 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×C3⋊Dic15C211yesmaximal, characteristic, complemented
4#67150C214yesmaximal, direct product factor
5C2×#1220—31nomaximal
6C2×#1201—31nomaximal
7C22×C32×C5C2211yescharacteristic
8C2×C3×D15—22no—
9C2×C3×D15C2212yescomplemented
10C2×C3×C5×D3—22no—
11C2×C3×C5×D3C2212yescomplemented
12C3⋊Dic15C2212yescomplemented
13C2×#625—51nomaximal, Hall
14C23×C3×C5—31no—
15C22×D15D311yescharacteristic, complemented
16C22×C5×D3D311yescharacteristic, complemented
17C22×C3×D5—31no—
18#1220—34no—
19C2×Dic15—31no—
20C2×Dic15—31no—
21C2×Dic15—61no—
22#1201—34no—
23C2×C32×C5D412yescomplemented
24C2×C32×C5C2311yescharacteristic
25C3×D15—22no—
26C3×C5×D3—22no—
27C2×C22⋊D5—91noHall
28C22×C3×D3—51no—
29C22×C3×D3D511yescharacteristic, complemented
30C2×C3⋊Dic3—51no—
31#625—54no—
32C22×C3×C5—21no—
33C22×C3×C5—32no—
34C22×C3×C5—62no—
35C22×C3×C5C2×D311yescharacteristic
36C22×C3×C5C2×D311yescharacteristic
37C2×D15—22no—
38C2×D15C2×D312yescomplemented
39C2×C5×D3—22no—
40C2×C5×D3C2×D312yescomplemented
41C2×C3×D5—32no—
42C2×C3×D5—62no—
43Dic15—32no—
44Dic15—32no—
45Dic15—62no—
46C2×#67—151no—
47C2×#67—151no—
48C32×C5C2×D411yescharacteristic, complemented, Hall
49C23×C5—31no—
50C22×D5—31no—
51C22⋊D5—94no—
52C2×Dic5—91no—
53C22×C32C2×D511yescharacteristic
54C2×C3×D3—22no—
55C2×C3×D3—52no—
56C2×C3×D3—102no—
57C2×C3×D3C2×D512yescomplemented
58C3⋊Dic3—52no—
59C2×C3×C5—22no—
60C2×C3×C5—21no—
61C2×C3×C5—62no—
62C2×C3×C5#6712yescomplemented
63C2×C3×C5#6712yescomplemented
64C2×C3×C5C22×D311yescharacteristic
65C2×C3×C5C22×D311yescharacteristic
66D15—22no—
67C3×D5—62no—
68C5×D3—22no—
69C23×C3—31no—
70C23×C3—151no—
71C22×D3—51no—
72C22×D3D1511yescharacteristic, complemented
73#67—154no—
74#67—154no—
75C2×Dic3—151no—
76C2×Dic3—151no—
77C2×Dic3—301no—
78C22×C5—32no—
79C22×C5—62no—
80C22×C5D3211yescharacteristic
81C2×D5—32no—
82C2×D5—62no—
83Dic5—92no—
84C2×C32C22⋊D512yescomplemented
85C2×C32C22×D511yescharacteristic
86C3×D3—22no—
87C3×D3—102no—
88C2×D4—451noSylow
89C3×C5—21no—
90C3×C5C2×#6711yescharacteristic, complemented
91C3×C5C2×#6711yescharacteristic, complemented
92C22×C3—21no—
93C22×C3—32no—
94C22×C3—62no—
95C22×C3—152no—
96C22×C3—302no—
97C22×C3D3×D511yescharacteristic
98C22×C3C2×D1511yescharacteristic
99C2×D3—22no—
100C2×D3—52no—
101C2×D3—102no—
102C2×D3C2×D1512yescomplemented
103Dic3—152no—
104Dic3—152no—
105Dic3—302no—
106C2×C5—62no—
107C2×C5#62512yescomplemented
108C2×C5C2×D3211yescharacteristic
109D5—62no—
110C32C2×C22⋊D511yescharacteristic, complemented, Sylow
111C23—31no—
112C23—151no—
113D4—454no—
114C2×C4—451no—
115C2×C3—22no—
116C2×C3—21no—
117C2×C3—62no—
118C2×C3—302no—
119C2×C3#120112yescomplemented
120C2×C3#122012yescomplemented
121C2×C3C2×D3×D511yescharacteristic
122C2×C3C22×D1511yescharacteristic
123D3—22no—
124D3—102no—
125C5C2×#62511yescharacteristic, complemented, Sylow
126C22—32no—
127C22—62no—
128C22—152no—
129C22—302no—
130C22D3×D1511yescharacteristic
131C4—452no—
132C3—21no—
133C3C2×#120111yescharacteristic, complemented
134C3C2×#122011yescharacteristic, complemented
135C2—62no—
136C2—302no—
137C2#6715012yesdirect product factor
138C2C2×D3×D1511yescharacteristic
subgroupG/H
center, hypercenterC22D3×D15
commutator, cosocleC2×C32×C5C23
FrattiniC2C2×D3×D15
Fitting, socleC22×C32×C5C22
hypocenterC32×C5C2×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

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×C3⋊Dic15C2semidirect
4. #67150C2direct (N × H)
9. C2×C3×D15C22semidirect
11. C2×C3×C5×D3C22semidirect
12. C3⋊Dic15C22semidirect
15. C22×D15D3semidirect
16. C22×C5×D3D3semidirect
23. C2×C32×C5D4semidirect
29. C22×C3×D3D5semidirect
38. C2×D15C2×D3semidirect
40. C2×C5×D3C2×D3semidirect
48. C32×C5C2×D4semidirect
57. C2×C3×D3C2×D5semidirect
62. C2×C3×C5#67semidirect
63. C2×C3×C5#67semidirect
72. C22×D3D15semidirect
84. C2×C32C22⋊D5semidirect
90. C3×C5C2×#67semidirect
91. C3×C5C2×#67semidirect
102. C2×D3C2×D15semidirect
107. C2×C5#625semidirect
110. C32C2×C22⋊D5semidirect
119. C2×C3#1201semidirect
120. C2×C3#1220semidirect
125. C5C2×#625semidirect
133. C3C2×#1201semidirect
134. C3C2×#1220semidirect
137. C2#67150direct (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×C3⋊Dic152
#671502
C2×#12203
C2×#12013
C2×#6255