Finite Group Databasebeta

← C2×C5×#1202 #11925587

Subgroups — C2×C5×#1202 #11925587

Order 1200.

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

normal subgroups only

This group has 248 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×C5×D15C211yesmaximal, characteristic, complemented
2C22×C3×C5×D5C211yesmaximal, characteristic, complemented
3C2×C52×Dic3C211yesmaximal, characteristic, complemented
4C5×#1202C214yesmaximal, direct product factor
5C2×C5×D20—31nomaximal, Hall
6C22×C3×C52C2211yescharacteristic
7C2×C5×D15—22no—
8C2×C5×D15C2212yescomplemented
9C2×C3×C5×D5—22no—
10C2×C3×C5×D5C2212yescomplemented
11C52×Dic3C2212yescomplemented
12C2×C5×#67—51nomaximal
13C2×#1202C511yesmaximal, characteristic, direct product factor
14C22×C5×D5—31no—
15C22×C5×D5D311yescharacteristic, complemented
16C2×C4×C52—31no—
17C5×D20—34no—
18C2×C3×C52D412yescomplemented
19C2×C3×C52C2311yescharacteristic
20C5×D15—22no—
21C3×C5×D5—22no—
22C23×C3×C5—51no—
23C22×D15C2×C511yescharacteristic, complemented
24C22×C5×D3—51no—
25C22×C3×D5C2×C511yescharacteristic, complemented
26C5×#67—54no—
27C2×C5×Dic3—22no—
28C2×C5×Dic3D511yescharacteristic, complemented
29C2×C5×Dic3C2×C511yescharacteristic, complemented
30#1202C2×C514yesdirect product factor
31C22×C52C2×D311yescharacteristic
32C2×C5×D5—22no—
33C2×C5×D5—32no—
34C2×C5×D5—62no—
35C2×C5×D5C2×D312yescomplemented
36C4×C52—32no—
37C2×C5×D4—151no—
38C2×D20—31no—
39C3×C52C2×D411yescharacteristic, complemented, Hall
40C22×C3×C5—22no—
41C22×C3×C5—52no—
42C22×C3×C5—102no—
43C22×C3×C5C2×D511yescharacteristic
44C22×C3×C5C22×C511yescharacteristic
45C2×D15—22no—
46C2×D15C22×C512yescomplemented
47C2×C5×D3—52no—
48C2×C5×D3—102no—
49C2×C3×D5—22no—
50C2×C3×D5C22×C512yescomplemented
51C5×Dic3—24no—
52C5×Dic3C2×D512yescomplemented
53C5×Dic3C22×C512yescomplemented
54C2×C52#6712yescomplemented
55C2×C52C22×D311yescharacteristic
56C5×D5—22no—
57C5×D5—62no—
58C2×#67—51noHall
59C23×C5—51no—
60C23×C5—151no—
61C22×D5—31no—
62C22×D5C5×D311yescharacteristic, complemented
63C5×D4—154no—
64C2×C4×C5—31no—
65C2×C4×C5—31no—
66C2×C4×C5—62no—
67D20—34no—
68C2×C3×C5—24no—
69C2×C3×C5—22no—
70C2×C3×C5—102no—
71C2×C3×C5D2012yescomplemented
72C2×C3×C5C5×D412yescomplemented
73C2×C3×C5C22×D511yescharacteristic
74C2×C3×C5C23×C511yescharacteristic
75D15—22no—
76C3×D5—22no—
77C5×D3—102no—
78C52C2×#6711yescharacteristic, complemented, Sylow
79C23×C3—51no—
80C22×D3—51no—
81#67—54no—
82C2×Dic3C5×D511yescharacteristic, complemented
83C22×C5—22no—
84C22×C5—52no—
85C22×C5—102no—
86C22×C5—152no—
87C22×C5—302no—
88C22×C5D3×D511yescharacteristic
89C22×C5C2×C5×D311yescharacteristic
90C2×D5—22no—
91C2×D5—32no—
92C2×D5—62no—
93C2×D5C2×C5×D312yescomplemented
94C4×C5—32no—
95C4×C5—32no—
96C4×C5—64no—
97C2×D4—151noSylow
98C3×C5—22no—
99C3×C5C2×D2011yescharacteristic, complemented
100C3×C5C2×C5×D411yescharacteristic, complemented
101C22×C3—52no—
102C22×C3—102no—
103C22×C3C2×C5×D511yescharacteristic
104C2×D3—52no—
105C2×D3—102no—
106Dic3C2×C5×D512yescomplemented
107C2×C5—22no—
108C2×C5—24no—
109C2×C5—102no—
110C2×C5—302no—
111C2×C5#120212yesdirect product factor
112C2×C5C5×#6712yescomplemented
113C2×C5C2×D3×D511yescharacteristic
114C2×C5C22×C5×D311yescharacteristic
115D5—22no—
116D5—62no—
117C23—51no—
118C23—151no—
119D4—154no—
120C2×C4—31no—
121C2×C3—102no—
122C2×C3C5×D2012yescomplemented
123C2×C3C22×C5×D511yescharacteristic
124D3—102no—
125C5—22no—
126C5C2×#120211yescharacteristic, direct product factor
127C5C2×C5×#6711yescharacteristic, complemented
128C22—52no—
129C22—102no—
130C22—152no—
131C22—302no—
132C22C5×D3×D511yescharacteristic
133C4—32no—
134C3C2×C5×D2011yescharacteristic, complemented, Sylow
135C2—102no—
136C2—302no—
137C2C5×#120212yesdirect product factor
138C2C2×C5×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC22×C5D3×D5
commutator, cosocleC2×C3×C5C23×C5
FrattiniC2C2×C5×D3×D5
Fitting, socleC22×C3×C52C22
hypocenterC3×C5C2×C5×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
1. C22×C5×D15C2semidirect
2. C22×C3×C5×D5C2semidirect
3. C2×C52×Dic3C2semidirect
4. C5×#1202C2direct (N × H)
8. C2×C5×D15C22semidirect
10. C2×C3×C5×D5C22semidirect
11. C52×Dic3C22semidirect
13. C2×#1202C5direct (N × H)
15. C22×C5×D5D3semidirect
18. C2×C3×C52D4semidirect
23. C22×D15C2×C5semidirect
25. C22×C3×D5C2×C5semidirect
28. C2×C5×Dic3D5semidirect
29. C2×C5×Dic3C2×C5semidirect
30. #1202C2×C5direct (N × H)
35. C2×C5×D5C2×D3semidirect
39. C3×C52C2×D4semidirect
46. C2×D15C22×C5semidirect
50. C2×C3×D5C22×C5semidirect
52. C5×Dic3C2×D5semidirect
53. C5×Dic3C22×C5semidirect
54. C2×C52#67semidirect
62. C22×D5C5×D3semidirect
71. C2×C3×C5D20semidirect
72. C2×C3×C5C5×D4semidirect
78. C52C2×#67semidirect
82. C2×Dic3C5×D5semidirect
93. C2×D5C2×C5×D3semidirect
99. C3×C5C2×D20semidirect
100. C3×C5C2×C5×D4semidirect
106. Dic3C2×C5×D5semidirect
111. C2×C5#1202direct (N × H)
112. C2×C5C5×#67semidirect
122. C2×C3C5×D20semidirect
126. C5C2×#1202direct (N × H)
127. C5C2×C5×#67semidirect
134. C3C2×C5×D20semidirect
137. C2C5×#1202direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D152
C22×C3×C5×D52
C2×C52×Dic32
C5×#12022
C2×C5×D203
C2×C5×#675
C2×#12025