Finite Group Databasebeta

← C2×C5×#1220 #11925714

Subgroups — C2×C5×#1220 #11925714

Order 1200.

280 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C23×C3×C52C211yesmaximal, characteristic, complemented
2C22×C5×D15C211yesmaximal, characteristic, complemented
3C5×#1220C214yesmaximal, direct product factor
4C2×C5×Dic15C211yesmaximal, characteristic, complemented
5C2×C5×C22⋊D5—31nomaximal, Hall
6C22×C3×C52—22no—
7C22×C3×C52C2211yescharacteristic
8C22×C3×C52C2212yescomplemented
9C2×C5×D15—22no—
10C2×C5×D15C2212yescomplemented
11C5×Dic15C2212yescomplemented
12C2×#1220C511yesmaximal, characteristic, direct product factor
13C2×C5×#67—51nomaximal
14C23×C52D311yescharacteristic, complemented
15C22×C5×D5—31no—
16C5×C22⋊D5—34no—
17C2×C5×Dic5—31no—
18C2×C3×C52—22no—
19C2×C3×C52D412yescomplemented
20C2×C3×C52C2311yescharacteristic
21C5×D15—22no—
22C23×C3×C5—22no—
23C23×C3×C5D511yescharacteristic, complemented
24C23×C3×C5C2×C511yescharacteristic, complemented
25C22×D15C2×C511yescharacteristic, complemented
26C22×C5×D3—51no—
27#1220C2×C514yesdirect product factor
28C2×Dic15C2×C511yescharacteristic, complemented
29C5×#67—54no—
30C2×C5×Dic3—51no—
31C22×C52—22no—
32C22×C52C2×D311yescharacteristic
33C22×C52C2×D312yescomplemented
34C2×C5×D5—32no—
35C2×C5×D5—62no—
36C5×Dic5—32no—
37C2×C5×D4—151no—
38C2×C22⋊D5—31no—
39C3×C52C2×D411yescharacteristic, complemented, Hall
40C22×C3×C5—28no—
41C22×C3×C5—22no—
42C22×C3×C5—22no—
43C22×C3×C5—22no—
44C22×C3×C5—24no—
45C22×C3×C5C2×D511yescharacteristic
46C22×C3×C5C2×D512yescomplemented
47C22×C3×C5C22×C511yescharacteristic
48C22×C3×C5C22×C512yescomplemented
49C2×D15—22no—
50C2×D15C22×C512yescomplemented
51C2×C5×D3—52no—
52C2×C5×D3—102no—
53Dic15C22×C512yescomplemented
54C5×Dic3—52no—
55C2×C52—22no—
56C2×C52#6712yescomplemented
57C2×C52C22×D311yescharacteristic
58C5×D5—62no—
59C2×#67—51noHall
60C23×C5—22no—
61C23×C5—151no—
62C23×C5C5×D311yescharacteristic, complemented
63C23×C5D1511yescharacteristic, complemented
64C22×D5—31no—
65C5×D4—154no—
66C2×C4×C5—151no—
67C22⋊D5—34no—
68C2×Dic5—31no—
69C2×C3×C5—28no—
70C2×C3×C5—22no—
71C2×C3×C5—22no—
72C2×C3×C5—24no—
73C2×C3×C5—22no—
74C2×C3×C5C22⋊D512yescomplemented
75C2×C3×C5C5×D412yescomplemented
76C2×C3×C5C22×D511yescharacteristic
77C2×C3×C5C23×C511yescharacteristic
78D15—22no—
79C5×D3—102no—
80C52C2×#6711yescharacteristic, complemented, Sylow
81C23×C3C5×D511yescharacteristic, complemented
82C22×D3—51no—
83#67—54no—
84C2×Dic3—51no—
85C22×C5—28no—
86C22×C5—22no—
87C22×C5—22no—
88C22×C5—22no—
89C22×C5—24no—
90C22×C5—152no—
91C22×C5—302no—
92C22×C5C2×C5×D311yescharacteristic
93C22×C5C2×C5×D312yescomplemented
94C22×C5C2×D1511yescharacteristic
95C22×C5C2×D1512yescomplemented
96C2×D5—32no—
97C2×D5—62no—
98C4×C5—152no—
99Dic5—32no—
100C2×D4—151noSylow
101C3×C5—22no—
102C3×C5C2×C22⋊D511yescharacteristic, complemented
103C3×C5C2×C5×D411yescharacteristic, complemented
104C22×C3—22no—
105C22×C3C2×C5×D511yescharacteristic
106C22×C3C2×C5×D512yescomplemented
107C2×D3—52no—
108C2×D3—102no—
109Dic3—52no—
110C2×C5—28no—
111C2×C5—22no—
112C2×C5—22no—
113C2×C5—24no—
114C2×C5—22no—
115C2×C5—302no—
116C2×C5C5×#6712yescomplemented
117C2×C5#122012yesdirect product factor
118C2×C5C22×C5×D311yescharacteristic
119C2×C5C22×D1511yescharacteristic
120D5—62no—
121C23—151no—
122C23C5×D1511yescharacteristic, complemented
123D4—154no—
124C2×C4—151no—
125C2×C3—22no—
126C2×C3C5×C22⋊D512yescomplemented
127C2×C3C22×C5×D511yescharacteristic
128D3—102no—
129C5—22no—
130C5C2×C5×#6711yescharacteristic, complemented
131C5C2×#122011yescharacteristic, direct product factor
132C22—22no—
133C22—152no—
134C22—302no—
135C22C2×C5×D1511yescharacteristic
136C22C2×C5×D1512yescomplemented
137C4—152no—
138C3C2×C5×C22⋊D511yescharacteristic, complemented, Sylow
139C2—22no—
140C2—302no—
141C2C5×#122012yesdirect product factor
142C2C22×C5×D1511yescharacteristic
subgroupG/H
centerC22×C5C2×D15
commutator, cosocleC2×C3×C5C23×C5
FrattiniC2C22×C5×D15
FittingC23×C3×C52C2
socleC22×C3×C52C22
hypercenterC23×C5D15
hypocenterC3×C5C2×C5×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C23×C3×C52C2semidirect
2. C22×C5×D15C2semidirect
3. C5×#1220C2direct (N × H)
4. C2×C5×Dic15C2semidirect
8. C22×C3×C52C22semidirect
10. C2×C5×D15C22semidirect
11. C5×Dic15C22semidirect
12. C2×#1220C5direct (N × H)
14. C23×C52D3semidirect
19. C2×C3×C52D4semidirect
23. C23×C3×C5D5semidirect
24. C23×C3×C5C2×C5semidirect
25. C22×D15C2×C5semidirect
27. #1220C2×C5direct (N × H)
28. C2×Dic15C2×C5semidirect
33. C22×C52C2×D3semidirect
39. C3×C52C2×D4semidirect
46. C22×C3×C5C2×D5semidirect
48. C22×C3×C5C22×C5semidirect
50. C2×D15C22×C5semidirect
53. Dic15C22×C5semidirect
56. C2×C52#67semidirect
62. C23×C5C5×D3semidirect
63. C23×C5D15semidirect
74. C2×C3×C5C22⋊D5semidirect
75. C2×C3×C5C5×D4semidirect
80. C52C2×#67semidirect
81. C23×C3C5×D5semidirect
93. C22×C5C2×C5×D3semidirect
95. C22×C5C2×D15semidirect
102. C3×C5C2×C22⋊D5semidirect
103. C3×C5C2×C5×D4semidirect
106. C22×C3C2×C5×D5semidirect
116. C2×C5C5×#67semidirect
117. C2×C5#1220direct (N × H)
122. C23C5×D15semidirect
126. C2×C3C5×C22⋊D5semidirect
130. C5C2×C5×#67semidirect
131. C5C2×#1220direct (N × H)
136. C22C2×C5×D15semidirect
138. C3C2×C5×C22⋊D5semidirect
141. C2C5×#1220direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C522
C22×C5×D152
C5×#12202
C2×C5×Dic152
C2×C5×C22⋊D53
C2×#12205
C2×C5×#675