Finite Group Databasebeta

← C2×C5×#1203 #11925588

Subgroups — C2×C5×#1203 #11925588

Order 1200.

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

normal subgroups only

This group has 280 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×C52×D3C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic5C211yesmaximal, characteristic, complemented
4C5×#1203C214yesmaximal, direct product factor
5C2×C5×C22⋊D5—31nomaximal, Hall
6C22×C3×C52C2211yescharacteristic
7C2×C5×D15—22no—
8C2×C5×D15C2212yescomplemented
9C2×C52×D3—22no—
10C2×C52×D3C2212yescomplemented
11C3×C5×Dic5C2212yescomplemented
12C2×C5×D12—51nomaximal
13C2×#1203C511yesmaximal, characteristic, direct product factor
14C23×C52—31no—
15C22×C5×D5—31no—
16C5×C22⋊D5—34no—
17C2×C5×Dic5D311yescharacteristic, complemented
18C2×C3×C52D412yescomplemented
19C2×C3×C52C2311yescharacteristic
20C5×D15—22no—
21C52×D3—22no—
22C22×D15C2×C511yescharacteristic, complemented
23C22×C5×D3—22no—
24C22×C5×D3—51no—
25C22×C5×D3D511yescharacteristic, complemented
26C22×C5×D3C2×C511yescharacteristic, complemented
27C2×C3×C4×C5—51no—
28C5×D12—54no—
29C2×C3×Dic5C2×C511yescharacteristic, complemented
30#1203C2×C514yesdirect product factor
31C22×C52—32no—
32C22×C52—62no—
33C22×C52C2×D311yescharacteristic
34C2×C5×D5—32no—
35C2×C5×D5—62no—
36C5×Dic5C2×D312yescomplemented
37C2×C5×D4—151no—
38C2×C22⋊D5—31no—
39C3×C52C2×D411yescharacteristic, complemented, Hall
40C22×C3×C5—22no—
41C22×C3×C5C2×D511yescharacteristic
42C22×C3×C5C22×C511yescharacteristic
43C2×D15—22no—
44C2×D15C22×C512yescomplemented
45C2×C5×D3—28no—
46C2×C5×D3—22no—
47C2×C5×D3—22no—
48C2×C5×D3—24no—
49C2×C5×D3—52no—
50C2×C5×D3—102no—
51C2×C5×D3C2×D512yescomplemented
52C2×C5×D3C22×C512yescomplemented
53C3×C4×C5—52no—
54C3×Dic5C22×C512yescomplemented
55C2×C52—62no—
56C2×C52D1212yescomplemented
57C2×C52C22×D311yescharacteristic
58C5×D5—62no—
59C2×D12—51noHall
60C23×C5—31no—
61C23×C5—31no—
62C23×C5—62no—
63C23×C5—151no—
64C22×D5—31no—
65C5×D4—154no—
66C2×C4×C5—51no—
67C22⋊D5—34no—
68C2×Dic5C5×D311yescharacteristic, complemented
69C2×C3×C5—24no—
70C2×C3×C5—22no—
71C2×C3×C5C22⋊D512yescomplemented
72C2×C3×C5C5×D412yescomplemented
73C2×C3×C5C22×D511yescharacteristic
74C2×C3×C5C23×C511yescharacteristic
75D15—22no—
76C5×D3—28no—
77C5×D3—22no—
78C5×D3—22no—
79C5×D3—102no—
80C52C2×D1211yescharacteristic, complemented, Sylow
81C22×D3—51no—
82C22×D3C5×D511yescharacteristic, complemented
83C2×C3×C4—51no—
84D12—54no—
85C22×C5—22no—
86C22×C5—32no—
87C22×C5—32no—
88C22×C5—64no—
89C22×C5—68no—
90C22×C5—62no—
91C22×C5—62no—
92C22×C5—152no—
93C22×C5—302no—
94C22×C5D3×D511yescharacteristic
95C22×C5C2×C5×D311yescharacteristic
96C2×D5—32no—
97C2×D5—62no—
98C4×C5—52no—
99Dic5C2×C5×D312yescomplemented
100C2×D4—151noSylow
101C3×C5—22no—
102C3×C5C2×C22⋊D511yescharacteristic, complemented
103C3×C5C2×C5×D411yescharacteristic, complemented
104C22×C3C2×C5×D511yescharacteristic
105C2×D3—22no—
106C2×D3—52no—
107C2×D3—102no—
108C2×D3C2×C5×D512yescomplemented
109C3×C4—52no—
110C2×C5—22no—
111C2×C5—24no—
112C2×C5—68no—
113C2×C5—62no—
114C2×C5—62no—
115C2×C5—302no—
116C2×C5#120312yesdirect product factor
117C2×C5C5×D1212yescomplemented
118C2×C5C2×D3×D511yescharacteristic
119C2×C5C22×C5×D311yescharacteristic
120D5—62no—
121C23—31no—
122C23—151no—
123D4—154no—
124C2×C4—51no—
125C2×C3C5×C22⋊D512yescomplemented
126C2×C3C22×C5×D511yescharacteristic
127D3—22no—
128D3—102no—
129C5—22no—
130C5C2×#120311yescharacteristic, direct product factor
131C5C2×C5×D1211yescharacteristic, complemented
132C22—32no—
133C22—62no—
134C22—152no—
135C22—302no—
136C22C5×D3×D511yescharacteristic
137C4—52no—
138C3C2×C5×C22⋊D511yescharacteristic, complemented, Sylow
139C2—62no—
140C2—302no—
141C2C5×#120312yesdirect product factor
142C2C2×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×C52×D3C2semidirect
3. C2×C3×C5×Dic5C2semidirect
4. C5×#1203C2direct (N × H)
8. C2×C5×D15C22semidirect
10. C2×C52×D3C22semidirect
11. C3×C5×Dic5C22semidirect
13. C2×#1203C5direct (N × H)
17. C2×C5×Dic5D3semidirect
18. C2×C3×C52D4semidirect
22. C22×D15C2×C5semidirect
25. C22×C5×D3D5semidirect
26. C22×C5×D3C2×C5semidirect
29. C2×C3×Dic5C2×C5semidirect
30. #1203C2×C5direct (N × H)
36. C5×Dic5C2×D3semidirect
39. C3×C52C2×D4semidirect
44. C2×D15C22×C5semidirect
51. C2×C5×D3C2×D5semidirect
52. C2×C5×D3C22×C5semidirect
54. C3×Dic5C22×C5semidirect
56. C2×C52D12semidirect
68. C2×Dic5C5×D3semidirect
71. C2×C3×C5C22⋊D5semidirect
72. C2×C3×C5C5×D4semidirect
80. C52C2×D12semidirect
82. C22×D3C5×D5semidirect
99. Dic5C2×C5×D3semidirect
102. C3×C5C2×C22⋊D5semidirect
103. C3×C5C2×C5×D4semidirect
108. C2×D3C2×C5×D5semidirect
116. C2×C5#1203direct (N × H)
117. C2×C5C5×D12semidirect
125. C2×C3C5×C22⋊D5semidirect
130. C5C2×#1203direct (N × H)
131. C5C2×C5×D12semidirect
138. C3C2×C5×C22⋊D5semidirect
141. C2C5×#1203direct (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×C52×D32
C2×C3×C5×Dic52
C5×#12032
C2×C5×C22⋊D53
C2×C5×D125
C2×#12035