Finite Group Databasebeta

← C5×#63710 #13085335

Subgroups — C5×#63710 #13085335

Order 1440.

498 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 204 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
1C2×C5×D4×C9C211yesmaximal, characteristic, complemented
2C2×C5×#611C214yesmaximal, complemented
3C5×#3699C214yesmaximal, complemented
4C5×D4×D9C214yesmaximal, complemented
5C5×#3697C212yesmaximal, complemented
6C5×#1004—31nomaximal
7C23×C5×C9C2212yescomplemented
8C22×C5×D9C2214yescomplemented
9C5×D4×C9C2214yescomplemented
10C2×C4×C5×C9C2211yescharacteristic, complemented
11C5×#611C2214yescomplemented
12C5×#611C2218yescomplemented
13C2×C5×Dic9C2214yescomplemented
14C5×D36C2212yescomplemented
15C4×C5×D9C2214yescomplemented
16C5×Dic18C2212yescomplemented
17#63710C511yesmaximal, characteristic, direct product factor, Hall
18C2×C3×C5×D4D311yescharacteristic
19C2×C5×#67—34no—
20C5×#237—34no—
21C5×D3×D4—34no—
22C5×#235—32no—
23C22×C5×C9—42no—
24C22×C5×C9C2314yes—
25C22×C5×C9C2311yescharacteristic
26C2×C5×D9—44no—
27C2×C5×D9C2314yes—
28C4×C5×C9C2312yes—
29C5×Dic9C2314yes—
30C5×H(5,𝔽2)—91noHall
31C2×D4×C9C2×C511yescharacteristic, complemented
32C2×#611C2×C514yescomplemented
33#3699C2×C514yescomplemented
34D4×D9C2×C514yescomplemented
35#3697C2×C512yescomplemented
36C23×C3×C5C2×D312yes—
37C22×C5×D3—34no—
38C3×C5×D4C2×D314yes—
39C2×C3×C4×C5C2×D311yescharacteristic
40C5×#67—34no—
41C5×#67—38no—
42C2×C5×Dic3—34no—
43C5×D12—32no—
44C4×C5×D3—34no—
45C5×Dic6—32no—
46#1004—31no—
47C2×C5×C9—24no—
48C2×C5×C9—21no—
49C2×C5×C9C2411yescharacteristic
50C5×D9—24no—
51C5×#41—94no—
52C5×#41—92no—
53C2×C5×D4—94no—
54C2×C5×D4—94no—
55C2×C5×D4D911yescharacteristic, complemented
56C23×C9C22×C512yescomplemented
57C22×D9C22×C514yescomplemented
58D4×C9C22×C514yescomplemented
59C2×C4×C9C22×C511yescharacteristic, complemented
60#611C22×C514yescomplemented
61#611C22×C518yescomplemented
62C2×Dic9C22×C514yescomplemented
63D36C22×C512yescomplemented
64C4×D9C22×C514yescomplemented
65Dic18C22×C512yescomplemented
66C22×C3×C5—42no—
67C22×C3×C5C22×D314yes—
68C22×C3×C5C22×D311yescharacteristic
69C2×C5×D3—34no—
70C2×C5×D3—124no—
71C3×C4×C5C22×D312yes—
72C5×Dic3—34no—
73C2×C3×D4C5×D311yescharacteristic
74C2×#67—34no—
75#237—34no—
76D3×D4—34no—
77#235—32no—
78C5×C9H(5,𝔽2)11yescharacteristic, complemented, Hall
79C23×C5—94no—
80C23×C5C2×D912yescomplemented
81C5×Q8—92no—
82C5×D4—98no—
83C5×D4—94no—
84C5×D4—92no—
85C5×D4C2×D914yescomplemented
86C2×C4×C5—94no—
87C2×C4×C5—94no—
88C2×C4×C5C2×D911yescharacteristic, complemented
89C22×C9—42no—
90C22×C9C23×C514yes—
91C22×C9C23×C511yescharacteristic
92C2×D9—44no—
93C2×D9C23×C514yes—
94C4×C9C23×C512yes—
95Dic9C23×C514yes—
96H(5,𝔽2)—91noSylow
97C2×C3×C5—24no—
98C2×C3×C5—21no—
99C2×C3×C5C23×D311yescharacteristic
100C5×D3—64no—
101C23×C3C2×C5×D312yes—
102C22×D3—34no—
103C3×D4C2×C5×D314yes—
104C2×C3×C4C2×C5×D311yescharacteristic
105#67—34no—
106#67—38no—
107C2×Dic3—34no—
108D12—32no—
109C4×D3—34no—
110Dic6—32no—
111C22×C5—42no—
112C22×C5—94no—
113C22×C5—364no—
114C22×C5C22×D914yes—
115C22×C5C22×D911yescharacteristic
116C4×C5—94no—
117C4×C5C22×D912yes—
118C2×C9—24no—
119C2×C9—21no—
120C2×C9C24×C511yescharacteristic
121D9—24no—
122#41—94no—
123#41—92no—
124C2×D4—94no—
125C2×D4—94no—
126C2×D4C5×D911yescharacteristic, complemented
127C3×C5#100411yescharacteristic
128C22×C3—42no—
129C22×C3C22×C5×D314yes—
130C22×C3C22×C5×D311yescharacteristic
131C2×D3—34no—
132C2×D3—124no—
133C3×C4C22×C5×D312yes—
134Dic3—34no—
135C2×C5—21no—
136C2×C5—24no—
137C2×C5—184no—
138C2×C5C23×D911yescharacteristic
139C9C5×H(5,𝔽2)11yescharacteristic, complemented, Sylow
140C23—94no—
141C23C2×C5×D912yescomplemented
142Q8—92no—
143D4—98no—
144D4—94no—
145D4—92no—
146D4C2×C5×D914yescomplemented
147C2×C4—94no—
148C2×C4—94no—
149C2×C4C2×C5×D911yescharacteristic, complemented
150C2×C3—24no—
151C2×C3—21no—
152C2×C3C23×C5×D311yescharacteristic
153D3—64no—
154C5#6371011yescharacteristic, direct product factor, Sylow
155C22—42no—
156C22—94no—
157C22—364no—
158C22C22×C5×D914yes—
159C22C22×C5×D911yescharacteristic
160C4—94no—
161C4C22×C5×D912yes—
162C3C5×#100411yescharacteristic
163C2—21no—
164C2—24no—
165C2—184no—
166C2C23×C5×D911yescharacteristic
subgroupG/H
centerC2×C5C23×D9
commutator, cosocleC2×C9C24×C5
FrattiniC2×C3C23×C5×D3
FittingC2×C5×D4×C9C2
socleC2×C3×C5C23×D3
hypercenterC2×C5×D4D9
hypocenterC9C5×H(5,𝔽2)
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×C9C2semidirect
2. C2×C5×#611C2semidirect
3. C5×#3699C2semidirect
4. C5×D4×D9C2semidirect
5. C5×#3697C2semidirect
7. C23×C5×C9C22semidirect
8. C22×C5×D9C22semidirect
9. C5×D4×C9C22semidirect
10. C2×C4×C5×C9C22semidirect
11. C5×#611C22semidirect
12. C5×#611C22semidirect
13. C2×C5×Dic9C22semidirect
14. C5×D36C22semidirect
15. C4×C5×D9C22semidirect
16. C5×Dic18C22semidirect
17. #63710C5direct (N × H)
31. C2×D4×C9C2×C5semidirect
32. C2×#611C2×C5semidirect
33. #3699C2×C5semidirect
34. D4×D9C2×C5semidirect
35. #3697C2×C5semidirect
55. C2×C5×D4D9semidirect
56. C23×C9C22×C5semidirect
57. C22×D9C22×C5semidirect
58. D4×C9C22×C5semidirect
59. C2×C4×C9C22×C5semidirect
60. #611C22×C5semidirect
61. #611C22×C5semidirect
62. C2×Dic9C22×C5semidirect
63. D36C22×C5semidirect
64. C4×D9C22×C5semidirect
65. Dic18C22×C5semidirect
78. C5×C9H(5,𝔽2)semidirect
80. C23×C5C2×D9semidirect
85. C5×D4C2×D9semidirect
88. C2×C4×C5C2×D9semidirect
126. C2×D4C5×D9semidirect
139. C9C5×H(5,𝔽2)semidirect
141. C23C2×C5×D9semidirect
146. D4C2×C5×D9semidirect
149. C2×C4C2×C5×D9semidirect
154. C5#63710direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×D4×C92
C2×C5×#6112
C5×#36992
C5×D4×D92
C5×#36972
C5×#10043
#637105