Finite Group Databasebeta

← C5×#64158 #13088132

Subgroups — C5×#64158 #13088132

Order 1440.

408 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C32×C5×#41C211yesmaximal, characteristic, complemented
2C2×C5×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C5×#3755C212yesmaximal, complemented
4C5×#3753C212yesmaximal, complemented
5C5×#3748C211yesmaximal, characteristic, complemented
6C5×#949—34nomaximal
7C22×C5×C3⋊D3—21no—
8C32×C5×Q8C2211yescharacteristic, complemented
9C32×C5×D4—21no—
10C32×C5×D4C2211yescharacteristic, complemented
11C2×C32×C4×C5—21no—
12C2×C32×C4×C5C2211yescharacteristic
13C5×C4⋊(C3⋊D3)—21no—
14C5×C4⋊(C3⋊D3)C2212yes—
15C5×C3⋊(C3⋊C8)C2212yescomplemented
16#64158C511yesmaximal, characteristic, direct product factor, Hall
17C3×C5×#41D314yescomplemented
18C2×C5×D12—34no—
19C5×#215—38no—
20C5×#213—38no—
21C5×#208—34no—
22C22×C32×C5—21no—
23C22×C32×C5D411yescharacteristic
24C2×C5×C3⋊D3—22no—
25C2×C5×C3⋊D3—42no—
26C32×C4×C5—21no—
27C32×C4×C5D411yescharacteristic
28C32×C4×C5C2311yescharacteristic
29C5×AGL(1,ℤ/8)—91noHall
30C32×#41C2×C511yescharacteristic, complemented
31C2×C4⋊(C3⋊D3)C2×C511yescharacteristic, complemented
32#3755C2×C512yescomplemented
33#3753C2×C512yescomplemented
34#3748C2×C511yescharacteristic, complemented
35C22×C5×D3—64no—
36C3×C5×Q8C2×D314yescomplemented
37C3×C5×D4—24no—
38C3×C5×D4C2×D314yescomplemented
39C2×C3×C4×C5—24no—
40C2×C3×C4×C5C2×D314yes—
41C5×D12—38no—
42C5×D12—64no—
43C5×C3⋊C8—38no—
44#949—34no—
45C2×C32×C5—21no—
46C2×C32×C5—41no—
47C2×C32×C5C2×D411yescharacteristic
48C5×C3⋊D3—42no—
49C5×#41C3⋊D311yescharacteristic, complemented
50C2×C5×D4—91no—
51C5×SD16—92no—
52C5×D8—92no—
53C5×M16—91no—
54C22×C3⋊D3—21no—
55C32×Q8C22×C511yescharacteristic, complemented
56C32×D4—21no—
57C32×D4C22×C511yescharacteristic, complemented
58C2×C32×C4—21no—
59C2×C32×C4C22×C511yescharacteristic
60C4⋊(C3⋊D3)—21no—
61C4⋊(C3⋊D3)C22×C512yes—
62C3⋊(C3⋊C8)C22×C512yescomplemented
63C22×C3×C5—24no—
64C22×C3×C5#6714yes—
65C2×C5×D3—68no—
66C2×C5×D3—128no—
67C3×C4×C5—24no—
68C3×C4×C5#6714yes—
69C3×C4×C5C22×D314yes—
70C3×#41C5×D314yescomplemented
71C2×D12—34no—
72#215—38no—
73#213—38no—
74#208—34no—
75C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
76C23×C5—181no—
77C5×Q8C2×C3⋊D311yescharacteristic, complemented
78C5×D4—21no—
79C5×D4—92no—
80C5×D4—181no—
81C5×D4C2×C3⋊D311yescharacteristic, complemented
82C2×C4×C5—21no—
83C2×C4×C5C2×C3⋊D311yescharacteristic
84C5×C8—92no—
85C22×C32—21no—
86C22×C32C5×D411yescharacteristic
87C2×C3⋊D3—22no—
88C2×C3⋊D3—42no—
89C32×C4—21no—
90C32×C4C5×D411yescharacteristic
91C32×C4C23×C511yescharacteristic
92AGL(1,ℤ/8)—91noSylow
93C2×C3×C5—24no—
94C2×C3×C5—44no—
95C2×C3×C5C2×#6714yes—
96C5×D3—128no—
97C22×D3—64no—
98C3×Q8C2×C5×D314yescomplemented
99C3×D4—24no—
100C3×D4C2×C5×D314yescomplemented
101C2×C3×C4—24no—
102C2×C3×C4C2×C5×D314yes—
103D12—38no—
104D12—64no—
105C3⋊C8—38no—
106C22×C5—21no—
107C22×C5—182no—
108C22×C5—362no—
109C22×C5#63811yescharacteristic
110C4×C5—21no—
111C4×C5#63811yescharacteristic
112C4×C5C22×C3⋊D311yescharacteristic
113C2×C32—21no—
114C2×C32—41no—
115C2×C32C2×C5×D411yescharacteristic
116C3⋊D3—42no—
117#41C5×C3⋊D311yescharacteristic, complemented
118C2×D4—91no—
119SD16—92no—
120D8—92no—
121M16—91no—
122C3×C5#94914yescomplemented
123C22×C3—24no—
124C22×C3C5×#6714yes—
125C2×D3—68no—
126C2×D3—128no—
127C3×C4—24no—
128C3×C4C5×#6714yes—
129C3×C4C22×C5×D314yes—
130C2×C5—21no—
131C2×C5—41no—
132C2×C5—362no—
133C2×C5C2×#63811yescharacteristic
134C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
135C23—181no—
136Q8C2×C5×C3⋊D311yescharacteristic, complemented
137D4—21no—
138D4—92no—
139D4—181no—
140D4C2×C5×C3⋊D311yescharacteristic, complemented
141C2×C4—21no—
142C2×C4C2×C5×C3⋊D311yescharacteristic
143C8—92no—
144C2×C3—24no—
145C2×C3—44no—
146C2×C3C2×C5×#6714yes—
147D3—128no—
148C5#6415811yescharacteristic, direct product factor, Sylow
149C22—21no—
150C22—182no—
151C22—362no—
152C22C5×#63811yescharacteristic
153C4—21no—
154C4C5×#63811yescharacteristic
155C4C22×C5×C3⋊D311yescharacteristic
156C3C5×#94914yescomplemented
157C2—21no—
158C2—41no—
159C2—362no—
160C2C2×C5×#63811yescharacteristic
subgroupG/H
centerC2×C5C2×#638
commutator, cosocleC32×C4C23×C5
FrattiniC4C22×C5×C3⋊D3
FittingC32×C5×#41C2
socleC2×C32×C5C2×D4
hypercenterC5×#41C3⋊D3
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#41

Complements

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

NH ≅ G/Ntype
1. C32×C5×#41C2semidirect
2. C2×C5×C4⋊(C3⋊D3)C2semidirect
3. C5×#3755C2semidirect
4. C5×#3753C2semidirect
5. C5×#3748C2semidirect
8. C32×C5×Q8C22semidirect
10. C32×C5×D4C22semidirect
15. C5×C3⋊(C3⋊C8)C22semidirect
16. #64158C5direct (N × H)
17. C3×C5×#41D3semidirect
30. C32×#41C2×C5semidirect
31. C2×C4⋊(C3⋊D3)C2×C5semidirect
32. #3755C2×C5semidirect
33. #3753C2×C5semidirect
34. #3748C2×C5semidirect
36. C3×C5×Q8C2×D3semidirect
38. C3×C5×D4C2×D3semidirect
49. C5×#41C3⋊D3semidirect
55. C32×Q8C22×C5semidirect
57. C32×D4C22×C5semidirect
62. C3⋊(C3⋊C8)C22×C5semidirect
70. C3×#41C5×D3semidirect
75. C32×C5AGL(1,ℤ/8)semidirect
77. C5×Q8C2×C3⋊D3semidirect
81. C5×D4C2×C3⋊D3semidirect
98. C3×Q8C2×C5×D3semidirect
100. C3×D4C2×C5×D3semidirect
117. #41C5×C3⋊D3semidirect
122. C3×C5#949semidirect
134. C32C5×AGL(1,ℤ/8)semidirect
136. Q8C2×C5×C3⋊D3semidirect
140. D4C2×C5×C3⋊D3semidirect
148. C5#64158direct (N × H)
156. C3C5×#949semidirect

Maximal subgroups

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

subgroupindex
C32×C5×#412
C2×C5×C4⋊(C3⋊D3)2
C5×#37552
C5×#37532
C5×#37482
C5×#9493
#641585