Finite Group Databasebeta

← C5×#64123 #13088097

Subgroups — C5×#64123 #13088097

Order 1440.

408 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C5×#3832C211yesmaximal, characteristic, complemented
2C5×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×C5×SD16C211yesmaximal, characteristic, complemented
4C5×#3755C211yesmaximal, characteristic, complemented
5C5×#3753C211yesmaximal, characteristic, complemented
6C5×#3745C211yesmaximal, characteristic, complemented
7C5×#3743C211yesmaximal, characteristic, complemented
8C5×#914—34nomaximal
9C22×C5×C3⋊D3—21no—
10C32×C5×Q8C2211yescharacteristic, complemented
11C32×C5×D4C2211yescharacteristic
12C5×#638—21no—
13C5×C4⋊(C3⋊D3)—21no—
14C5×C4⋊(C3⋊D3)C2211yescharacteristic
15C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C4×C5×C3⋊D3—21no—
17C4×C5×C3⋊D3C2211yescharacteristic
18C32×C5×C8C2211yescharacteristic, complemented
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20#64123C511yesmaximal, characteristic, direct product factor, Hall
21C5×#239—34no—
22C5×D3×D4—34no—
23C3×C5×SD16D314yescomplemented
24C5×#215—34no—
25C5×#213—34no—
26C5×D24—34no—
27C5×ΓL(1,𝔽25)—34no—
28C22×C32×C5—21no—
29C2×C5×C3⋊D3—21no—
30C2×C5×C3⋊D3—21no—
31C2×C5×C3⋊D3—42no—
32C2×C5×C3⋊D3D411yescharacteristic
33C32×C4×C5—21no—
34C32×C4×C5C2311yescharacteristic
35C5×C3⋊Dic3D411yescharacteristic
36C5×AGL(1,ℤ/8)—91noHall
37#3832C2×C511yescharacteristic, complemented
38D4×C3⋊D3C2×C511yescharacteristic, complemented
39C32×SD16C2×C511yescharacteristic, complemented
40#3755C2×C511yescharacteristic, complemented
41#3753C2×C511yescharacteristic, complemented
42#3745C2×C511yescharacteristic, complemented
43#3743C2×C511yescharacteristic, complemented
44C22×C5×D3—64no—
45C3×C5×Q8C2×D314yescomplemented
46C3×C5×D4C2×D314yes—
47C5×#67—64no—
48C5×D12—34no—
49C5×D12—34no—
50C5×D12—64no—
51C4×C5×D3—34no—
52C4×C5×D3—64no—
53C3×C5×C8C2×D314yescomplemented
54C5×C3⋊C8—34no—
55#914—34no—
56C2×C32×C5—41no—
57C2×C32×C5C2×D411yescharacteristic
58C5×C3⋊D3—21no—
59C5×C3⋊D3—41no—
60C5×C3⋊D3—41no—
61C5×#41—91no—
62C2×C5×D4—91no—
63C5×SD16—91no—
64C5×SD16C3⋊D311yescharacteristic, complemented
65C5×D8—91no—
66C5×D8—91no—
67C5×M16—91no—
68C22×C3⋊D3—21no—
69C32×Q8C22×C511yescharacteristic, complemented
70C32×D4C22×C511yescharacteristic
71#638—21no—
72C4⋊(C3⋊D3)—21no—
73C4⋊(C3⋊D3)C22×C511yescharacteristic
74C4⋊(C3⋊D3)C22×C511yescharacteristic, complemented
75C4×C3⋊D3—21no—
76C4×C3⋊D3C22×C511yescharacteristic
77C32×C8C22×C511yescharacteristic, complemented
78C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
79C22×C3×C5—24no—
80C2×C5×D3—34no—
81C2×C5×D3—64no—
82C2×C5×D3—64no—
83C2×C5×D3—128no—
84C3×C4×C5—24no—
85C3×C4×C5C22×D314yes—
86C5×Dic3—34no—
87#239—34no—
88D3×D4—34no—
89C3×SD16C5×D314yescomplemented
90#215—34no—
91#213—34no—
92D24—34no—
93ΓL(1,𝔽25)—34no—
94C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
95C23×C5—181no—
96C5×Q8C2×C3⋊D311yescharacteristic, complemented
97C5×D4—91no—
98C5×D4—91no—
99C5×D4—181no—
100C5×D4—181no—
101C5×D4C2×C3⋊D311yescharacteristic
102C2×C4×C5—91no—
103C2×C4×C5—181no—
104C5×C8—91no—
105C5×C8C2×C3⋊D311yescharacteristic, complemented
106C22×C32—21no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3—21no—
109C2×C3⋊D3—42no—
110C2×C3⋊D3C5×D411yescharacteristic
111C32×C4—21no—
112C32×C4C23×C511yescharacteristic
113C3⋊Dic3C5×D411yescharacteristic
114AGL(1,ℤ/8)—91noSylow
115C2×C3×C5—44no—
116C2×C3×C5D3×D414yes—
117C5×D3—64no—
118C5×D3—124no—
119C5×D3—124no—
120C22×D3—64no—
121C3×Q8C2×C5×D314yescomplemented
122C3×D4C2×C5×D314yes—
123#67—64no—
124D12—34no—
125D12—34no—
126D12—64no—
127C4×D3—34no—
128C4×D3—64no—
129C3×C8C2×C5×D314yescomplemented
130C3⋊C8—34no—
131C22×C5—21no—
132C22×C5—91no—
133C22×C5—181no—
134C22×C5—181no—
135C22×C5—362no—
136C4×C5—21no—
137C4×C5—91no—
138C4×C5C22×C3⋊D311yescharacteristic
139C2×C32—41no—
140C2×C32C2×C5×D411yescharacteristic
141C3⋊D3—21no—
142C3⋊D3—41no—
143C3⋊D3—41no—
144#41—91no—
145C2×D4—91no—
146SD16—91no—
147SD16C5×C3⋊D311yescharacteristic, complemented
148D8—91no—
149D8—91no—
150M16—91no—
151C3×C5#91414yescomplemented
152C22×C3—24no—
153C2×D3—34no—
154C2×D3—64no—
155C2×D3—64no—
156C2×D3—128no—
157C3×C4—24no—
158C3×C4C22×C5×D314yes—
159Dic3—34no—
160C2×C5—41no—
161C2×C5—181no—
162C2×C5—361no—
163C2×C5—361no—
164C2×C5D4×C3⋊D311yescharacteristic
165C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
166C23—181no—
167Q8C2×C5×C3⋊D311yescharacteristic, complemented
168D4—91no—
169D4—91no—
170D4—181no—
171D4—181no—
172D4C2×C5×C3⋊D311yescharacteristic
173C2×C4—91no—
174C2×C4—181no—
175C8—91no—
176C8C2×C5×C3⋊D311yescharacteristic, complemented
177C2×C3—44no—
178C2×C3C5×D3×D414yes—
179D3—64no—
180D3—124no—
181D3—124no—
182C5#6412311yescharacteristic, direct product factor, Sylow
183C22—21no—
184C22—91no—
185C22—181no—
186C22—181no—
187C22—362no—
188C4—21no—
189C4—91no—
190C4C22×C5×C3⋊D311yescharacteristic
191C3C5×#91414yescomplemented
192C2—41no—
193C2—181no—
194C2—361no—
195C2—361no—
196C2C5×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C5D4×C3⋊D3
commutator, cosocleC32×C4C23×C5
FrattiniC4C22×C5×C3⋊D3
FittingC32×C5×SD16C2
socleC2×C32×C5C2×D4
hypercenterC5×SD16C3⋊D3
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×SD16

Complements

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

NH ≅ G/Ntype
1. C5×#3832C2semidirect
2. C5×D4×C3⋊D3C2semidirect
3. C32×C5×SD16C2semidirect
4. C5×#3755C2semidirect
5. C5×#3753C2semidirect
6. C5×#3745C2semidirect
7. C5×#3743C2semidirect
10. C32×C5×Q8C22semidirect
15. C5×C4⋊(C3⋊D3)C22semidirect
18. C32×C5×C8C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. #64123C5direct (N × H)
23. C3×C5×SD16D3semidirect
37. #3832C2×C5semidirect
38. D4×C3⋊D3C2×C5semidirect
39. C32×SD16C2×C5semidirect
40. #3755C2×C5semidirect
41. #3753C2×C5semidirect
42. #3745C2×C5semidirect
43. #3743C2×C5semidirect
45. C3×C5×Q8C2×D3semidirect
53. C3×C5×C8C2×D3semidirect
64. C5×SD16C3⋊D3semidirect
69. C32×Q8C22×C5semidirect
74. C4⋊(C3⋊D3)C22×C5semidirect
77. C32×C8C22×C5semidirect
78. C3⋊(C3⋊C8)C22×C5semidirect
89. C3×SD16C5×D3semidirect
94. C32×C5AGL(1,ℤ/8)semidirect
96. C5×Q8C2×C3⋊D3semidirect
105. C5×C8C2×C3⋊D3semidirect
121. C3×Q8C2×C5×D3semidirect
129. C3×C8C2×C5×D3semidirect
147. SD16C5×C3⋊D3semidirect
151. C3×C5#914semidirect
165. C32C5×AGL(1,ℤ/8)semidirect
167. Q8C2×C5×C3⋊D3semidirect
176. C8C2×C5×C3⋊D3semidirect
182. C5#64123direct (N × H)
191. C3C5×#914semidirect

Maximal subgroups

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

subgroupindex
C5×#38322
C5×D4×C3⋊D32
C32×C5×SD162
C5×#37552
C5×#37532
C5×#37452
C5×#37432
C5×#9143
#641235