Finite Group Databasebeta

← C5×#64319 #13089332

Subgroups — C5×#64319 #13089332

Order 1440.

694 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 220 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×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×C5×#239C212yesmaximal, complemented
3C5×#3802C213yesmaximal, complemented
4C4×C5×D32C213yesmaximal, complemented
5C5×#3795C216yesmaximal, complemented
6C5×D3×#41—32nomaximal
7C2×C5×D32C2213yes—
8C32×C5×Q8C2211yescharacteristic, complemented
9C4×C5×C3⋊D3C2213yes—
10C5×#634C2213yescomplemented
11C3×C5×D12C2216yescomplemented
12C3×C4×C5×D3C2216yescomplemented
13C5×#625C2216yescomplemented
14C5×#624C2211yescharacteristic
15C5×D3×Dic3C2216yescomplemented
16#64319C511yesmaximal, characteristic, direct product factor, Hall
17C3×C5×#41—32no—
18C5×#239D312yescomplemented
19C5×D3×Q8—32no—
20C5×D3×Q8—61no—
21C5×#237—36no—
22C5×D3×D4—36no—
23C5×#235—36no—
24C2×C4×C5×D3—36no—
25C2×C5×C3⋊D3C2311yescharacteristic
26C2×C3×C5×D3C2316yes—
27C5×D32—26no—
28C32×C4×C5C2313yes—
29C5×C3⋊Dic3C2313yes—
30C3×C5×Dic3C2312yes—
31C2×C5×#41—91noHall
32Q8×C3⋊D3C2×C511yescharacteristic, complemented
33C3×#239C2×C512yescomplemented
34#3802C2×C513yescomplemented
35C4×D32C2×C513yescomplemented
36#3795C2×C516yescomplemented
37C22×C5×D3—36no—
38C3×C5×Q8—21no—
39C3×C5×Q8C2×D312yescomplemented
40C3×C5×D4—36no—
41C2×C3×C4×C5—36no—
42C5×#67—312no—
43C2×C5×Dic3—36no—
44C5×D12C2×D316yescomplemented
45C4×C5×D3—36no—
46C4×C5×D3—36no—
47C4×C5×D3—32no—
48C4×C5×D3—63no—
49C4×C5×D3C2×D316yescomplemented
50C5×Dic6—36no—
51C5×Dic6—63no—
52D3×#41—32no—
53C2×C32×C5C2411yescharacteristic
54C5×C3⋊D3#4112yescomplemented
55C3×C5×D3—26no—
56C5×#41—32no—
57C5×#41—96no—
58C2×C5×Q8—91no—
59C2×C5×D4—93no—
60C22×C4×C5—93no—
61C2×D32C22×C513yes—
62C32×Q8C22×C511yescharacteristic, complemented
63C4×C3⋊D3C22×C513yes—
64#634C22×C513yescomplemented
65C3×D12C22×C516yescomplemented
66C3×C4×D3C22×C516yescomplemented
67#625C22×C516yescomplemented
68#624C22×C511yescharacteristic
69D3×Dic3C22×C516yescomplemented
70C22×C3×C5—36no—
71C2×C5×D3—32no—
72C2×C5×D3—61no—
73C2×C5×D3—612no—
74C2×C5×D3C22×D316yes—
75C3×C4×C5—23no—
76C3×C4×C5—32no—
77C3×C4×C5C22×D316yes—
78C5×Dic3—36no—
79C5×Dic3—63no—
80C5×Dic3C22×D312yes—
81C3×#41—32no—
82#239C5×D312yescomplemented
83D3×Q8—32no—
84D3×Q8—61no—
85#237—36no—
86D3×D4—36no—
87#235—36no—
88C2×C4×D3—36no—
89C32×C5C2×#4111yescharacteristic, complemented, Hall
90C23×C5—93no—
91C5×Q8—93no—
92C5×Q8D3211yescharacteristic, complemented
93C5×D4—36no—
94C5×D4—96no—
95C2×C4×C5—36no—
96C2×C4×C5—91no—
97C2×C4×C5—96no—
98C2×C4×C5—93no—
99C2×C3⋊D3C23×C511yescharacteristic
100C2×C3×D3C23×C516yes—
101D32—26no—
102C32×C4C23×C513yes—
103C3⋊Dic3C23×C513yes—
104C3×Dic3C23×C512yes—
105C2×#41—91noSylow
106C2×C3×C5—21no—
107C2×C3×C5—66no—
108C2×C3×C5C23×D312yes—
109C5×D3—26no—
110C5×D3—34no—
111C5×D3—62no—
112C22×D3—36no—
113C3×Q8—21no—
114C3×Q8C2×C5×D312yescomplemented
115C3×D4—36no—
116C2×C3×C4—36no—
117#67—312no—
118C2×Dic3—36no—
119D12C2×C5×D316yescomplemented
120C4×D3—36no—
121C4×D3—36no—
122C4×D3—32no—
123C4×D3—63no—
124C4×D3C2×C5×D316yescomplemented
125Dic6—36no—
126Dic6—63no—
127C22×C5—36no—
128C22×C5—91no—
129C22×C5—186no—
130C4×C5—32no—
131C4×C5—93no—
132C4×C5C2×D3213yes—
133C2×C32C24×C511yescharacteristic
134C3⋊D3C5×#4112yescomplemented
135C3×D3—26no—
136#41—32no—
137#41—96no—
138C2×Q8—91no—
139C2×D4—93no—
140C22×C4—93no—
141C3×C5—21no—
142C3×C5D3×#4112yescomplemented
143C22×C3—36no—
144C2×D3—32no—
145C2×D3—61no—
146C2×D3—612no—
147C2×D3C22×C5×D316yes—
148C3×C4—23no—
149C3×C4—32no—
150C3×C4C22×C5×D316yes—
151Dic3—36no—
152Dic3—63no—
153Dic3C22×C5×D312yes—
154C2×C5—66no—
155C2×C5—92no—
156C2×C5C22×D3211yescharacteristic
157C32C2×C5×#4111yescharacteristic, complemented, Sylow
158C23—93no—
159Q8—93no—
160Q8C5×D3211yescharacteristic, complemented
161D4—36no—
162D4—96no—
163C2×C4—36no—
164C2×C4—91no—
165C2×C4—96no—
166C2×C4—93no—
167C2×C3—21no—
168C2×C3—66no—
169C2×C3C23×C5×D312yes—
170D3—26no—
171D3—34no—
172D3—62no—
173C5#6431911yescharacteristic, direct product factor, Sylow
174C22—36no—
175C22—91no—
176C22—186no—
177C4—32no—
178C4—93no—
179C4C2×C5×D3213yes—
180C3—21no—
181C3C5×D3×#4112yescomplemented
182C2—66no—
183C2—92no—
184C2C22×C5×D3211yescharacteristic
subgroupG/H
centerC2×C5C22×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC32×C5×Q8C22
socleC2×C32×C5C24
hypercenterC5×Q8D32
hypocenterC32C2×C5×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C5×Q8

Complements

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

NH ≅ G/Ntype
1. C5×Q8×C3⋊D3C2semidirect
2. C3×C5×#239C2semidirect
3. C5×#3802C2semidirect
4. C4×C5×D32C2semidirect
5. C5×#3795C2semidirect
8. C32×C5×Q8C22semidirect
10. C5×#634C22semidirect
11. C3×C5×D12C22semidirect
12. C3×C4×C5×D3C22semidirect
13. C5×#625C22semidirect
15. C5×D3×Dic3C22semidirect
16. #64319C5direct (N × H)
18. C5×#239D3semidirect
32. Q8×C3⋊D3C2×C5semidirect
33. C3×#239C2×C5semidirect
34. #3802C2×C5semidirect
35. C4×D32C2×C5semidirect
36. #3795C2×C5semidirect
39. C3×C5×Q8C2×D3semidirect
44. C5×D12C2×D3semidirect
49. C4×C5×D3C2×D3semidirect
54. C5×C3⋊D3#41semidirect
62. C32×Q8C22×C5semidirect
64. #634C22×C5semidirect
65. C3×D12C22×C5semidirect
66. C3×C4×D3C22×C5semidirect
67. #625C22×C5semidirect
69. D3×Dic3C22×C5semidirect
82. #239C5×D3semidirect
89. C32×C5C2×#41semidirect
92. C5×Q8D32semidirect
114. C3×Q8C2×C5×D3semidirect
119. D12C2×C5×D3semidirect
124. C4×D3C2×C5×D3semidirect
134. C3⋊D3C5×#41semidirect
142. C3×C5D3×#41semidirect
157. C32C2×C5×#41semidirect
160. Q8C5×D32semidirect
173. C5#64319direct (N × H)
181. C3C5×D3×#41semidirect

Maximal subgroups

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

subgroupindex
C5×Q8×C3⋊D32
C3×C5×#2392
C5×#38022
C4×C5×D322
C5×#37952
C5×D3×#413
#643195