Finite Group Databasebeta

← C3×C5×#560 #11756649

Subgroups — C3×C5×#560 #11756649

Order 960.

504 conjugacy classes of subgroups; only the 284 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 284 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
1C3×C5×#127C211yesmaximal, characteristic, complemented
2C3×C5×#126C212yesmaximal, complemented
3C3×C5×#123C213yesmaximal, complemented
4C3×C5×#121C216yesmaximal, complemented
5C3×C5×#120C213yesmaximal, complemented
6C5×#560C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×C5C2211yescharacteristic
8C2×C3×C5×D4C2216yes—
9C2×C3×C5×D4C2213yes—
10C2×C3×C5×D4C2213yescomplemented
11C22×C3×C4×C5C2213yescomplemented
12C3×C5×C4⋊C4C2216yescomplemented
13C3×C5×C22⋊C4C2216yes—
14C3×C5×C22⋊C4C2213yes—
15C3×C5×C22⋊C4C2213yescomplemented
16C3×C42×C5C2211yescharacteristic, complemented
17C3×#560C511yesmaximal, characteristic, direct product factor, Hall
18C5×#127C2×C311yescharacteristic, complemented
19C5×#126C2×C312yescomplemented
20C5×#123C2×C313yescomplemented
21C5×#121C2×C316yescomplemented
22C5×#120C2×C313yescomplemented
23C23×C3×C5—43no—
24C23×C3×C5C2312yes—
25C23×C3×C5C2311yescharacteristic
26C23×C3×C5C2313yes—
27C3×C5×D4—46no—
28C3×C5×D4—43no—
29C3×C5×D4—43no—
30C2×C3×C4×C5—43no—
31C2×C3×C4×C5C2313yes—
32C2×C3×C4×C5C2316yes—
33C3×#127C2×C511yescharacteristic, complemented
34C3×#126C2×C512yescomplemented
35C3×#123C2×C513yescomplemented
36C3×#121C2×C516yescomplemented
37C3×#120C2×C513yescomplemented
38C24×C5C22×C311yescharacteristic
39C2×C5×D4C22×C316yes—
40C2×C5×D4C22×C313yes—
41C2×C5×D4C22×C313yescomplemented
42C22×C4×C5C22×C313yescomplemented
43C5×C4⋊C4C22×C316yescomplemented
44C5×C22⋊C4C22×C316yes—
45C5×C22⋊C4C22×C313yes—
46C5×C22⋊C4C22×C313yescomplemented
47C42×C5C22×C311yescharacteristic, complemented
48#560C3×C511yescharacteristic, direct product factor, Sylow
49C22×C3×C5—23no—
50C22×C3×C5—26no—
51C22×C3×C5—26no—
52C22×C3×C5—23no—
53C22×C3×C5—44no—
54C22×C3×C5C2411yescharacteristic
55C3×C4×C5—26no—
56C3×C4×C5—23no—
57C24×C3C22×C511yescharacteristic
58C2×C3×D4C22×C516yes—
59C2×C3×D4C22×C513yes—
60C2×C3×D4C22×C513yescomplemented
61C22×C3×C4C22×C513yescomplemented
62C3×C4⋊C4C22×C516yescomplemented
63C3×C22⋊C4C22×C516yes—
64C3×C22⋊C4C22×C513yes—
65C3×C22⋊C4C22×C513yescomplemented
66C3×C42C22×C511yescharacteristic, complemented
67C23×C5—43no—
68C23×C5C23×C312yes—
69C23×C5C23×C311yescharacteristic
70C23×C5C23×C313yes—
71C5×D4—46no—
72C5×D4—43no—
73C5×D4—43no—
74C2×C4×C5—43no—
75C2×C4×C5C23×C313yes—
76C2×C4×C5C23×C316yes—
77#127C2×C3×C511yescharacteristic, complemented
78#126C2×C3×C512yescomplemented
79#123C2×C3×C513yescomplemented
80#121C2×C3×C516yescomplemented
81#120C2×C3×C513yescomplemented
82C2×C3×C5—43no—
83C2×C3×C5—42no—
84C2×C3×C5—41no—
85C2×C3×C5H(5,𝔽2)13yes—
86C23×C3—43no—
87C23×C3C23×C512yes—
88C23×C3C23×C511yescharacteristic
89C23×C3C23×C513yes—
90C3×D4—46no—
91C3×D4—43no—
92C3×D4—43no—
93C2×C3×C4—43no—
94C2×C3×C4C23×C513yes—
95C2×C3×C4C23×C516yes—
96C22×C5—23no—
97C22×C5—26no—
98C22×C5—26no—
99C22×C5—23no—
100C22×C5—44no—
101C22×C5C24×C311yescharacteristic
102C4×C5—26no—
103C4×C5—23no—
104C24C22×C3×C511yescharacteristic
105C2×D4C22×C3×C516yes—
106C2×D4C22×C3×C513yes—
107C2×D4C22×C3×C513yescomplemented
108C22×C4C22×C3×C513yescomplemented
109C4⋊C4C22×C3×C516yescomplemented
110C22⋊C4C22×C3×C516yes—
111C22⋊C4C22×C3×C513yes—
112C22⋊C4C22×C3×C513yescomplemented
113C42C22×C3×C511yescharacteristic, complemented
114C3×C5#56011yescharacteristic, direct product factor, Hall
115C22×C3—23no—
116C22×C3—26no—
117C22×C3—26no—
118C22×C3—23no—
119C22×C3—44no—
120C22×C3C24×C511yescharacteristic
121C3×C4—26no—
122C3×C4—23no—
123C2×C5—42no—
124C2×C5—41no—
125C2×C5—43no—
126C2×C5C3×H(5,𝔽2)13yes—
127C23—43no—
128C23C23×C3×C512yes—
129C23C23×C3×C511yescharacteristic
130C23C23×C3×C513yes—
131D4—46no—
132D4—43no—
133D4—43no—
134C2×C4—43no—
135C2×C4C23×C3×C513yes—
136C2×C4C23×C3×C516yes—
137C2×C3—42no—
138C2×C3—41no—
139C2×C3—43no—
140C2×C3C5×H(5,𝔽2)13yes—
141C5C3×#56011yescharacteristic, direct product factor, Sylow
142C22—23no—
143C22—26no—
144C22—26no—
145C22—23no—
146C22—44no—
147C22C24×C3×C511yescharacteristic
148C4—26no—
149C4—23no—
150C3C5×#56011yescharacteristic, direct product factor, Sylow
151C2—42no—
152C2—41no—
153C2—43no—
154C2C3×C5×H(5,𝔽2)13yes—
subgroupG/H
center, socleC22×C3×C5C24
commutator, Frattini, cosocleC22C24×C3×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#127C2semidirect
2. C3×C5×#126C2semidirect
3. C3×C5×#123C2semidirect
4. C3×C5×#121C2semidirect
5. C3×C5×#120C2semidirect
6. C5×#560C3direct (N × H)
10. C2×C3×C5×D4C22semidirect
11. C22×C3×C4×C5C22semidirect
12. C3×C5×C4⋊C4C22semidirect
15. C3×C5×C22⋊C4C22semidirect
16. C3×C42×C5C22semidirect
17. C3×#560C5direct (N × H)
18. C5×#127C2×C3semidirect
19. C5×#126C2×C3semidirect
20. C5×#123C2×C3semidirect
21. C5×#121C2×C3semidirect
22. C5×#120C2×C3semidirect
33. C3×#127C2×C5semidirect
34. C3×#126C2×C5semidirect
35. C3×#123C2×C5semidirect
36. C3×#121C2×C5semidirect
37. C3×#120C2×C5semidirect
41. C2×C5×D4C22×C3semidirect
42. C22×C4×C5C22×C3semidirect
43. C5×C4⋊C4C22×C3semidirect
46. C5×C22⋊C4C22×C3semidirect
47. C42×C5C22×C3semidirect
48. #560C3×C5direct (N × H)
60. C2×C3×D4C22×C5semidirect
61. C22×C3×C4C22×C5semidirect
62. C3×C4⋊C4C22×C5semidirect
65. C3×C22⋊C4C22×C5semidirect
66. C3×C42C22×C5semidirect
77. #127C2×C3×C5semidirect
78. #126C2×C3×C5semidirect
79. #123C2×C3×C5semidirect
80. #121C2×C3×C5semidirect
81. #120C2×C3×C5semidirect
107. C2×D4C22×C3×C5semidirect
108. C22×C4C22×C3×C5semidirect
109. C4⋊C4C22×C3×C5semidirect
112. C22⋊C4C22×C3×C5semidirect
113. C42C22×C3×C5semidirect
114. C3×C5#560direct (N × H)
141. C5C3×#560direct (N × H)
150. C3C5×#560direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#1272
C3×C5×#1262
C3×C5×#1232
C3×C5×#1212
C3×C5×#1202
C5×#5603
C3×#5605