Finite Group Databasebeta

← C3×#66017 #11755694

Subgroups — C3×#66017 #11755694

Order 960.

520 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 182 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×#124C211yesmaximal, characteristic, complemented
2C3×#4093C212yesmaximal, complemented
3C3×#4083C212yesmaximal, complemented
4C3×#4031C214yesmaximal, complemented
5C3×#4028C214yesmaximal, complemented
6C3×#4021C212yesmaximal, complemented
7#66017C311yesmaximal, characteristic, direct product factor, Hall
8C23×C3×D5C2212yescomplemented
9C2×C3×C5×Q8C2211yescharacteristic, complemented
10C2×C3×C5×D4C2211yescharacteristic
11C2×C3×C22⋊D5C2214yes—
12C2×C3×D20C2212yes—
13C2×C3×D20C2212yes—
14C2×C3×Dic10C2212yescomplemented
15C3×C5×C22⋊C4C2214yescomplemented
16C3×C42×C5C2211yescharacteristic, complemented
17C3×C22⋊Dic5C2212yescomplemented
18C3×#685C2214yescomplemented
19C3×#685C2214yescomplemented
20C3×#685C2214yescomplemented
21C3×C4×Dic5C2212yescomplemented
22C3×H(3,𝔽4)—51nomaximal, Hall
23C5×#124C2×C311yescharacteristic, complemented
24#4093C2×C312yescomplemented
25#4083C2×C312yescomplemented
26#4031C2×C314yescomplemented
27#4028C2×C314yescomplemented
28#4021C2×C312yescomplemented
29C23×C3×C5C2312yes—
30C22×C3×D5—42no—
31C22×C3×D5—44no—
32C22×C3×D5C2314yes—
33C3×C5×Q8—41no—
34C3×C5×D4—41no—
35C2×C3×C4×C5C2312yes—
36C2×C3×C4×C5C2311yescharacteristic
37C2×C3×C4×C5C2312yes—
38C3×C22⋊D5—44no—
39C2×C3×Dic5C2314yes—
40C3×D20—42no—
41C3×D20—42no—
42C3×Dic10—42no—
43C3×#124—52no—
44C3×#124—54no—
45C3×#124—52no—
46C3×#124D511yescharacteristic, complemented
47C3×#120—52no—
48C3×#120—54no—
49C23×D5C22×C312yescomplemented
50C2×C5×Q8C22×C311yescharacteristic, complemented
51C2×C5×D4C22×C311yescharacteristic
52C2×C22⋊D5C22×C314yes—
53C2×D20C22×C312yes—
54C2×D20C22×C312yes—
55C2×Dic10C22×C312yescomplemented
56C5×C22⋊C4C22×C314yescomplemented
57C42×C5C22×C311yescharacteristic, complemented
58C22⋊Dic5C22×C312yescomplemented
59#685C22×C314yescomplemented
60#685C22×C314yescomplemented
61#685C22×C314yescomplemented
62C4×Dic5C22×C312yescomplemented
63H(3,𝔽4)—51noSylow
64C22×C3×C5—24no—
65C22×C3×C5—22no—
66C22×C3×C5C2411yescharacteristic
67C2×C3×D5—24no—
68C2×C3×D5—24no—
69C2×C3×D5—24no—
70C2×C3×D5—48no—
71C3×C4×C5—22no—
72C3×C4×C5—22no—
73C3×C4×C5—21no—
74C3×Dic5—24no—
75C24×C3—52no—
76C2×C3×Q8—52no—
77C2×C3×Q8C2×D511yescharacteristic, complemented
78C2×C3×D4—54no—
79C2×C3×D4—52no—
80C2×C3×D4—52no—
81C2×C3×D4C2×D511yescharacteristic
82C3×C22⋊C4—52no—
83C3×C22⋊C4—54no—
84C3×C22⋊C4—54no—
85C3×C22⋊C4—54no—
86C3×C22⋊C4C2×D514yescomplemented
87C3×C42—52no—
88C3×C42C2×D511yescharacteristic, complemented
89C23×C5C23×C312yes—
90C22×D5—42no—
91C22×D5—44no—
92C22×D5C23×C314yes—
93C5×Q8—41no—
94C5×D4—41no—
95C2×C4×C5C23×C312yes—
96C2×C4×C5C23×C311yescharacteristic
97C2×C4×C5C23×C312yes—
98C22⋊D5—44no—
99C2×Dic5C23×C314yes—
100D20—42no—
101D20—42no—
102Dic10—42no—
103#124—52no—
104#124—54no—
105#124—52no—
106#124C3×D511yescharacteristic, complemented
107#120—52no—
108#120—54no—
109C2×C3×C5—42no—
110C2×C3×C5H(5,𝔽2)11yescharacteristic
111C2×C3×C5H(5,𝔽2)12yes—
112C3×D5—44no—
113C23×C3—54no—
114C23×C3—202no—
115C23×C3—204no—
116C23×C3C22×D512yes—
117C3×Q8—41no—
118C3×Q8—202no—
119C3×D4—41no—
120C3×D4—202no—
121C3×D4—202no—
122C3×D4—204no—
123C2×C3×C4—54no—
124C2×C3×C4C22×D512yes—
125C2×C3×C4C22×D511yescharacteristic
126C2×C3×C4C22×D512yes—
127C22×C5—24no—
128C22×C5—22no—
129C22×C5C24×C311yescharacteristic
130C2×D5—24no—
131C2×D5—24no—
132C2×D5—24no—
133C2×D5—48no—
134C4×C5—22no—
135C4×C5—22no—
136C4×C5—21no—
137Dic5—24no—
138C24—52no—
139C2×Q8—52no—
140C2×Q8C2×C3×D511yescharacteristic, complemented
141C2×D4—54no—
142C2×D4—52no—
143C2×D4—52no—
144C2×D4C2×C3×D511yescharacteristic
145C22⋊C4—52no—
146C22⋊C4—54no—
147C22⋊C4—54no—
148C22⋊C4—54no—
149C22⋊C4C2×C3×D514yescomplemented
150C42—52no—
151C42C2×C3×D511yescharacteristic, complemented
152C3×C5H(3,𝔽4)11yescharacteristic, complemented, Hall
153C22×C3—22no—
154C22×C3—24no—
155C22×C3—104no—
156C22×C3—104no—
157C22×C3—104no—
158C22×C3—208no—
159C22×C3C23×D511yescharacteristic
160C3×C4—21no—
161C3×C4—22no—
162C3×C4—22no—
163C3×C4—104no—
164C2×C5—42no—
165C2×C5C3×H(5,𝔽2)11yescharacteristic
166C2×C5C3×H(5,𝔽2)12yes—
167D5—44no—
168C23—54no—
169C23—202no—
170C23—204no—
171C23C22×C3×D512yes—
172Q8—41no—
173Q8—202no—
174D4—41no—
175D4—202no—
176D4—202no—
177D4—204no—
178C2×C4—54no—
179C2×C4C22×C3×D512yes—
180C2×C4C22×C3×D511yescharacteristic
181C2×C4C22×C3×D512yes—
182C2×C3—42no—
183C2×C3—204no—
184C2×C3#414411yescharacteristic
185C2×C3#414912yes—
186C5C3×H(3,𝔽4)11yescharacteristic, complemented, Sylow
187C22—22no—
188C22—24no—
189C22—104no—
190C22—104no—
191C22—104no—
192C22—208no—
193C22C23×C3×D511yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—104no—
198C3#6601711yescharacteristic, direct product factor, Sylow
199C2—42no—
200C2—204no—
201C2C3×#414411yescharacteristic
202C2C3×#414912yes—
subgroupG/H
centerC22×C3C23×D5
commutator, cosocleC22×C5C24×C3
FrattiniC22C23×C3×D5
FittingC3×C5×#124C2
socleC22×C3×C5C24
hypercenterC3×#124D5
hypocenterC5C3×H(3,𝔽4)
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C3×C5×#124C2semidirect
2. C3×#4093C2semidirect
3. C3×#4083C2semidirect
4. C3×#4031C2semidirect
5. C3×#4028C2semidirect
6. C3×#4021C2semidirect
7. #66017C3direct (N × H)
8. C23×C3×D5C22semidirect
9. C2×C3×C5×Q8C22semidirect
14. C2×C3×Dic10C22semidirect
15. C3×C5×C22⋊C4C22semidirect
16. C3×C42×C5C22semidirect
17. C3×C22⋊Dic5C22semidirect
18. C3×#685C22semidirect
19. C3×#685C22semidirect
20. C3×#685C22semidirect
21. C3×C4×Dic5C22semidirect
23. C5×#124C2×C3semidirect
24. #4093C2×C3semidirect
25. #4083C2×C3semidirect
26. #4031C2×C3semidirect
27. #4028C2×C3semidirect
28. #4021C2×C3semidirect
46. C3×#124D5semidirect
49. C23×D5C22×C3semidirect
50. C2×C5×Q8C22×C3semidirect
55. C2×Dic10C22×C3semidirect
56. C5×C22⋊C4C22×C3semidirect
57. C42×C5C22×C3semidirect
58. C22⋊Dic5C22×C3semidirect
59. #685C22×C3semidirect
60. #685C22×C3semidirect
61. #685C22×C3semidirect
62. C4×Dic5C22×C3semidirect
77. C2×C3×Q8C2×D5semidirect
86. C3×C22⋊C4C2×D5semidirect
88. C3×C42C2×D5semidirect
106. #124C3×D5semidirect
140. C2×Q8C2×C3×D5semidirect
149. C22⋊C4C2×C3×D5semidirect
151. C42C2×C3×D5semidirect
152. C3×C5H(3,𝔽4)semidirect
186. C5C3×H(3,𝔽4)semidirect
198. C3#66017direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#1242
C3×#40932
C3×#40832
C3×#40312
C3×#40282
C3×#40212
#660173
C3×H(3,𝔽4)5