Finite Group Databasebeta

← C3×C5×#932 #13087948

Subgroups — C3×C5×#932 #13087948

Order 1440.

326 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C2×C32×C5×D4C211yesmaximal, characteristic, complemented
2C3×C5×#235C211yesmaximal, characteristic, complemented
3C3×C5×#214C212yesmaximal, complemented
4C3×C5×#213C212yesmaximal, complemented
5C3×C5×#208C211yesmaximal, characteristic, complemented
6C3×C5×AGL(1,ℤ/8)—31nomaximal
7C5×#932C311yesmaximal, characteristic, direct product factor
8C23×C32×C5—21no—
9C32×C5×D4—21no—
10C32×C5×D4C2212yes—
11C2×C32×C4×C5C2211yescharacteristic
12C3×C5×#67—21no—
13C3×C5×D12C2211yescharacteristic, complemented
14C3×C4×C5×D3—21no—
15C3×C5×Dic6C2211yescharacteristic, complemented
16C3×C5×C3⋊C8C2212yescomplemented
17C3×#932C511yesmaximal, characteristic, direct product factor, Hall
18C3×C5×#41—31no—
19C2×C3×C5×D4—21no—
20C2×C3×C5×D4D311yescharacteristic, complemented
21C2×C3×C5×D4C2×C311yescharacteristic, complemented
22C5×#235C2×C311yescharacteristic, complemented
23C3×C5×SD16—32no—
24C3×C5×D8—32no—
25C3×C5×M16—31no—
26C5×#214C2×C312yescomplemented
27C5×#213C2×C312yescomplemented
28C5×#208C2×C311yescharacteristic, complemented
29C22×C32×C5—22no—
30C22×C32×C5—42no—
31C22×C32×C5D411yescharacteristic
32C2×C3×C5×D3—21no—
33C32×C4×C5D411yescharacteristic
34C32×C4×C5C2311yescharacteristic
35C3×C5×Dic3—21no—
36C5×AGL(1,ℤ/8)—31noHall
37C2×C32×D4C2×C511yescharacteristic, complemented
38C3×#235C2×C511yescharacteristic, complemented
39C3×#214C2×C512yescomplemented
40C3×#213C2×C512yescomplemented
41C3×#208C2×C511yescharacteristic, complemented
42C23×C3×C5—22no—
43C23×C3×C5—21no—
44C23×C3×C5—21no—
45C3×C5×Q8—31no—
46C3×C5×D4—22no—
47C3×C5×D4—22no—
48C3×C5×D4—21no—
49C3×C5×D4—21no—
50C3×C5×D4—31no—
51C3×C5×D4—61no—
52C3×C5×D4C2×D312yes—
53C3×C5×D4C22×C312yes—
54C2×C3×C4×C5—21no—
55C2×C3×C4×C5—61no—
56C2×C3×C4×C5C2×D311yescharacteristic
57C2×C3×C4×C5C22×C311yescharacteristic
58C5×#67—21no—
59C5×D12C22×C311yescharacteristic, complemented
60C4×C5×D3—21no—
61C5×Dic6C22×C311yescharacteristic, complemented
62C3×C5×C8—32no—
63C5×C3⋊C8C22×C312yescomplemented
64C3×AGL(1,ℤ/8)—31no—
65#932C3×C511yescharacteristic, direct product factor
66C2×C32×C5—21no—
67C2×C32×C5—42no—
68C2×C32×C5C2×D411yescharacteristic
69C3×C5×D3—41no—
70C5×#41—31no—
71C2×C5×D4C3×D311yescharacteristic, complemented
72C5×SD16—32no—
73C5×D8—32no—
74C5×M16—31no—
75C23×C32—21no—
76C32×D4—21no—
77C32×D4C22×C512yes—
78C2×C32×C4C22×C511yescharacteristic
79C3×#67—21no—
80C3×D12C22×C511yescharacteristic, complemented
81C3×C4×D3—21no—
82C3×Dic6C22×C511yescharacteristic, complemented
83C3×C3⋊C8C22×C512yescomplemented
84C22×C3×C5—24no—
85C22×C3×C5—22no—
86C22×C3×C5—22no—
87C22×C3×C5—21no—
88C22×C3×C5—44no—
89C22×C3×C5—42no—
90C22×C3×C5—42no—
91C22×C3×C5—61no—
92C22×C3×C5#6711yescharacteristic
93C22×C3×C5C3×D411yescharacteristic
94C2×C5×D3—21no—
95C3×C4×C5—21no—
96C3×C4×C5—21no—
97C3×C4×C5—61no—
98C3×C4×C5#6711yescharacteristic
99C3×C4×C5C3×D411yescharacteristic
100C3×C4×C5C22×D311yescharacteristic
101C3×C4×C5C23×C311yescharacteristic
102C5×Dic3—21no—
103C3×#41—31no—
104C2×C3×D4—21no—
105C2×C3×D4C5×D311yescharacteristic, complemented
106C2×C3×D4C2×C3×C511yescharacteristic, complemented
107#235C2×C3×C511yescharacteristic, complemented
108C3×SD16—32no—
109C3×D8—32no—
110C3×M16—31no—
111#214C2×C3×C512yescomplemented
112#213C2×C3×C512yescomplemented
113#208C2×C3×C511yescharacteristic, complemented
114C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
115C23×C5—21no—
116C5×Q8—31no—
117C5×D4—21no—
118C5×D4—31no—
119C5×D4—61no—
120C5×D4C2×C3×D312yes—
121C2×C4×C5—61no—
122C2×C4×C5C2×C3×D311yescharacteristic
123C5×C8—32no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32C5×D411yescharacteristic
127C2×C3×D3—21no—
128C32×C4C5×D411yescharacteristic
129C32×C4C23×C511yescharacteristic
130C3×Dic3—21no—
131AGL(1,ℤ/8)—31noSylow
132C2×C3×C5—22no—
133C2×C3×C5—21no—
134C2×C3×C5—21no—
135C2×C3×C5—21no—
136C2×C3×C5—44no—
137C2×C3×C5—42no—
138C2×C3×C5—42no—
139C2×C3×C5—121no—
140C2×C3×C5C2×#6711yescharacteristic
141C2×C3×C5C2×C3×D411yescharacteristic
142C5×D3—41no—
143C23×C3—22no—
144C23×C3—21no—
145C23×C3—21no—
146C3×Q8—31no—
147C3×D4—22no—
148C3×D4—21no—
149C3×D4—21no—
150C3×D4—22no—
151C3×D4—31no—
152C3×D4—61no—
153C3×D4C2×C5×D312yes—
154C3×D4C22×C3×C512yes—
155C2×C3×C4—21no—
156C2×C3×C4—61no—
157C2×C3×C4C2×C5×D311yescharacteristic
158C2×C3×C4C22×C3×C511yescharacteristic
159#67—21no—
160D12C22×C3×C511yescharacteristic, complemented
161C4×D3—21no—
162Dic6C22×C3×C511yescharacteristic, complemented
163C3×C8—32no—
164C3⋊C8C22×C3×C512yescomplemented
165C22×C5—22no—
166C22×C5—42no—
167C22×C5—61no—
168C22×C5C3×#6711yescharacteristic
169C4×C5—61no—
170C4×C5C3×#6711yescharacteristic
171C4×C5C22×C3×D311yescharacteristic
172C2×C32—21no—
173C2×C32—42no—
174C2×C32C2×C5×D411yescharacteristic
175C3×D3—41no—
176#41—31no—
177C2×D4C3×C5×D311yescharacteristic, complemented
178SD16—32no—
179D8—32no—
180M16—31no—
181C3×C5—21no—
182C3×C5#93211yescharacteristic, direct product factor
183C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
184C22×C3—24no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—21no—
188C22×C3—44no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—61no—
192C22×C3C5×#6711yescharacteristic
193C22×C3C3×C5×D411yescharacteristic
194C2×D3—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—61no—
198C3×C4C5×#6711yescharacteristic
199C3×C4C3×C5×D411yescharacteristic
200C3×C4C22×C5×D311yescharacteristic
201C3×C4C23×C3×C511yescharacteristic
202Dic3—21no—
203C2×C5—21no—
204C2×C5—42no—
205C2×C5—121no—
206C2×C5C2×C3×#6711yescharacteristic
207C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
208C23—21no—
209Q8—31no—
210D4—21no—
211D4—31no—
212D4—61no—
213D4C2×C3×C5×D312yes—
214C2×C4—61no—
215C2×C4C2×C3×C5×D311yescharacteristic
216C8—32no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—44no—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3—121no—
225C2×C3C2×C5×#6711yescharacteristic
226C2×C3C2×C3×C5×D411yescharacteristic
227D3—41no—
228C5C3×#93211yescharacteristic, direct product factor, Sylow
229C22—22no—
230C22—42no—
231C22—61no—
232C22C3×C5×#6711yescharacteristic
233C4—61no—
234C4C3×C5×#6711yescharacteristic
235C4C22×C3×C5×D311yescharacteristic
236C3—21no—
237C3C5×#93211yescharacteristic, direct product factor
238C3C3×C5×AGL(1,ℤ/8)11yescharacteristic, complemented
239C2—21no—
240C2—42no—
241C2—121no—
242C2C2×C3×C5×#6711yescharacteristic
subgroupG/H
centerC2×C3×C5C2×#67
commutator, cosocleC3×C4C23×C3×C5
FrattiniC4C22×C3×C5×D3
FittingC2×C32×C5×D4C2
socleC2×C32×C5C2×D4
hypercenterC2×C3×C5×D4D3
hypocenterC3C3×C5×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×D4C2semidirect
2. C3×C5×#235C2semidirect
3. C3×C5×#214C2semidirect
4. C3×C5×#213C2semidirect
5. C3×C5×#208C2semidirect
7. C5×#932C3direct (N × H)
13. C3×C5×D12C22semidirect
15. C3×C5×Dic6C22semidirect
16. C3×C5×C3⋊C8C22semidirect
17. C3×#932C5direct (N × H)
20. C2×C3×C5×D4D3semidirect
21. C2×C3×C5×D4C2×C3semidirect
22. C5×#235C2×C3semidirect
26. C5×#214C2×C3semidirect
27. C5×#213C2×C3semidirect
28. C5×#208C2×C3semidirect
37. C2×C32×D4C2×C5semidirect
38. C3×#235C2×C5semidirect
39. C3×#214C2×C5semidirect
40. C3×#213C2×C5semidirect
41. C3×#208C2×C5semidirect
59. C5×D12C22×C3semidirect
61. C5×Dic6C22×C3semidirect
63. C5×C3⋊C8C22×C3semidirect
65. #932C3×C5direct (N × H)
71. C2×C5×D4C3×D3semidirect
80. C3×D12C22×C5semidirect
82. C3×Dic6C22×C5semidirect
83. C3×C3⋊C8C22×C5semidirect
105. C2×C3×D4C5×D3semidirect
106. C2×C3×D4C2×C3×C5semidirect
107. #235C2×C3×C5semidirect
111. #214C2×C3×C5semidirect
112. #213C2×C3×C5semidirect
113. #208C2×C3×C5semidirect
114. C32×C5AGL(1,ℤ/8)semidirect
160. D12C22×C3×C5semidirect
162. Dic6C22×C3×C5semidirect
164. C3⋊C8C22×C3×C5semidirect
177. C2×D4C3×C5×D3semidirect
182. C3×C5#932direct (N × H)
183. C3×C5C3×AGL(1,ℤ/8)semidirect
207. C32C5×AGL(1,ℤ/8)semidirect
228. C5C3×#932direct (N × H)
237. C3C5×#932direct (N × H)
238. C3C3×C5×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C5×D42
C3×C5×#2352
C3×C5×#2142
C3×C5×#2132
C3×C5×#2082
C3×C5×AGL(1,ℤ/8)3
C5×#9323
C3×#9325