Finite Group Databasebeta

← C5×#63932 #13087204

Subgroups — C5×#63932 #13087204

Order 1440.

292 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 80 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×D3×D4C211yesmaximal, characteristic, complemented
2C5×#3795C211yesmaximal, characteristic, complemented
3C5×#3754C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5C5×#3716C211yesmaximal, characteristic, complemented
6C5×#3713C211yesmaximal, characteristic, complemented
7C5×#3711C211yesmaximal, characteristic, complemented
8C5×#932—31nomaximal
9C5×#911—31nomaximal
10C22×C3×C5×D3—21no—
11C32×C5×D4C2211yescharacteristic
12C5×#634C2211yescharacteristic, complemented
13C3×C5×#67—21no—
14C3×C5×D12C2211yescharacteristic
15C3×C5×D12C2211yescharacteristic, complemented
16C3×C4×C5×D3C2211yescharacteristic
17C5×#625—21no—
18C5×D3×Dic3—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63932C511yesmaximal, characteristic, direct product factor, Hall
22C2×C3×C5×D4—31no—
23C5×#237—31no—
24C5×D3×D4D311yescharacteristic, complemented
25C5×#235—31no—
26C3×C5×D8—31no—
27C5×#214—31no—
28C5×#214—31no—
29C5×#214—31no—
30C5×#214—61no—
31C5×#213—31no—
32C5×#213—31no—
33C5×#213D311yescharacteristic, complemented
34C5×#208—31no—
35C5×#204—31no—
36C5×ΓL(1,𝔽25)—31no—
37C22×C32×C5—21no—
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—42no—
41C2×C3×C5×D3D411yescharacteristic
42C32×C4×C5C2311yescharacteristic
43C5×C3⋊Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45C5×AGL(1,ℤ/8)—91noHall
46C3×D3×D4C2×C511yescharacteristic, complemented
47#3795C2×C511yescharacteristic, complemented
48#3754C2×C511yescharacteristic, complemented
49C3×#213C2×C511yescharacteristic, complemented
50#3716C2×C511yescharacteristic, complemented
51#3713C2×C511yescharacteristic, complemented
52#3711C2×C511yescharacteristic, complemented
53C23×C3×C5—61no—
54C22×C5×D3—21no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4—31no—
58C3×C5×D4—61no—
59C3×C5×D4C2×D311yescharacteristic
60C3×C5×D4C2×D311yescharacteristic
61C2×C3×C4×C5—31no—
62C5×#67—21no—
63C5×#67—61no—
64C5×#67—61no—
65C2×C5×Dic3—61no—
66C5×D12C2×D311yescharacteristic
67C5×D12C2×D311yescharacteristic, complemented
68C4×C5×D3—61no—
69C4×C5×D3C2×D311yescharacteristic
70C5×Dic6—31no—
71C5×Dic6—31no—
72C5×Dic6—61no—
73C3×C5×C8—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#932—31no—
79#911—31no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×C5×D3—21no—
83C3×C5×D3—41no—
84C3×C5×D3—41no—
85C5×#41—91no—
86C2×C5×D4—31no—
87C5×SD16—91no—
88C5×SD16—91no—
89C5×D8—31no—
90C5×D8—91no—
91C5×M16—91no—
92C22×C3×D3—21no—
93C32×D4C22×C511yescharacteristic
94#634C22×C511yescharacteristic, complemented
95C3×#67—21no—
96C3×D12C22×C511yescharacteristic
97C3×D12C22×C511yescharacteristic, complemented
98C3×C4×D3C22×C511yescharacteristic
99#625—21no—
100D3×Dic3—21no—
101C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
102C3×C3⋊C8C22×C511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—31no—
106C22×C3×C5—41no—
107C22×C3×C5—61no—
108C22×C3×C5—61no—
109C22×C3×C5—122no—
110C2×C5×D3—21no—
111C2×C5×D3—21no—
112C2×C5×D3—42no—
113C2×C5×D3#6711yescharacteristic
114C3×C4×C5—21no—
115C3×C4×C5—31no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118C5×Dic3—61no—
119C5×Dic3—61no—
120C5×Dic3—121no—
121C5×Dic3#6711yescharacteristic
122C2×C3×D4—31no—
123#237—31no—
124D3×D4C5×D311yescharacteristic, complemented
125#235—31no—
126C3×D8—31no—
127#214—31no—
128#214—31no—
129#214—31no—
130#214—61no—
131#213—31no—
132#213—31no—
133#213C5×D311yescharacteristic, complemented
134#208—31no—
135#204—31no—
136ΓL(1,𝔽25)—31no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—61no—
139C5×Q8—91no—
140C5×D4—31no—
141C5×D4—31no—
142C5×D4—61no—
143C5×D4—181no—
144C5×D4D3211yescharacteristic
145C2×C4×C5—31no—
146C2×C4×C5—181no—
147C5×C8—31no—
148C5×C8—91no—
149C22×C32—21no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—42no—
153C2×C3×D3C5×D411yescharacteristic
154C32×C4C23×C511yescharacteristic
155C3⋊Dic3—21no—
156C3×Dic3C5×D411yescharacteristic
157AGL(1,ℤ/8)—91noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—61no—
162C2×C3×C5—81no—
163C2×C3×C5—121no—
164C2×C3×C5—121no—
165C2×C3×C5D3×D411yescharacteristic
166C2×C3×C5C2×#6711yescharacteristic
167C5×D3—21no—
168C5×D3—41no—
169C5×D3—41no—
170C23×C3—61no—
171C22×D3—21no—
172C3×D4—21no—
173C3×D4—31no—
174C3×D4—31no—
175C3×D4—61no—
176C3×D4C2×C5×D311yescharacteristic
177C3×D4C2×C5×D311yescharacteristic
178C2×C3×C4—31no—
179#67—21no—
180#67—61no—
181#67—61no—
182C2×Dic3—61no—
183D12C2×C5×D311yescharacteristic
184D12C2×C5×D311yescharacteristic, complemented
185C4×D3—61no—
186C4×D3C2×C5×D311yescharacteristic
187Dic6—31no—
188Dic6—31no—
189Dic6—61no—
190C3×C8—31no—
191C3⋊C8—31no—
192C3⋊C8—31no—
193C3⋊C8—61no—
194C3⋊C8C2×C5×D311yescharacteristic, complemented
195C22×C5—21no—
196C22×C5—31no—
197C22×C5—61no—
198C22×C5—61no—
199C22×C5—122no—
200C4×C5—31no—
201C4×C5—181no—
202C4×C5C2×D3211yescharacteristic
203C2×C32—41no—
204C2×C32C2×C5×D411yescharacteristic
205C3×D3—21no—
206C3×D3—41no—
207C3×D3—41no—
208#41—91no—
209C2×D4—31no—
210SD16—91no—
211SD16—91no—
212D8—31no—
213D8—91no—
214M16—91no—
215C3×C5—21no—
216C3×C5#91111yescharacteristic, complemented
217C3×C5#93211yescharacteristic, complemented
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—31no—
221C22×C3—41no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—122no—
225C2×D3—21no—
226C2×D3—21no—
227C2×D3—42no—
228C2×D3C5×#6711yescharacteristic
229C3×C4—21no—
230C3×C4—31no—
231C3×C4C22×C5×D311yescharacteristic
232C3×C4C22×C5×D311yescharacteristic
233Dic3—61no—
234Dic3—61no—
235Dic3—121no—
236Dic3C5×#6711yescharacteristic
237C2×C5—41no—
238C2×C5—61no—
239C2×C5—121no—
240C2×C5—121no—
241C2×C5D3×#6711yescharacteristic
242C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
243C23—61no—
244Q8—91no—
245D4—31no—
246D4—31no—
247D4—61no—
248D4—181no—
249D4C5×D3211yescharacteristic
250C2×C4—31no—
251C2×C4—181no—
252C8—31no—
253C8—91no—
254C2×C3—21no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—61no—
258C2×C3—81no—
259C2×C3—121no—
260C2×C3—121no—
261C2×C3C5×D3×D411yescharacteristic
262C2×C3C2×C5×#6711yescharacteristic
263D3—21no—
264D3—41no—
265D3—41no—
266C5#6393211yescharacteristic, direct product factor, Sylow
267C22—21no—
268C22—31no—
269C22—61no—
270C22—61no—
271C22—122no—
272C4—31no—
273C4—181no—
274C4C2×C5×D3211yescharacteristic
275C3—21no—
276C3C5×#91111yescharacteristic, complemented
277C3C5×#93211yescharacteristic, complemented
278C2—41no—
279C2—61no—
280C2—121no—
281C2—121no—
282C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC2×C5D3×#67
commutator, cosocleC32×C4C23×C5
FrattiniC4C2×C5×D32
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterC5×D4D32
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×D4

Complements

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

NH ≅ G/Ntype
1. C3×C5×D3×D4C2semidirect
2. C5×#3795C2semidirect
3. C5×#3754C2semidirect
4. C3×C5×#213C2semidirect
5. C5×#3716C2semidirect
6. C5×#3713C2semidirect
7. C5×#3711C2semidirect
12. C5×#634C22semidirect
15. C3×C5×D12C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. #63932C5direct (N × H)
24. C5×D3×D4D3semidirect
33. C5×#213D3semidirect
46. C3×D3×D4C2×C5semidirect
47. #3795C2×C5semidirect
48. #3754C2×C5semidirect
49. C3×#213C2×C5semidirect
50. #3716C2×C5semidirect
51. #3713C2×C5semidirect
52. #3711C2×C5semidirect
67. C5×D12C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
94. #634C22×C5semidirect
97. C3×D12C22×C5semidirect
101. C3⋊(C3⋊C8)C22×C5semidirect
102. C3×C3⋊C8C22×C5semidirect
124. D3×D4C5×D3semidirect
133. #213C5×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
184. D12C2×C5×D3semidirect
194. C3⋊C8C2×C5×D3semidirect
216. C3×C5#911semidirect
217. C3×C5#932semidirect
242. C32C5×AGL(1,ℤ/8)semidirect
266. C5#63932direct (N × H)
276. C3C5×#911semidirect
277. C3C5×#932semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×D42
C5×#37952
C5×#37542
C3×C5×#2132
C5×#37162
C5×#37132
C5×#37112
C5×#9323
C5×#9113
#639325