Finite Group Databasebeta

← C5×#63939 #13087211

Subgroups — C5×#63939 #13087211

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×#239C211yesmaximal, characteristic, complemented
2C5×D3×D12C211yesmaximal, characteristic, complemented
3C5×#3755C211yesmaximal, characteristic, complemented
4C3×C5×#215C211yesmaximal, characteristic, complemented
5C5×#3714C211yesmaximal, characteristic, complemented
6C5×#3713C211yesmaximal, characteristic, complemented
7C5×#3711C211yesmaximal, characteristic, complemented
8C5×#949—31nomaximal
9C5×#914—31nomaximal
10C2×C5×D32—21no—
11C32×C5×Q8C2211yescharacteristic, complemented
12C5×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C5×D12—21no—
14C3×C5×D12C2211yescharacteristic
15C3×C5×D12C2211yescharacteristic, complemented
16C3×C4×C5×D3—21no—
17C3×C4×C5×D3C2211yescharacteristic
18C5×#626—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63939C511yesmaximal, characteristic, direct product factor, Hall
22C3×C5×#41—31no—
23C5×#239D311yescharacteristic, complemented
24C5×D3×D4—31no—
25C2×C5×D12—31no—
26C3×C5×SD16—31no—
27C5×#215—31no—
28C5×#215—31no—
29C5×#215—61no—
30C5×#215D311yescharacteristic, complemented
31C5×#213—31no—
32C5×#213—31no—
33C5×#213—31no—
34C5×#208—31no—
35C5×D24—31no—
36C5×ΓL(1,𝔽25)—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3D411yescharacteristic
41C5×D32—42no—
42C32×C4×C5—21no—
43C32×C4×C5C2311yescharacteristic
44C3×C5×Dic3D411yescharacteristic
45C5×AGL(1,ℤ/8)—91noHall
46C3×#239C2×C511yescharacteristic, complemented
47D3×D12C2×C511yescharacteristic, complemented
48#3755C2×C511yescharacteristic, complemented
49C3×#215C2×C511yescharacteristic, complemented
50#3714C2×C511yescharacteristic, complemented
51#3713C2×C511yescharacteristic, complemented
52#3711C2×C511yescharacteristic, complemented
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C3×C5×Q8—21no—
56C3×C5×Q8C2×D311yescharacteristic, complemented
57C3×C5×Q8C2×D311yescharacteristic, complemented
58C3×C5×D4—31no—
59C3×C5×D4—31no—
60C3×C5×D4—61no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—61no—
63C5×#67—61no—
64C5×D12—21no—
65C5×D12—31no—
66C5×D12—31no—
67C5×D12—61no—
68C5×D12—61no—
69C5×D12C2×D311yescharacteristic
70C5×D12C2×D311yescharacteristic, complemented
71C4×C5×D3—21no—
72C4×C5×D3C2×D311yescharacteristic
73C3×C5×C8—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#949—31no—
79#914—31no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C3×C5×D3—21no—
83C3×C5×D3—41no—
84C3×C5×D3—41no—
85C5×#41—31no—
86C2×C5×D4—91no—
87C5×SD16—31no—
88C5×SD16—91no—
89C5×D8—91no—
90C5×D8—91no—
91C5×M16—91no—
92C2×D32—21no—
93C32×Q8C22×C511yescharacteristic, complemented
94C4⋊(C3⋊D3)C22×C511yescharacteristic
95C3×D12—21no—
96C3×D12C22×C511yescharacteristic
97C3×D12C22×C511yescharacteristic, complemented
98C3×C4×D3—21no—
99C3×C4×D3C22×C511yescharacteristic
100#626—21no—
101C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
102C3×C3⋊C8C22×C511yescharacteristic, complemented
103C22×C3×C5—31no—
104C22×C3×C5—61no—
105C22×C3×C5—61no—
106C2×C5×D3—21no—
107C2×C5×D3—21no—
108C2×C5×D3—61no—
109C2×C5×D3—61no—
110C2×C5×D3—121no—
111C2×C5×D3—122no—
112C2×C5×D3—122no—
113C2×C5×D3#6711yescharacteristic
114C3×C4×C5—21no—
115C3×C4×C5—21no—
116C3×C4×C5—21no—
117C3×C4×C5—31no—
118C3×C4×C5—41no—
119C3×C4×C5C22×D311yescharacteristic
120C3×C4×C5C22×D311yescharacteristic
121C5×Dic3#6711yescharacteristic
122C3×#41—31no—
123#239C5×D311yescharacteristic, complemented
124D3×D4—31no—
125C2×D12—31no—
126C3×SD16—31no—
127#215—31no—
128#215—31no—
129#215—61no—
130#215C5×D311yescharacteristic, complemented
131#213—31no—
132#213—31no—
133#213—31no—
134#208—31no—
135D24—31no—
136ΓL(1,𝔽25)—31no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—181no—
139C5×Q8D3211yescharacteristic, complemented
140C5×D4—31no—
141C5×D4—31no—
142C5×D4—61no—
143C5×D4—91no—
144C5×D4—181no—
145C2×C4×C5—31no—
146C2×C4×C5—61no—
147C5×C8—31no—
148C5×C8—91no—
149C2×C3⋊D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3C5×D411yescharacteristic
153D32—42no—
154C32×C4—21no—
155C32×C4C23×C511yescharacteristic
156C3×Dic3C5×D411yescharacteristic
157AGL(1,ℤ/8)—91noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5—121no—
161C2×C3×C5—121no—
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5C2×#6711yescharacteristic
164C5×D3—21no—
165C5×D3—41no—
166C5×D3—41no—
167C5×D3—121no—
168C5×D3—121no—
169C5×D3—241no—
170C22×D3—61no—
171C22×D3—61no—
172C3×Q8—21no—
173C3×Q8C2×C5×D311yescharacteristic, complemented
174C3×Q8C2×C5×D311yescharacteristic, complemented
175C3×D4—31no—
176C3×D4—31no—
177C3×D4—61no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180#67—61no—
181D12—21no—
182D12—31no—
183D12—31no—
184D12—61no—
185D12—61no—
186D12C2×C5×D311yescharacteristic
187D12C2×C5×D311yescharacteristic, complemented
188C4×D3—21no—
189C4×D3C2×C5×D311yescharacteristic
190C3×C8—31no—
191C3⋊C8—31no—
192C3⋊C8—31no—
193C3⋊C8—61no—
194C3⋊C8C2×C5×D311yescharacteristic, complemented
195C22×C5—31no—
196C22×C5—61no—
197C22×C5—61no—
198C22×C5—181no—
199C22×C5—362no—
200C4×C5—21no—
201C4×C5—31no—
202C4×C5C2×D3211yescharacteristic
203C2×C32C2×C5×D411yescharacteristic
204C3⋊D3—41no—
205C3×D3—21no—
206C3×D3—41no—
207C3×D3—41no—
208#41—31no—
209C2×D4—91no—
210SD16—31no—
211SD16—91no—
212D8—91no—
213D8—91no—
214M16—91no—
215C3×C5—21no—
216C3×C5#91411yescharacteristic, complemented
217C3×C5#94911yescharacteristic, complemented
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—61no—
221C2×D3—21no—
222C2×D3—21no—
223C2×D3—61no—
224C2×D3—61no—
225C2×D3—121no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3C5×#6711yescharacteristic
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—41no—
234C3×C4C22×C5×D311yescharacteristic
235C3×C4C22×C5×D311yescharacteristic
236Dic3C5×#6711yescharacteristic
237C2×C5—61no—
238C2×C5—121no—
239C2×C5—121no—
240C2×C5—361no—
241C2×C5D3×#6711yescharacteristic
242C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
243C23—181no—
244Q8C5×D3211yescharacteristic, complemented
245D4—31no—
246D4—31no—
247D4—61no—
248D4—91no—
249D4—181no—
250C2×C4—31no—
251C2×C4—61no—
252C8—31no—
253C8—91no—
254C2×C3—21no—
255C2×C3—61no—
256C2×C3—121no—
257C2×C3—121no—
258C2×C3C5×D3×D411yescharacteristic
259C2×C3C2×C5×#6711yescharacteristic
260D3—21no—
261D3—41no—
262D3—41no—
263D3—121no—
264D3—121no—
265D3—241no—
266C5#6393911yescharacteristic, direct product factor, Sylow
267C22—31no—
268C22—61no—
269C22—61no—
270C22—181no—
271C22—362no—
272C4—21no—
273C4—31no—
274C4C2×C5×D3211yescharacteristic
275C3—21no—
276C3C5×#91411yescharacteristic, complemented
277C3C5×#94911yescharacteristic, complemented
278C2—61no—
279C2—121no—
280C2—121no—
281C2—361no—
282C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC2×C5D3×#67
commutator, cosocleC32×C4C23×C5
FrattiniC4C2×C5×D32
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
hypercenterC5×Q8D32
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C5×#239C2semidirect
2. C5×D3×D12C2semidirect
3. C5×#3755C2semidirect
4. C3×C5×#215C2semidirect
5. C5×#3714C2semidirect
6. C5×#3713C2semidirect
7. C5×#3711C2semidirect
11. C32×C5×Q8C22semidirect
15. C3×C5×D12C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. #63939C5direct (N × H)
23. C5×#239D3semidirect
30. C5×#215D3semidirect
46. C3×#239C2×C5semidirect
47. D3×D12C2×C5semidirect
48. #3755C2×C5semidirect
49. C3×#215C2×C5semidirect
50. #3714C2×C5semidirect
51. #3713C2×C5semidirect
52. #3711C2×C5semidirect
56. C3×C5×Q8C2×D3semidirect
57. C3×C5×Q8C2×D3semidirect
70. C5×D12C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
93. C32×Q8C22×C5semidirect
97. C3×D12C22×C5semidirect
101. C3⋊(C3⋊C8)C22×C5semidirect
102. C3×C3⋊C8C22×C5semidirect
123. #239C5×D3semidirect
130. #215C5×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
139. C5×Q8D32semidirect
173. C3×Q8C2×C5×D3semidirect
174. C3×Q8C2×C5×D3semidirect
187. D12C2×C5×D3semidirect
194. C3⋊C8C2×C5×D3semidirect
216. C3×C5#914semidirect
217. C3×C5#949semidirect
242. C32C5×AGL(1,ℤ/8)semidirect
244. Q8C5×D32semidirect
266. C5#63939direct (N × H)
276. C3C5×#914semidirect
277. C3C5×#949semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#2392
C5×D3×D122
C5×#37552
C3×C5×#2152
C5×#37142
C5×#37132
C5×#37112
C5×#9493
C5×#9143
#639395