Finite Group Databasebeta

← C5×#63794 #13087066

Subgroups — C5×#63794 #13087066

Order 1440.

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

normal subgroups only

This group has 294 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
1C5×D3×D12C211yesmaximal, characteristic, complemented
2C5×#3799C211yesmaximal, characteristic, complemented
3C5×#3745C211yesmaximal, characteristic, complemented
4C3×C5×#204C211yesmaximal, characteristic, complemented
5C3×C5×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
6C5×#3717C211yesmaximal, characteristic, complemented
7C5×#3714C211yesmaximal, characteristic, complemented
8C5×#914—31nomaximal
9C5×#908—31nomaximal
10C2×C5×D32—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic
12C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
13C3×C5×D12C2211yescharacteristic
14C3×C4×C5×D3C2211yescharacteristic
15C3×C5×Dic6C2211yescharacteristic, complemented
16C5×#626—21no—
17C5×#626—21no—
18C5×#624—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63794C511yesmaximal, characteristic, direct product factor, Hall
22C5×#239—31no—
23C5×D3×D4—31no—
24C5×#235—31no—
25C2×C5×D12—31no—
26C3×C5×SD16—31no—
27C3×C5×M16—31no—
28C5×#215—31no—
29C5×#213—31no—
30C5×D24—31no—
31C5×D24—31no—
32C5×D24—31no—
33C5×D24—61no—
34C5×#204—31no—
35C5×#204D311yescharacteristic, complemented
36C5×ΓL(1,𝔽25)D311yescharacteristic, complemented
37C2×C5×C3⋊D3—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3D411yescharacteristic
41C5×D32—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45C5×AGL(1,ℤ/8)—91noHall
46D3×D12C2×C511yescharacteristic, complemented
47#3799C2×C511yescharacteristic, complemented
48#3745C2×C511yescharacteristic, complemented
49C3×#204C2×C511yescharacteristic, complemented
50C3×ΓL(1,𝔽25)C2×C511yescharacteristic, complemented
51#3717C2×C511yescharacteristic, complemented
52#3714C2×C511yescharacteristic, complemented
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57C2×C3×C4×C5—31no—
58C5×#67—61no—
59C5×#67—61no—
60C5×D12—31no—
61C5×D12—31no—
62C5×D12—31no—
63C5×D12—31no—
64C5×D12—61no—
65C5×D12—61no—
66C5×D12—61no—
67C5×D12—61no—
68C5×D12C2×D311yescharacteristic
69C4×C5×D3—61no—
70C4×C5×D3—61no—
71C4×C5×D3C2×D311yescharacteristic
72C5×Dic6C2×D311yescharacteristic, complemented
73C3×C5×C8—21no—
74C3×C5×C8—31no—
75C3×C5×C8C2×D311yescharacteristic, complemented
76C3×C5×C8C2×D311yescharacteristic, complemented
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#914—31no—
79#908—31no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C5×C3⋊D3—41no—
83C3×C5×D3—21no—
84C3×C5×D3—41no—
85C5×#41—91no—
86C2×C5×D4—91no—
87C5×SD16—31no—
88C5×SD16—91no—
89C5×D8—91no—
90C5×D8—91no—
91C5×M16—31no—
92C2×D32—21no—
93C4⋊(C3⋊D3)C22×C511yescharacteristic
94C4⋊(C3⋊D3)C22×C511yescharacteristic, complemented
95C3×D12C22×C511yescharacteristic
96C3×C4×D3C22×C511yescharacteristic
97C3×Dic6C22×C511yescharacteristic, complemented
98#626—21no—
99#626—21no—
100#624—21no—
101C32×C8C22×C511yescharacteristic, complemented
102C3×C3⋊C8C22×C511yescharacteristic, complemented
103C22×C3×C5—31no—
104C22×C3×C5—61no—
105C2×C5×D3—21no—
106C2×C5×D3—61no—
107C2×C5×D3—61no—
108C2×C5×D3—61no—
109C2×C5×D3—61no—
110C2×C5×D3—121no—
111C2×C5×D3—121no—
112C2×C5×D3—122no—
113C2×C5×D3—122no—
114C2×C5×D3D1211yescharacteristic
115C3×C4×C5—21no—
116C3×C4×C5—31no—
117C3×C4×C5—61no—
118C3×C4×C5C22×D311yescharacteristic
119C3×C4×C5C22×D311yescharacteristic
120C5×Dic3—21no—
121C5×Dic3D1211yescharacteristic
122#239—31no—
123D3×D4—31no—
124#235—31no—
125C2×D12—31no—
126C3×SD16—31no—
127C3×M16—31no—
128#215—31no—
129#213—31no—
130D24—31no—
131D24—31no—
132D24—31no—
133D24—61no—
134#204—31no—
135#204C5×D311yescharacteristic, complemented
136ΓL(1,𝔽25)C5×D311yescharacteristic, complemented
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—181no—
139C5×Q8—31no—
140C5×D4—31no—
141C5×D4—91no—
142C5×D4—91no—
143C5×D4—181no—
144C5×D4—181no—
145C2×C4×C5—31no—
146C2×C4×C5—181no—
147C5×C8—31no—
148C5×C8D3211yescharacteristic, complemented
149C2×C3⋊D3—21no—
150C2×C3⋊D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3C5×D411yescharacteristic
153D32—42no—
154C32×C4C23×C511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3C5×D411yescharacteristic
157AGL(1,ℤ/8)—91noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5—121no—
161C2×C3×C5C2×D1211yescharacteristic
162C2×C3×C5D3×D411yescharacteristic
163C5×D3—21no—
164C5×D3—41no—
165C5×D3—121no—
166C5×D3—121no—
167C5×D3—121no—
168C5×D3—121no—
169C5×D3—241no—
170C5×D3—241no—
171C22×D3—61no—
172C22×D3—61no—
173C3×Q8—31no—
174C3×D4—31no—
175C2×C3×C4—31no—
176#67—61no—
177#67—61no—
178D12—31no—
179D12—31no—
180D12—31no—
181D12—31no—
182D12—61no—
183D12—61no—
184D12—61no—
185D12—61no—
186D12C2×C5×D311yescharacteristic
187C4×D3—61no—
188C4×D3—61no—
189C4×D3C2×C5×D311yescharacteristic
190Dic6C2×C5×D311yescharacteristic, complemented
191C3×C8—21no—
192C3×C8—31no—
193C3×C8C2×C5×D311yescharacteristic, complemented
194C3×C8C2×C5×D311yescharacteristic, complemented
195C3⋊C8C2×C5×D311yescharacteristic, complemented
196C22×C5—31no—
197C22×C5—61no—
198C22×C5—181no—
199C22×C5—181no—
200C22×C5—362no—
201C4×C5—31no—
202C4×C5—61no—
203C4×C5C2×D3211yescharacteristic
204C2×C32C2×C5×D411yescharacteristic
205C3⋊D3—41no—
206C3⋊D3—41no—
207C3×D3—21no—
208C3×D3—41no—
209#41—91no—
210C2×D4—91no—
211SD16—31no—
212SD16—91no—
213D8—91no—
214D8—91no—
215M16—31no—
216C3×C5—21no—
217C3×C5#90811yescharacteristic, complemented
218C3×C5#91411yescharacteristic, complemented
219C22×C3—31no—
220C22×C3—61no—
221C2×D3—21no—
222C2×D3—61no—
223C2×D3—61no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—121no—
227C2×D3—121no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3C5×D1211yescharacteristic
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4C22×C5×D311yescharacteristic
235C3×C4C22×C5×D311yescharacteristic
236Dic3—21no—
237Dic3C5×D1211yescharacteristic
238C2×C5—61no—
239C2×C5—121no—
240C2×C5—361no—
241C2×C5—361no—
242C2×C5D3×D1211yescharacteristic
243C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
244C23—181no—
245Q8—31no—
246D4—31no—
247D4—91no—
248D4—91no—
249D4—181no—
250D4—181no—
251C2×C4—31no—
252C2×C4—181no—
253C8—31no—
254C8C5×D3211yescharacteristic, complemented
255C2×C3—21no—
256C2×C3—61no—
257C2×C3—121no—
258C2×C3C2×C5×D1211yescharacteristic
259C2×C3C5×D3×D411yescharacteristic
260D3—21no—
261D3—41no—
262D3—121no—
263D3—121no—
264D3—121no—
265D3—121no—
266D3—241no—
267D3—241no—
268C5#6379411yescharacteristic, direct product factor, Sylow
269C22—31no—
270C22—61no—
271C22—181no—
272C22—181no—
273C22—362no—
274C4—31no—
275C4—61no—
276C4C2×C5×D3211yescharacteristic
277C3—21no—
278C3C5×#90811yescharacteristic, complemented
279C3C5×#91411yescharacteristic, complemented
280C2—61no—
281C2—121no—
282C2—361no—
283C2—361no—
284C2C5×D3×D1211yescharacteristic
subgroupG/H
centerC2×C5D3×D12
commutator, cosocleC32×C4C23×C5
FrattiniC4C2×C5×D32
FittingC32×C5×C8C22
socleC2×C32×C5C2×D4
hypercenterC5×C8D32
hypocenterC32C5×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8

Complements

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

NH ≅ G/Ntype
1. C5×D3×D12C2semidirect
2. C5×#3799C2semidirect
3. C5×#3745C2semidirect
4. C3×C5×#204C2semidirect
5. C3×C5×ΓL(1,𝔽25)C2semidirect
6. C5×#3717C2semidirect
7. C5×#3714C2semidirect
12. C5×C4⋊(C3⋊D3)C22semidirect
15. C3×C5×Dic6C22semidirect
19. C32×C5×C8C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. #63794C5direct (N × H)
35. C5×#204D3semidirect
36. C5×ΓL(1,𝔽25)D3semidirect
46. D3×D12C2×C5semidirect
47. #3799C2×C5semidirect
48. #3745C2×C5semidirect
49. C3×#204C2×C5semidirect
50. C3×ΓL(1,𝔽25)C2×C5semidirect
51. #3717C2×C5semidirect
52. #3714C2×C5semidirect
72. C5×Dic6C2×D3semidirect
75. C3×C5×C8C2×D3semidirect
76. C3×C5×C8C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
94. C4⋊(C3⋊D3)C22×C5semidirect
97. C3×Dic6C22×C5semidirect
101. C32×C8C22×C5semidirect
102. C3×C3⋊C8C22×C5semidirect
135. #204C5×D3semidirect
136. ΓL(1,𝔽25)C5×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
148. C5×C8D32semidirect
190. Dic6C2×C5×D3semidirect
193. C3×C8C2×C5×D3semidirect
194. C3×C8C2×C5×D3semidirect
195. C3⋊C8C2×C5×D3semidirect
217. C3×C5#908semidirect
218. C3×C5#914semidirect
243. C32C5×AGL(1,ℤ/8)semidirect
254. C8C5×D32semidirect
268. C5#63794direct (N × H)
278. C3C5×#908semidirect
279. C3C5×#914semidirect

Maximal subgroups

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

subgroupindex
C5×D3×D122
C5×#37992
C5×#37452
C3×C5×#2042
C3×C5×ΓL(1,𝔽25)2
C5×#37172
C5×#37142
C5×#9143
C5×#9083
#637945