Finite Group Databasebeta

← C5×#63798 #13087069

Subgroups — C5×#63798 #13087069

Order 1440.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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×#3802C211yesmaximal, characteristic, complemented
2C5×#3796C211yesmaximal, characteristic, complemented
3C5×#3743C211yesmaximal, characteristic, complemented
4C3×C5×D24C211yesmaximal, characteristic, complemented
5C3×C5×#204C211yesmaximal, characteristic, complemented
6C5×#3715C211yesmaximal, characteristic, complemented
7C5×#3713C211yesmaximal, characteristic, complemented
8C5×#914—31nomaximal
9C5×#911—31nomaximal
10C2×C5×D32—21no—
11C4×C5×C3⋊D3C2211yescharacteristic
12C3×C5×D12C2211yescharacteristic
13C3×C5×D12C2211yescharacteristic
14C3×C5×D12C2211yescharacteristic, complemented
15C3×C5×Dic6C2211yescharacteristic, complemented
16C5×#626—21no—
17C5×#625—21no—
18C5×D3×Dic3—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21#63798C511yesmaximal, characteristic, direct product factor, Hall
22C5×#239—31no—
23C5×#237—31no—
24C5×D3×D4—31no—
25C5×D3×D4—31no—
26C3×C5×SD16—31no—
27C3×C5×D8—31no—
28C5×#215—31no—
29C5×#214—31no—
30C5×#213—31no—
31C5×#213—31no—
32C5×D24D311yescharacteristic, complemented
33C5×#204D311yescharacteristic, complemented
34C5×ΓL(1,𝔽25)—31no—
35C5×ΓL(1,𝔽25)—31no—
36C5×ΓL(1,𝔽25)—61no—
37C2×C5×C3⋊D3D411yescharacteristic
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—21no—
41C5×D32—42no—
42C32×C4×C5C2311yescharacteristic
43C5×C3⋊Dic3D411yescharacteristic
44C3×C5×Dic3—21no—
45C5×AGL(1,ℤ/8)—91noHall
46#3802C2×C511yescharacteristic, complemented
47#3796C2×C511yescharacteristic, complemented
48#3743C2×C511yescharacteristic, complemented
49C3×D24C2×C511yescharacteristic, complemented
50C3×#204C2×C511yescharacteristic, complemented
51#3715C2×C511yescharacteristic, complemented
52#3713C2×C511yescharacteristic, complemented
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57C3×C5×D4—31no—
58C3×C5×D4—31no—
59C5×#67—61no—
60C5×#67—61no—
61C5×#67—61no—
62C2×C5×Dic3—61no—
63C5×D12—61no—
64C5×D12C2×D311yescharacteristic
65C5×D12C2×D311yescharacteristic
66C5×D12C2×D311yescharacteristic, complemented
67C4×C5×D3—31no—
68C4×C5×D3—31no—
69C4×C5×D3—61no—
70C4×C5×D3—61no—
71C5×Dic6C2×D311yescharacteristic, complemented
72C3×C5×C8—21no—
73C3×C5×C8C2×D311yescharacteristic, complemented
74C3×C5×C8C2×D311yescharacteristic, complemented
75C5×C3⋊C8—31no—
76C5×C3⋊C8—31no—
77C5×C3⋊C8—61no—
78#914—31no—
79#911—31no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—21no—
82C3×C5×D3—41no—
83C3×C5×D3—41no—
84C3×C5×D3—41no—
85C5×#41—91no—
86C2×C5×D4—91no—
87C5×SD16—31no—
88C5×SD16—91no—
89C5×D8—31no—
90C5×D8—91no—
91C5×M16—91no—
92C2×D32—21no—
93C4×C3⋊D3C22×C511yescharacteristic
94C3×D12C22×C511yescharacteristic
95C3×D12C22×C511yescharacteristic
96C3×D12C22×C511yescharacteristic, complemented
97C3×Dic6C22×C511yescharacteristic, complemented
98#626—21no—
99#625—21no—
100D3×Dic3—21no—
101C32×C8C22×C511yescharacteristic, complemented
102C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
103C22×C3×C5—61no—
104C22×C3×C5—61no—
105C22×C3×C5—61no—
106C2×C5×D3—21no—
107C2×C5×D3—21no—
108C2×C5×D3—21no—
109C2×C5×D3—31no—
110C2×C5×D3—31no—
111C2×C5×D3—61no—
112C2×C5×D3—122no—
113C2×C5×D3—122no—
114C3×C4×C5—21no—
115C3×C4×C5—61no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118C5×Dic3—21no—
119C5×Dic3—31no—
120C5×Dic3—31no—
121C5×Dic3—61no—
122#239—31no—
123#237—31no—
124D3×D4—31no—
125D3×D4—31no—
126C3×SD16—31no—
127C3×D8—31no—
128#215—31no—
129#214—31no—
130#213—31no—
131#213—31no—
132D24C5×D311yescharacteristic, complemented
133#204C5×D311yescharacteristic, complemented
134ΓL(1,𝔽25)—31no—
135ΓL(1,𝔽25)—31no—
136ΓL(1,𝔽25)—61no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—181no—
139C5×Q8—31no—
140C5×D4—31no—
141C5×D4—31no—
142C5×D4—31no—
143C5×D4—181no—
144C5×D4—181no—
145C2×C4×C5—91no—
146C2×C4×C5—181no—
147C5×C8—91no—
148C5×C8D3211yescharacteristic, complemented
149C2×C3⋊D3C5×D411yescharacteristic
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—21no—
153D32—42no—
154C32×C4C23×C511yescharacteristic
155C3⋊Dic3C5×D411yescharacteristic
156C3×Dic3—21no—
157AGL(1,ℤ/8)—91noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5—121no—
161C2×C3×C5—121no—
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5D3×D411yescharacteristic
164C5×D3—41no—
165C5×D3—41no—
166C5×D3—41no—
167C5×D3—61no—
168C5×D3—61no—
169C5×D3—121no—
170C22×D3—61no—
171C22×D3—61no—
172C3×Q8—31no—
173C3×D4—31no—
174C3×D4—31no—
175C3×D4—31no—
176#67—61no—
177#67—61no—
178#67—61no—
179C2×Dic3—61no—
180D12—61no—
181D12C2×C5×D311yescharacteristic
182D12C2×C5×D311yescharacteristic
183D12C2×C5×D311yescharacteristic, complemented
184C4×D3—31no—
185C4×D3—31no—
186C4×D3—61no—
187C4×D3—61no—
188Dic6C2×C5×D311yescharacteristic, complemented
189C3×C8—21no—
190C3×C8C2×C5×D311yescharacteristic, complemented
191C3×C8C2×C5×D311yescharacteristic, complemented
192C3⋊C8—31no—
193C3⋊C8—31no—
194C3⋊C8—61no—
195C22×C5—61no—
196C22×C5—61no—
197C22×C5—61no—
198C22×C5—91no—
199C22×C5—362no—
200C4×C5—61no—
201C4×C5—91no—
202C4×C5C2×D3211yescharacteristic
203C2×C32C2×C5×D411yescharacteristic
204C3⋊D3—21no—
205C3×D3—41no—
206C3×D3—41no—
207C3×D3—41no—
208#41—91no—
209C2×D4—91no—
210SD16—31no—
211SD16—91no—
212D8—31no—
213D8—91no—
214M16—91no—
215C3×C5—21no—
216C3×C5#91111yescharacteristic, complemented
217C3×C5#91411yescharacteristic, complemented
218C22×C3—61no—
219C22×C3—61no—
220C22×C3—61no—
221C2×D3—21no—
222C2×D3—21no—
223C2×D3—21no—
224C2×D3—31no—
225C2×D3—31no—
226C2×D3—61no—
227C2×D3—122no—
228C2×D3—122no—
229C3×C4—21no—
230C3×C4—61no—
231C3×C4C22×C5×D311yescharacteristic
232C3×C4C22×C5×D311yescharacteristic
233Dic3—21no—
234Dic3—31no—
235Dic3—31no—
236Dic3—61no—
237C2×C5—121no—
238C2×C5—121no—
239C2×C5—121no—
240C2×C5—181no—
241C2×C5#380211yescharacteristic
242C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
243C23—181no—
244Q8—31no—
245D4—31no—
246D4—31no—
247D4—31no—
248D4—181no—
249D4—181no—
250C2×C4—91no—
251C2×C4—181no—
252C8—91no—
253C8C5×D3211yescharacteristic, complemented
254C2×C3—21no—
255C2×C3—121no—
256C2×C3—121no—
257C2×C3—121no—
258C2×C3C5×D3×D411yescharacteristic
259C2×C3C5×D3×D411yescharacteristic
260D3—41no—
261D3—41no—
262D3—41no—
263D3—61no—
264D3—61no—
265D3—121no—
266C5#6379811yescharacteristic, direct product factor, Sylow
267C22—61no—
268C22—61no—
269C22—61no—
270C22—91no—
271C22—362no—
272C4—61no—
273C4—91no—
274C4C2×C5×D3211yescharacteristic
275C3—21no—
276C3C5×#91111yescharacteristic, complemented
277C3C5×#91411yescharacteristic, complemented
278C2—121no—
279C2—121no—
280C2—121no—
281C2—181no—
282C2C5×#380211yescharacteristic
subgroupG/H
centerC2×C5#3802
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×#3802C2semidirect
2. C5×#3796C2semidirect
3. C5×#3743C2semidirect
4. C3×C5×D24C2semidirect
5. C3×C5×#204C2semidirect
6. C5×#3715C2semidirect
7. C5×#3713C2semidirect
14. C3×C5×D12C22semidirect
15. C3×C5×Dic6C22semidirect
19. C32×C5×C8C22semidirect
20. C5×C3⋊(C3⋊C8)C22semidirect
21. #63798C5direct (N × H)
32. C5×D24D3semidirect
33. C5×#204D3semidirect
46. #3802C2×C5semidirect
47. #3796C2×C5semidirect
48. #3743C2×C5semidirect
49. C3×D24C2×C5semidirect
50. C3×#204C2×C5semidirect
51. #3715C2×C5semidirect
52. #3713C2×C5semidirect
66. C5×D12C2×D3semidirect
71. C5×Dic6C2×D3semidirect
73. C3×C5×C8C2×D3semidirect
74. C3×C5×C8C2×D3semidirect
96. C3×D12C22×C5semidirect
97. C3×Dic6C22×C5semidirect
101. C32×C8C22×C5semidirect
102. C3⋊(C3⋊C8)C22×C5semidirect
132. D24C5×D3semidirect
133. #204C5×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
148. C5×C8D32semidirect
183. D12C2×C5×D3semidirect
188. Dic6C2×C5×D3semidirect
190. C3×C8C2×C5×D3semidirect
191. C3×C8C2×C5×D3semidirect
216. C3×C5#911semidirect
217. C3×C5#914semidirect
242. C32C5×AGL(1,ℤ/8)semidirect
253. C8C5×D32semidirect
266. C5#63798direct (N × H)
276. C3C5×#911semidirect
277. C3C5×#914semidirect

Maximal subgroups

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

subgroupindex
C5×#38022
C5×#37962
C5×#37432
C3×C5×D242
C3×C5×#2042
C5×#37152
C5×#37132
C5×#9143
C5×#9113
#637985