Finite Group Databasebeta

← C5×#63937 #13087209

Subgroups — C5×#63937 #13087209

Order 1440.

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

normal subgroups only

This group has 326 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×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C5×#3796C211yesmaximal, characteristic, complemented
3C3×C5×#214C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5C5×#3717C211yesmaximal, characteristic, complemented
6C5×#3714C211yesmaximal, characteristic, complemented
7C5×#3712C211yesmaximal, characteristic, complemented
8C5×#914—31nomaximal
9C5×#911—31nomaximal
10C22×C5×C3⋊D3—21no—
11C32×C5×D4C2211yescharacteristic
12C5×#638—21no—
13C5×C4⋊(C3⋊D3)C2211yescharacteristic
14C4×C5×C3⋊D3C2211yescharacteristic
15C3×C5×D12C2211yescharacteristic, complemented
16C3×C5×Dic6C2211yescharacteristic, complemented
17C5×#626—21no—
18C5×D3×Dic3—21no—
19C3×C5×C3⋊C8C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63937C511yesmaximal, characteristic, direct product factor, Hall
22C5×#239—31no—
23C5×#237—31no—
24C5×D3×D4—31no—
25C5×D3×D4—31no—
26C5×D3×D4—61no—
27C3×C5×SD16—31no—
28C3×C5×D8—31no—
29C5×#215—31no—
30C5×#214D311yescharacteristic, complemented
31C5×#213—31no—
32C5×#213D311yescharacteristic, complemented
33C5×D24—31no—
34C5×#204—31no—
35C5×ΓL(1,𝔽25)—31no—
36C5×ΓL(1,𝔽25)—31no—
37C22×C32×C5—21no—
38C2×C5×C3⋊D3—21no—
39C2×C5×C3⋊D3—42no—
40C2×C5×C3⋊D3D411yescharacteristic
41C2×C3×C5×D3—21no—
42C32×C4×C5C2311yescharacteristic
43C5×C3⋊Dic3D411yescharacteristic
44C3×C5×Dic3—21no—
45C5×AGL(1,ℤ/8)—91noHall
46D4×C3⋊D3C2×C511yescharacteristic, complemented
47#3796C2×C511yescharacteristic, complemented
48C3×#214C2×C511yescharacteristic, complemented
49C3×#213C2×C511yescharacteristic, complemented
50#3717C2×C511yescharacteristic, complemented
51#3714C2×C511yescharacteristic, complemented
52#3712C2×C511yescharacteristic, complemented
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C22×C5×D3—62no—
56C3×C5×Q8—31no—
57C3×C5×D4—21no—
58C3×C5×D4—31no—
59C3×C5×D4C2×D311yescharacteristic
60C3×C5×D4C2×D311yescharacteristic
61C5×#67—61no—
62C5×#67—61no—
63C5×#67—61no—
64C5×#67—62no—
65C2×C5×Dic3—61no—
66C5×D12—31no—
67C5×D12—31no—
68C5×D12—61no—
69C5×D12—61no—
70C5×D12C2×D311yescharacteristic, complemented
71C4×C5×D3—31no—
72C4×C5×D3—31no—
73C4×C5×D3—61no—
74C4×C5×D3—61no—
75C5×Dic6C2×D311yescharacteristic, complemented
76C3×C5×C8—31no—
77C3×C5×C8—31no—
78C5×C3⋊C8C2×D311yescharacteristic, complemented
79C5×C3⋊C8C2×D311yescharacteristic, complemented
80#914—31no—
81#911—31no—
82C2×C32×C5—41no—
83C2×C32×C5C2×D411yescharacteristic
84C5×C3⋊D3—21no—
85C5×C3⋊D3—41no—
86C3×C5×D3—41no—
87C5×#41—91no—
88C2×C5×D4—91no—
89C5×SD16—31no—
90C5×SD16—91no—
91C5×D8—31no—
92C5×D8—91no—
93C5×M16—91no—
94C22×C3⋊D3—21no—
95C32×D4C22×C511yescharacteristic
96#638—21no—
97C4⋊(C3⋊D3)C22×C511yescharacteristic
98C4×C3⋊D3C22×C511yescharacteristic
99C3×D12C22×C511yescharacteristic, complemented
100C3×Dic6C22×C511yescharacteristic, complemented
101#626—21no—
102D3×Dic3—21no—
103C3×C3⋊C8C22×C511yescharacteristic, complemented
104C3×C3⋊C8C22×C511yescharacteristic, complemented
105C22×C3×C5—22no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—61no—
109C2×C5×D3—21no—
110C2×C5×D3—31no—
111C2×C5×D3—31no—
112C2×C5×D3—61no—
113C2×C5×D3—62no—
114C2×C5×D3—61no—
115C2×C5×D3—61no—
116C2×C5×D3—124no—
117C2×C5×D3—122no—
118C2×C5×D3—122no—
119C3×C4×C5—21no—
120C3×C4×C5—61no—
121C3×C4×C5C22×D311yescharacteristic
122C3×C4×C5C22×D311yescharacteristic
123C5×Dic3—21no—
124C5×Dic3—31no—
125C5×Dic3—31no—
126C5×Dic3—61no—
127#239—31no—
128#237—31no—
129D3×D4—31no—
130D3×D4—31no—
131D3×D4—61no—
132C3×SD16—31no—
133C3×D8—31no—
134#215—31no—
135#214C5×D311yescharacteristic, complemented
136#213—31no—
137#213C5×D311yescharacteristic, complemented
138D24—31no—
139#204—31no—
140ΓL(1,𝔽25)—31no—
141ΓL(1,𝔽25)—31no—
142C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
143C23×C5—181no—
144C5×Q8—31no—
145C5×D4—31no—
146C5×D4—91no—
147C5×D4—181no—
148C5×D4—181no—
149C5×D4D3211yescharacteristic
150C2×C4×C5—91no—
151C2×C4×C5—181no—
152C5×C8—31no—
153C5×C8—31no—
154C22×C32—21no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3C5×D411yescharacteristic
158C2×C3×D3—21no—
159C32×C4C23×C511yescharacteristic
160C3⋊Dic3C5×D411yescharacteristic
161C3×Dic3—21no—
162AGL(1,ℤ/8)—91noSylow
163C2×C3×C5—21no—
164C2×C3×C5—42no—
165C2×C3×C5—41no—
166C2×C3×C5—41no—
167C2×C3×C5—121no—
168C2×C3×C5D3×D411yescharacteristic
169C2×C3×C5D3×D411yescharacteristic
170C5×D3—41no—
171C5×D3—62no—
172C5×D3—61no—
173C5×D3—61no—
174C5×D3—122no—
175C5×D3—121no—
176C5×D3—121no—
177C22×D3—61no—
178C22×D3—61no—
179C22×D3—62no—
180C3×Q8—31no—
181C3×D4—21no—
182C3×D4—31no—
183C3×D4C2×C5×D311yescharacteristic
184C3×D4C2×C5×D311yescharacteristic
185#67—61no—
186#67—61no—
187#67—61no—
188#67—62no—
189C2×Dic3—61no—
190D12—31no—
191D12—31no—
192D12—61no—
193D12—61no—
194D12C2×C5×D311yescharacteristic, complemented
195C4×D3—31no—
196C4×D3—31no—
197C4×D3—61no—
198C4×D3—61no—
199Dic6C2×C5×D311yescharacteristic, complemented
200C3×C8—31no—
201C3×C8—31no—
202C3⋊C8C2×C5×D311yescharacteristic, complemented
203C3⋊C8C2×C5×D311yescharacteristic, complemented
204C22×C5—21no—
205C22×C5—61no—
206C22×C5—91no—
207C22×C5—181no—
208C22×C5—362no—
209C4×C5—61no—
210C4×C5—91no—
211C4×C5C2×D3211yescharacteristic
212C2×C32—41no—
213C2×C32C2×C5×D411yescharacteristic
214C3⋊D3—21no—
215C3⋊D3—41no—
216C3×D3—41no—
217#41—91no—
218C2×D4—91no—
219SD16—31no—
220SD16—91no—
221D8—31no—
222D8—91no—
223M16—91no—
224C3×C5—21no—
225C3×C5#91111yescharacteristic, complemented
226C3×C5#91411yescharacteristic, complemented
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—61no—
231C2×D3—21no—
232C2×D3—31no—
233C2×D3—31no—
234C2×D3—61no—
235C2×D3—62no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—124no—
239C2×D3—122no—
240C2×D3—122no—
241C3×C4—21no—
242C3×C4—61no—
243C3×C4C22×C5×D311yescharacteristic
244C3×C4C22×C5×D311yescharacteristic
245Dic3—21no—
246Dic3—31no—
247Dic3—31no—
248Dic3—61no—
249C2×C5—41no—
250C2×C5—121no—
251C2×C5—181no—
252C2×C5—361no—
253C2×C5#381111yescharacteristic
254C32C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
255C23—181no—
256Q8—31no—
257D4—31no—
258D4—91no—
259D4—181no—
260D4—181no—
261D4C5×D3211yescharacteristic
262C2×C4—91no—
263C2×C4—181no—
264C8—31no—
265C8—31no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—121no—
271C2×C3C5×D3×D411yescharacteristic
272C2×C3C5×D3×D411yescharacteristic
273D3—41no—
274D3—62no—
275D3—61no—
276D3—61no—
277D3—122no—
278D3—121no—
279D3—121no—
280C5#6393711yescharacteristic, direct product factor, Sylow
281C22—21no—
282C22—61no—
283C22—91no—
284C22—181no—
285C22—362no—
286C4—61no—
287C4—91no—
288C4C2×C5×D3211yescharacteristic
289C3—21no—
290C3C5×#91111yescharacteristic, complemented
291C3C5×#91411yescharacteristic, complemented
292C2—41no—
293C2—121no—
294C2—181no—
295C2—361no—
296C2C5×#381111yescharacteristic
subgroupG/H
centerC2×C5#3811
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. C5×D4×C3⋊D3C2semidirect
2. C5×#3796C2semidirect
3. C3×C5×#214C2semidirect
4. C3×C5×#213C2semidirect
5. C5×#3717C2semidirect
6. C5×#3714C2semidirect
7. C5×#3712C2semidirect
15. C3×C5×D12C22semidirect
16. C3×C5×Dic6C22semidirect
19. C3×C5×C3⋊C8C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. #63937C5direct (N × H)
30. C5×#214D3semidirect
32. C5×#213D3semidirect
46. D4×C3⋊D3C2×C5semidirect
47. #3796C2×C5semidirect
48. C3×#214C2×C5semidirect
49. C3×#213C2×C5semidirect
50. #3717C2×C5semidirect
51. #3714C2×C5semidirect
52. #3712C2×C5semidirect
70. C5×D12C2×D3semidirect
75. C5×Dic6C2×D3semidirect
78. C5×C3⋊C8C2×D3semidirect
79. C5×C3⋊C8C2×D3semidirect
99. C3×D12C22×C5semidirect
100. C3×Dic6C22×C5semidirect
103. C3×C3⋊C8C22×C5semidirect
104. C3×C3⋊C8C22×C5semidirect
135. #214C5×D3semidirect
137. #213C5×D3semidirect
142. C32×C5AGL(1,ℤ/8)semidirect
194. D12C2×C5×D3semidirect
199. Dic6C2×C5×D3semidirect
202. C3⋊C8C2×C5×D3semidirect
203. C3⋊C8C2×C5×D3semidirect
225. C3×C5#911semidirect
226. C3×C5#914semidirect
254. C32C5×AGL(1,ℤ/8)semidirect
280. C5#63937direct (N × H)
290. C3C5×#911semidirect
291. C3C5×#914semidirect

Maximal subgroups

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

subgroupindex
C5×D4×C3⋊D32
C5×#37962
C3×C5×#2142
C3×C5×#2132
C5×#37172
C5×#37142
C5×#37122
C5×#9143
C5×#9113
#639375