Finite Group Databasebeta

← C5×#63880 #13087164

Subgroups — C5×#63880 #13087164

Order 1440.

410 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#626C211yesmaximal, characteristic
2C2×C5×#624C211yesmaximal, characteristic, complemented
3C2×C5×D3×Dic3C211yesmaximal, characteristic, complemented
4C5×#3752C211yesmaximal, characteristic, complemented
5C3×C5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C3×C4×C5×Dic3C211yesmaximal, characteristic, complemented
7C5×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
8C5×#67⋊C4—31nomaximal
9C4×C5×D12—31nomaximal
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×C5×Dic3C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C5×#626C414yescomplemented
18C5×#624—22no—
19C5×D3×Dic3—22no—
20#63880C511yesmaximal, characteristic, direct product factor, Hall
21C2×C5×#67—31no—
22C22×C5×Dic3—31no—
23C2×C5×D12—31no—
24C2×C4×C5×D3—31no—
25C2×C4×C5×D3—31no—
26C2×C4×C5×D3—31no—
27C3×C5×C22⋊C4—31no—
28C3×C42×C5—31no—
29C5×(C2×D3)⋊C4—31no—
30C5×(C2×D3)⋊C4—31no—
31C5×(C2×D3)⋊C4—61no—
32C5×(C2×D3)⋊C4D311yescharacteristic, complemented
33C5×C4⋊Dic3—31no—
34C5×Dic3⋊C4—31no—
35C4×C5×Dic3D311yescharacteristic, complemented
36C22×C32×C5C2311yescharacteristic
37C2×C5×C3⋊D3—21no—
38C2×C5×C3⋊D3—21no—
39C2×C5×C3⋊D3C2×C412yescomplemented
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C32×C4×C5—21no—
44C5×C3⋊Dic3—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3C2×C412yescomplemented
47C3×C5×Dic3D412yescomplemented
48C4×C5×D4—91noHall
49C2×#626C2×C511yescharacteristic
50C2×#624C2×C511yescharacteristic, complemented
51C2×D3×Dic3C2×C511yescharacteristic, complemented
52#3752C2×C511yescharacteristic, complemented
53C3×(C2×D3)⋊C4C2×C511yescharacteristic, complemented
54C3×C4×Dic3C2×C511yescharacteristic, complemented
55Dic3⋊Dic3C2×C511yescharacteristic, complemented
56C23×C3×C5—31no—
57C22×C5×D3—31no—
58C22×C5×D3—31no—
59C22×C5×D3—61no—
60C22×C5×D3C2×D311yescharacteristic
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5—31no—
64C2×C3×C4×C5—31no—
65C2×C3×C4×C5C2×D311yescharacteristic
66C2×C3×C4×C5C2×D311yescharacteristic
67C5×#67—34no—
68C2×C5×Dic3—31no—
69C2×C5×Dic3—31no—
70C2×C5×Dic3—62no—
71C2×C5×Dic3—61no—
72C2×C5×Dic3C2×D311yescharacteristic
73C2×C5×Dic3C2×D311yescharacteristic
74C2×C5×Dic3C2×D311yescharacteristic
75C5×D12—34no—
76C4×C5×D3—62no—
77C4×C5×D3—62no—
78C4×C5×D3—62no—
79#67⋊C4—31no—
80C4×D12—31no—
81C2×C32×C5C22×C411yescharacteristic
82C2×C32×C5C2×D411yescharacteristic
83C2×C32×C5#4111yescharacteristic
84C5×C3⋊D3—22no—
85C3×C5×D3—22no—
86C2×C5×D4—91no—
87C22×C4×C5—91no—
88C22×C4×C5—91no—
89C5×C4⋊C4—91no—
90C5×C22⋊C4—31no—
91C5×C22⋊C4—91no—
92C42×C5—31no—
93C22×C3⋊D3C22×C511yescharacteristic
94C22×C3×D3C22×C511yescharacteristic
95C2×C32×C4C22×C511yescharacteristic
96C2×C3⋊Dic3C22×C511yescharacteristic
97C2×C3×Dic3C22×C511yescharacteristic
98C2×C3×Dic3C22×C511yescharacteristic
99C2×C3×Dic3C22×C511yescharacteristic
100#626C4×C514yescomplemented
101#624—22no—
102D3×Dic3—22no—
103C22×C3×C5—21no—
104C22×C3×C5—32no—
105C22×C3×C5—61no—
106C22×C3×C5—61no—
107C22×C3×C5C22×D311yescharacteristic
108C22×C3×C5C22×D311yescharacteristic
109C2×C5×D3—21no—
110C2×C5×D3—21no—
111C2×C5×D3—32no—
112C2×C5×D3—32no—
113C2×C5×D3—62no—
114C2×C5×D3—61no—
115C2×C5×D3—61no—
116C2×C5×D3—61no—
117C2×C5×D3—61no—
118C2×C5×D3—121no—
119C2×C5×D3—121no—
120C2×C5×D3C4×D312yescomplemented
121C3×C4×C5—21no—
122C3×C4×C5—21no—
123C3×C4×C5—32no—
124C3×C4×C5—32no—
125C3×C4×C5—41no—
126C3×C4×C5—61no—
127C5×Dic3—21no—
128C5×Dic3—61no—
129C5×Dic3—61no—
130C5×Dic3—121no—
131C5×Dic3C4×D312yescomplemented
132C5×Dic3D1212yescomplemented
133C2×#67—31no—
134C22×Dic3—31no—
135C2×D12—31no—
136C2×C4×D3—31no—
137C2×C4×D3—31no—
138C2×C4×D3—31no—
139C3×C22⋊C4—31no—
140C3×C42—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4C5×D311yescharacteristic, complemented
145C4⋊Dic3—31no—
146Dic3⋊C4—31no—
147C4×Dic3C5×D311yescharacteristic, complemented
148C32×C5C4×D411yescharacteristic, complemented, Hall
149C23×C5—31no—
150C23×C5—91no—
151C5×D4—94no—
152C2×C4×C5—31no—
153C2×C4×C5—31no—
154C2×C4×C5—31no—
155C2×C4×C5—91no—
156C2×C4×C5—182no—
157C2×C4×C5—182no—
158C2×C4×C5D3211yescharacteristic
159C22×C32C23×C511yescharacteristic
160C2×C3⋊D3—21no—
161C2×C3⋊D3—21no—
162C2×C3⋊D3C2×C4×C512yescomplemented
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3C2×C4×C512yescomplemented
166C32×C4—21no—
167C3⋊Dic3—21no—
168C3×Dic3—21no—
169C3×Dic3C2×C4×C512yescomplemented
170C3×Dic3C5×D412yescomplemented
171C4×D4—91noSylow
172C2×C3×C5—21no—
173C2×C3×C5—21no—
174C2×C3×C5—21no—
175C2×C3×C5—62no—
176C2×C3×C5C2×C4×D311yescharacteristic
177C2×C3×C5C2×C4×D311yescharacteristic
178C2×C3×C5C2×D1211yescharacteristic
179C2×C3×C5#23511yescharacteristic
180C2×C3×C5D3×D411yescharacteristic
181C2×C3×C5#23711yescharacteristic
182C5×D3—22no—
183C5×D3—62no—
184C5×D3—62no—
185C5×D3—122no—
186C23×C3—31no—
187C22×D3—31no—
188C22×D3—31no—
189C22×D3—61no—
190C22×D3C2×C5×D311yescharacteristic
191C2×C3×C4—21no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4C2×C5×D311yescharacteristic
196C2×C3×C4C2×C5×D311yescharacteristic
197#67—34no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3C2×C5×D311yescharacteristic
203C2×Dic3C2×C5×D311yescharacteristic
204C2×Dic3C2×C5×D311yescharacteristic
205D12—34no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—62no—
209C22×C5—32no—
210C22×C5—61no—
211C22×C5—61no—
212C22×C5—92no—
213C22×C5—181no—
214C22×C5—181no—
215C22×C5C2×D3211yescharacteristic
216C4×C5—21no—
217C4×C5—32no—
218C4×C5—32no—
219C4×C5—61no—
220C4×C5—181no—
221C2×C32C22×C4×C511yescharacteristic
222C2×C32C2×C5×D411yescharacteristic
223C2×C32C5×#4111yescharacteristic
224C3⋊D3—22no—
225C3×D3—22no—
226C2×D4—91no—
227C22×C4—91no—
228C22×C4—91no—
229C4⋊C4—91no—
230C22⋊C4—31no—
231C22⋊C4—91no—
232C42—31no—
233C3×C5—21no—
234C3×C5C4×D1211yescharacteristic, complemented
235C3×C5#67⋊C411yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—32no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3C22×C5×D311yescharacteristic
241C22×C3C22×C5×D311yescharacteristic
242C2×D3—21no—
243C2×D3—21no—
244C2×D3—32no—
245C2×D3—32no—
246C2×D3—62no—
247C2×D3—61no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—121no—
252C2×D3—121no—
253C2×D3C4×C5×D312yescomplemented
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—32no—
257C3×C4—32no—
258C3×C4—41no—
259C3×C4—61no—
260Dic3—21no—
261Dic3—61no—
262Dic3—61no—
263Dic3—121no—
264Dic3C4×C5×D312yescomplemented
265Dic3C5×D1212yescomplemented
266C2×C5—62no—
267C2×C5—182no—
268C2×C5C4×D3211yescharacteristic
269C2×C5D3×D1211yescharacteristic
270C2×C5#380411yescharacteristic
271C32C4×C5×D411yescharacteristic, complemented, Sylow
272C23—31no—
273C23—91no—
274D4—94no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—91no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4C5×D3211yescharacteristic
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—62no—
286C2×C3C2×C4×C5×D311yescharacteristic
287C2×C3C2×C4×C5×D311yescharacteristic
288C2×C3C2×C5×D1211yescharacteristic
289C2×C3C5×#23511yescharacteristic
290C2×C3C5×D3×D411yescharacteristic
291C2×C3C5×#23711yescharacteristic
292D3—22no—
293D3—62no—
294D3—62no—
295D3—122no—
296C5#6388011yescharacteristic, direct product factor, Sylow
297C22—32no—
298C22—61no—
299C22—61no—
300C22—92no—
301C22—181no—
302C22—181no—
303C22C2×C5×D3211yescharacteristic
304C4—21no—
305C4—32no—
306C4—32no—
307C4—61no—
308C4—181no—
309C3—21no—
310C3C4×C5×D1211yescharacteristic, complemented
311C3C5×#67⋊C411yescharacteristic, complemented
312C2—62no—
313C2—182no—
314C2C4×C5×D3211yescharacteristic
315C2C5×D3×D1211yescharacteristic
316C2C5×#380411yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutatorC2×C32C22×C4×C5
FrattiniC22C2×C5×D32
FittingC2×C32×C4×C5C22
socleC22×C32×C5C23
prime radicalC2×C5×#626C2
cosocleC22×C32C23×C5
hypercenterC2×C4×C5D32
hypocenterC32C4×C5×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C5×#624C2semidirect
3. C2×C5×D3×Dic3C2semidirect
4. C5×#3752C2semidirect
5. C3×C5×(C2×D3)⋊C4C2semidirect
6. C3×C4×C5×Dic3C2semidirect
7. C5×Dic3⋊Dic3C2semidirect
17. C5×#626C4semidirect
20. #63880C5direct (N × H)
32. C5×(C2×D3)⋊C4D3semidirect
35. C4×C5×Dic3D3semidirect
39. C2×C5×C3⋊D3C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
46. C3×C5×Dic3C2×C4semidirect
47. C3×C5×Dic3D4semidirect
50. C2×#624C2×C5semidirect
51. C2×D3×Dic3C2×C5semidirect
52. #3752C2×C5semidirect
53. C3×(C2×D3)⋊C4C2×C5semidirect
54. C3×C4×Dic3C2×C5semidirect
55. Dic3⋊Dic3C2×C5semidirect
100. #626C4×C5semidirect
120. C2×C5×D3C4×D3semidirect
131. C5×Dic3C4×D3semidirect
132. C5×Dic3D12semidirect
144. (C2×D3)⋊C4C5×D3semidirect
147. C4×Dic3C5×D3semidirect
148. C32×C5C4×D4semidirect
162. C2×C3⋊D3C2×C4×C5semidirect
165. C2×C3×D3C2×C4×C5semidirect
169. C3×Dic3C2×C4×C5semidirect
170. C3×Dic3C5×D4semidirect
234. C3×C5C4×D12semidirect
235. C3×C5#67⋊C4semidirect
253. C2×D3C4×C5×D3semidirect
264. Dic3C4×C5×D3semidirect
265. Dic3C5×D12semidirect
271. C32C4×C5×D4semidirect
296. C5#63880direct (N × H)
310. C3C4×C5×D12semidirect
311. C3C5×#67⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6262
C2×C5×#6242
C2×C5×D3×Dic32
C5×#37522
C3×C5×(C2×D3)⋊C42
C3×C4×C5×Dic32
C5×Dic3⋊Dic32
C5×#67⋊C43
C4×C5×D123
#638805