Finite Group Databasebeta

← C5×Dic3×#67 #13087237

Subgroups — C5×Dic3×#67 #13087237

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×C3×C5×#67C211yesmaximal, characteristic
2C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
3C2×C5×D3×Dic3C211yesmaximal, characteristic, complemented
4C5×#3757C211yesmaximal, characteristic, complemented
5C5×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6C5×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7C5×Dic32C211yesmaximal, characteristic, complemented
8C5×D4×Dic3—31nomaximal
9C4×C5×#67—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C3×C5×#67C414yescomplemented
15C2×C3×C5×Dic3—22no—
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19C5×D3×Dic3—22no—
20Dic3×#67C511yesmaximal, characteristic, direct product factor, Hall
21C2×C3×C5×D4—31no—
22C22×C3×C4×C5—31no—
23C2×C5×#67D311yescharacteristic
24C22×C5×Dic3—31no—
25C22×C5×Dic3D311yescharacteristic, complemented
26C2×C4×C5×D3—31no—
27C5×#217—31no—
28C5×#217—31no—
29C5×#217—31no—
30C5×#217—61no—
31C5×(C2×D3)⋊C4—31no—
32C5×C4⋊Dic3—31no—
33C5×Dic3⋊C4—31no—
34C4×C5×Dic3—31no—
35C4×C5×Dic3—31no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2×C412yescomplemented
39C22×C32×C5C2311yescharacteristic
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C5×C3⋊Dic3—21no—
44C5×C3⋊Dic3—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3C2×C412yescomplemented
47C3×C5×Dic3D412yescomplemented
48C4×C5×D4—91noHall
49C2×C3×#67C2×C511yescharacteristic
50C22×C3×Dic3C2×C511yescharacteristic, complemented
51C2×D3×Dic3C2×C511yescharacteristic, complemented
52#3757C2×C511yescharacteristic, complemented
53Dic3⋊Dic3C2×C511yescharacteristic, complemented
54(C2×D3)⋊Dic3C2×C511yescharacteristic, complemented
55Dic32C2×C511yescharacteristic, complemented
56C23×C3×C5—21no—
57C23×C3×C5—31no—
58C23×C3×C5C2×D311yescharacteristic
59C23×C3×C5C2×D311yescharacteristic
60C22×C5×D3C2×D311yescharacteristic
61C3×C5×D4—34no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5—31no—
64C2×C3×C4×C5—31no—
65C2×C3×C4×C5—62no—
66C5×#67Dic314yesdirect product factor
67C2×C5×Dic3—22no—
68C2×C5×Dic3—31no—
69C2×C5×Dic3—31no—
70C2×C5×Dic3—31no—
71C2×C5×Dic3—31no—
72C2×C5×Dic3—62no—
73C2×C5×Dic3—61no—
74C2×C5×Dic3—61no—
75C2×C5×Dic3C2×D311yescharacteristic
76C2×C5×Dic3C2×D311yescharacteristic
77C2×C5×Dic3C2×D311yescharacteristic
78C4×C5×D3—62no—
79D4×Dic3—31no—
80C4×#67—31no—
81C2×C32×C5—22no—
82C2×C32×C5C22×C411yescharacteristic
83C2×C32×C5C2×D411yescharacteristic
84C2×C32×C5#4111yescharacteristic
85C3×C5×D3—22no—
86C2×C5×D4—31no—
87C22×C4×C5—31no—
88C22×C4×C5—91no—
89C5×C4⋊C4—91no—
90C5×C22⋊C4—91no—
91C5×C22⋊C4—91no—
92C42×C5—91no—
93C23×C32C22×C511yescharacteristic
94C22×C3×D3C22×C511yescharacteristic
95C2×C3⋊Dic3C22×C511yescharacteristic
96C2×C3⋊Dic3C22×C511yescharacteristic
97C3×#67C4×C514yescomplemented
98C2×C3×Dic3—22no—
99C2×C3×Dic3C22×C511yescharacteristic
100C2×C3×Dic3C22×C511yescharacteristic
101C2×C3×Dic3C22×C511yescharacteristic
102D3×Dic3—22no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—22no—
108C22×C3×C5—21no—
109C22×C3×C5—32no—
110C22×C3×C5—41no—
111C22×C3×C5—41no—
112C22×C3×C5—61no—
113C22×C3×C5—61no—
114C22×C3×C5C4×D312yescomplemented
115C22×C3×C5C2×Dic312yescomplemented
116C22×C3×C5C22×D311yescharacteristic
117C22×C3×C5C22×D311yescharacteristic
118C2×C5×D3—21no—
119C2×C5×D3—21no—
120C2×C5×D3C2×Dic312yescomplemented
121C3×C4×C5—32no—
122C3×C4×C5—32no—
123C3×C4×C5—61no—
124C5×Dic3—21no—
125C5×Dic3—61no—
126C5×Dic3—61no—
127C5×Dic3—61no—
128C5×Dic3—61no—
129C5×Dic3—121no—
130C5×Dic3—121no—
131C5×Dic3C2×Dic312yescomplemented
132C5×Dic3#6712yesdirect product factor
133C2×C3×D4—31no—
134C22×C3×C4—31no—
135C2×#67C5×D311yescharacteristic
136C22×Dic3—31no—
137C22×Dic3C5×D311yescharacteristic, complemented
138C2×C4×D3—31no—
139#217—31no—
140#217—31no—
141#217—31no—
142#217—61no—
143(C2×D3)⋊C4—31no—
144C4⋊Dic3—31no—
145Dic3⋊C4—31no—
146C4×Dic3—31no—
147C4×Dic3—31no—
148C32×C5C4×D411yescharacteristic, complemented, Hall
149C23×C5—31no—
150C23×C5D3211yescharacteristic
151C5×D4—34no—
152C2×C4×C5—31no—
153C2×C4×C5—31no—
154C2×C4×C5—31no—
155C2×C4×C5—62no—
156C2×C4×C5—91no—
157C2×C4×C5—91no—
158C2×C4×C5—182no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32C2×C4×C512yescomplemented
162C22×C32C23×C511yescharacteristic
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3C2×C4×C512yescomplemented
166C3⋊Dic3—21no—
167C3⋊Dic3—21no—
168C3×Dic3—21no—
169C3×Dic3C2×C4×C512yescomplemented
170C3×Dic3C5×D412yescomplemented
171C4×D4—91noSylow
172C2×C3×C5—22no—
173C2×C3×C5—22no—
174C2×C3×C5—21no—
175C2×C3×C5—21no—
176C2×C3×C5—21no—
177C2×C3×C5—42no—
178C2×C3×C5—62no—
179C2×C3×C5C2×C4×D311yescharacteristic
180C2×C3×C5#23511yescharacteristic
181C2×C3×C5D3×D411yescharacteristic
182C2×C3×C5#23711yescharacteristic
183C2×C3×C5C22×Dic311yescharacteristic
184C2×C3×C5C2×#6711yescharacteristic
185C5×D3—22no—
186C23×C3—21no—
187C23×C3—31no—
188C23×C3C2×C5×D311yescharacteristic
189C23×C3C2×C5×D311yescharacteristic
190C22×D3C2×C5×D311yescharacteristic
191C3×D4—34no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—62no—
196#67C5×Dic314yesdirect product factor
197C2×Dic3—22no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—62no—
205C2×Dic3C2×C5×D311yescharacteristic
206C2×Dic3C2×C5×D311yescharacteristic
207C2×Dic3C2×C5×D311yescharacteristic
208C4×D3—62no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—32no—
212C22×C5—61no—
213C22×C5—61no—
214C22×C5D3×Dic312yescomplemented
215C22×C5C2×D3211yescharacteristic
216C4×C5—32no—
217C4×C5—32no—
218C4×C5—61no—
219C4×C5—181no—
220C4×C5—181no—
221C2×C32—22no—
222C2×C32C22×C4×C511yescharacteristic
223C2×C32C2×C5×D411yescharacteristic
224C2×C32C5×#4111yescharacteristic
225C3×D3—22no—
226C2×D4—31no—
227C22×C4—31no—
228C22×C4—91no—
229C4⋊C4—91no—
230C22⋊C4—91no—
231C22⋊C4—91no—
232C42—91no—
233C3×C5—21no—
234C3×C5C4×#6711yescharacteristic, complemented
235C3×C5D4×Dic311yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—32no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3C4×C5×D312yescomplemented
248C22×C3C2×C5×Dic312yescomplemented
249C22×C3C22×C5×D311yescharacteristic
250C22×C3C22×C5×D311yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3C2×C5×Dic312yescomplemented
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—61no—
257Dic3—21no—
258Dic3—61no—
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—121no—
263Dic3—121no—
264Dic3C2×C5×Dic312yescomplemented
265Dic3C5×#6712yesdirect product factor
266C2×C5—22no—
267C2×C5—62no—
268C2×C5C2×D3×Dic311yescharacteristic
269C2×C5#380411yescharacteristic
270C2×C5D3×#6711yescharacteristic
271C32C4×C5×D411yescharacteristic, complemented, Sylow
272C23—31no—
273C23C5×D3211yescharacteristic
274D4—34no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—62no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—182no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—42no—
288C2×C3—62no—
289C2×C3C2×C4×C5×D311yescharacteristic
290C2×C3C5×#23511yescharacteristic
291C2×C3C5×D3×D411yescharacteristic
292C2×C3C5×#23711yescharacteristic
293C2×C3C22×C5×Dic311yescharacteristic
294C2×C3C2×C5×#6711yescharacteristic
295D3—22no—
296C5Dic3×#6711yescharacteristic, direct product factor, Sylow
297C22—21no—
298C22—21no—
299C22—32no—
300C22—61no—
301C22—61no—
302C22C5×D3×Dic312yescomplemented
303C22C2×C5×D3211yescharacteristic
304C4—32no—
305C4—32no—
306C4—61no—
307C4—181no—
308C4—181no—
309C3—21no—
310C3C4×C5×#6711yescharacteristic, complemented
311C3C5×D4×Dic311yescharacteristic, complemented
312C2—22no—
313C2—62no—
314C2C2×C5×D3×Dic311yescharacteristic
315C2C5×#380411yescharacteristic
316C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutatorC2×C32C22×C4×C5
FrattiniC22C2×C5×D32
FittingC23×C32×C5C22
socleC22×C32×C5C23
prime radicalC2×C3×C5×#67C2
cosocleC22×C32C23×C5
hypercenterC23×C5D32
hypocenterC32C4×C5×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×Dic3C2semidirect
3. C2×C5×D3×Dic3C2semidirect
4. C5×#3757C2semidirect
5. C5×Dic3⋊Dic3C2semidirect
6. C5×(C2×D3)⋊Dic3C2semidirect
7. C5×Dic32C2semidirect
14. C3×C5×#67C4semidirect
20. Dic3×#67C5direct (N × H)
25. C22×C5×Dic3D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
46. C3×C5×Dic3C2×C4semidirect
47. C3×C5×Dic3D4semidirect
50. C22×C3×Dic3C2×C5semidirect
51. C2×D3×Dic3C2×C5semidirect
52. #3757C2×C5semidirect
53. Dic3⋊Dic3C2×C5semidirect
54. (C2×D3)⋊Dic3C2×C5semidirect
55. Dic32C2×C5semidirect
66. C5×#67Dic3direct (N × H)
97. C3×#67C4×C5semidirect
114. C22×C3×C5C4×D3semidirect
115. C22×C3×C5C2×Dic3semidirect
120. C2×C5×D3C2×Dic3semidirect
131. C5×Dic3C2×Dic3semidirect
132. C5×Dic3#67direct (N × H)
137. C22×Dic3C5×D3semidirect
148. C32×C5C4×D4semidirect
161. C22×C32C2×C4×C5semidirect
165. C2×C3×D3C2×C4×C5semidirect
169. C3×Dic3C2×C4×C5semidirect
170. C3×Dic3C5×D4semidirect
196. #67C5×Dic3direct (N × H)
214. C22×C5D3×Dic3semidirect
234. C3×C5C4×#67semidirect
235. C3×C5D4×Dic3semidirect
247. C22×C3C4×C5×D3semidirect
248. C22×C3C2×C5×Dic3semidirect
253. C2×D3C2×C5×Dic3semidirect
264. Dic3C2×C5×Dic3semidirect
265. Dic3C5×#67direct (N × H)
271. C32C4×C5×D4semidirect
296. C5Dic3×#67direct (N × H)
302. C22C5×D3×Dic3semidirect
310. C3C4×C5×#67semidirect
311. C3C5×D4×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#672
C22×C3×C5×Dic32
C2×C5×D3×Dic32
C5×#37572
C5×Dic3⋊Dic32
C5×(C2×D3)⋊Dic32
C5×Dic322
C5×D4×Dic33
C4×C5×#673
Dic3×#675