Finite Group Databasebeta

← C3×C5×#883 #13087895

Subgroups — C3×C5×#883 #13087895

Order 1440.

346 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#67C211yesmaximal, characteristic
2C2×C3×C4×C5×D3C211yesmaximal, characteristic, complemented
3C32×C5×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5C3×C5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C3×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
7C3×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
8C3×C5×#123—31nomaximal
9C5×#883C311yesmaximal, characteristic, direct product factor
10C23×C32×C5C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C3×C5×#67—22no—
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18C3×C4×C5×D3—22no—
19C3×#883C511yesmaximal, characteristic, direct product factor, Hall
20C2×C3×C5×D4—31no—
21C22×C3×C4×C5—31no—
22C2×C5×#67C2×C311yescharacteristic
23C2×C4×C5×D3C2×C311yescharacteristic, complemented
24C3×C5×C4⋊C4—31no—
25C3×C5×C4⋊C4—31no—
26C3×C5×C22⋊C4—21no—
27C3×C5×C22⋊C4—31no—
28C3×C5×C22⋊C4—31no—
29C3×C5×C22⋊C4D311yescharacteristic, complemented
30C3×C5×C22⋊C4C2×C311yescharacteristic, complemented
31C5×#217C2×C311yescharacteristic, complemented
32C5×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
33C5×C4⋊Dic3C2×C311yescharacteristic, complemented
34C5×Dic3⋊C4C2×C311yescharacteristic, complemented
35C22×C32×C5—21no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2311yescharacteristic
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3D412yes—
42C32×C4×C5—21no—
43C32×C4×C5—21no—
44C3×C5×Dic3—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3—21no—
47C5×#123—31noHall
48C2×C3×#67C2×C511yescharacteristic
49C2×C3×C4×D3C2×C511yescharacteristic, complemented
50C32×C22⋊C4C2×C511yescharacteristic, complemented
51C3×#217C2×C511yescharacteristic, complemented
52C3×(C2×D3)⋊C4C2×C511yescharacteristic, complemented
53C3×C4⋊Dic3C2×C511yescharacteristic, complemented
54C3×Dic3⋊C4C2×C511yescharacteristic, complemented
55C23×C3×C5—21no—
56C23×C3×C5—31no—
57C23×C3×C5C2×D311yescharacteristic
58C23×C3×C5C22×C311yescharacteristic
59C22×C5×D3C22×C311yescharacteristic
60C3×C5×D4—62no—
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5—21no—
63C2×C3×C4×C5—31no—
64C2×C3×C4×C5—31no—
65C2×C3×C4×C5—31no—
66C2×C3×C4×C5—62no—
67C2×C3×C4×C5C2×D311yescharacteristic
68C2×C3×C4×C5C2×D311yescharacteristic
69C2×C3×C4×C5C22×C311yescharacteristic
70C2×C3×C4×C5C22×C311yescharacteristic
71C5×#67—22no—
72C2×C5×Dic3C22×C311yescharacteristic
73C2×C5×Dic3C22×C311yescharacteristic
74C2×C5×Dic3C22×C311yescharacteristic
75C4×C5×D3—22no—
76C3×#123—31no—
77#883C3×C511yescharacteristic, direct product factor
78C2×C32×C5—41no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C3×C5×D3—22no—
83C2×C5×D4—31no—
84C22×C4×C5—31no—
85C5×C4⋊C4—31no—
86C5×C4⋊C4—31no—
87C5×C22⋊C4—31no—
88C5×C22⋊C4—31no—
89C5×C22⋊C4C3×D311yescharacteristic, complemented
90C23×C32C22×C511yescharacteristic
91C22×C3×D3C22×C511yescharacteristic
92C2×C32×C4C22×C511yescharacteristic
93C2×C32×C4C22×C511yescharacteristic
94C3×#67—22no—
95C2×C3×Dic3C22×C511yescharacteristic
96C2×C3×Dic3C22×C511yescharacteristic
97C2×C3×Dic3C22×C511yescharacteristic
98C3×C4×D3—22no—
99C22×C3×C5—22no—
100C22×C3×C5—21no—
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—32no—
108C22×C3×C5—41no—
109C22×C3×C5—41no—
110C22×C3×C5—61no—
111C22×C3×C5—61no—
112C22×C3×C5C22×D311yescharacteristic
113C22×C3×C5C23×C311yescharacteristic
114C2×C5×D3—21no—
115C2×C5×D3—21no—
116C2×C5×D3C3×D412yes—
117C3×C4×C5—21no—
118C3×C4×C5—21no—
119C3×C4×C5—21no—
120C3×C4×C5—21no—
121C3×C4×C5—41no—
122C3×C4×C5—41no—
123C3×C4×C5—61no—
124C3×C4×C5—61no—
125C3×C4×C5—61no—
126C5×Dic3—21no—
127C5×Dic3—21no—
128C5×Dic3—21no—
129C2×C3×D4—31no—
130C22×C3×C4—31no—
131C2×#67C2×C3×C511yescharacteristic
132C2×C4×D3C2×C3×C511yescharacteristic, complemented
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4C5×D311yescharacteristic, complemented
139C3×C22⋊C4C2×C3×C511yescharacteristic, complemented
140#217C2×C3×C511yescharacteristic, complemented
141(C2×D3)⋊C4C2×C3×C511yescharacteristic, complemented
142C4⋊Dic3C2×C3×C511yescharacteristic, complemented
143Dic3⋊C4C2×C3×C511yescharacteristic, complemented
144C32×C5#12311yescharacteristic, complemented, Hall
145C23×C5—31no—
146C23×C5C2×C3×D311yescharacteristic
147C5×D4—62no—
148C2×C4×C5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—31no—
151C2×C4×C5—62no—
152C2×C4×C5C2×C3×D311yescharacteristic
153C2×C4×C5C2×C3×D311yescharacteristic
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32C23×C511yescharacteristic
158C2×C3×D3—21no—
159C2×C3×D3—21no—
160C2×C3×D3C5×D412yes—
161C32×C4—21no—
162C32×C4—21no—
163C3×Dic3—21no—
164C3×Dic3—21no—
165C3×Dic3—21no—
166#123—31noSylow
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5—42no—
171C2×C3×C5—41no—
172C2×C3×C5—41no—
173C2×C3×C5—62no—
174C2×C3×C5#23511yescharacteristic
175C2×C3×C5D3×D411yescharacteristic
176C2×C3×C5#23711yescharacteristic
177C2×C3×C5C2×C3×D411yescharacteristic
178C2×C3×C5C3×#4111yescharacteristic
179C2×C3×C5C3×#4111yescharacteristic
180C5×D3—22no—
181C23×C3—21no—
182C23×C3—31no—
183C23×C3C2×C5×D311yescharacteristic
184C23×C3C22×C3×C511yescharacteristic
185C22×D3C22×C3×C511yescharacteristic
186C3×D4—62no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—62no—
193C2×C3×C4C2×C5×D311yescharacteristic
194C2×C3×C4C2×C5×D311yescharacteristic
195C2×C3×C4C22×C3×C511yescharacteristic
196C2×C3×C4C22×C3×C511yescharacteristic
197#67—22no—
198C2×Dic3C22×C3×C511yescharacteristic
199C2×Dic3C22×C3×C511yescharacteristic
200C2×Dic3C22×C3×C511yescharacteristic
201C4×D3—22no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—32no—
206C22×C5—61no—
207C22×C5—61no—
208C22×C5C22×C3×D311yescharacteristic
209C4×C5—21no—
210C4×C5—21no—
211C4×C5—61no—
212C4×C5—61no—
213C4×C5—61no—
214C2×C32—41no—
215C2×C32C2×C5×D411yescharacteristic
216C2×C32C5×#4111yescharacteristic
217C2×C32C5×#4111yescharacteristic
218C3×D3—22no—
219C2×D4—31no—
220C22×C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C22⋊C4—31no—
224C22⋊C4—31no—
225C22⋊C4C3×C5×D311yescharacteristic, complemented
226C3×C5—21no—
227C3×C5#88311yescharacteristic, direct product factor
228C3×C5C3×#12311yescharacteristic, complemented
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—32no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3C22×C5×D311yescharacteristic
243C22×C3C23×C3×C511yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3C3×C5×D412yes—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256Dic3—21no—
257Dic3—21no—
258Dic3—21no—
259C2×C5—41no—
260C2×C5—62no—
261C2×C5C3×#23511yescharacteristic
262C2×C5C3×D3×D411yescharacteristic
263C2×C5C3×#23711yescharacteristic
264C32C5×#12311yescharacteristic, complemented, Sylow
265C23—31no—
266C23C2×C3×C5×D311yescharacteristic
267D4—62no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—62no—
272C2×C4C2×C3×C5×D311yescharacteristic
273C2×C4C2×C3×C5×D311yescharacteristic
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—62no—
281C2×C3C5×#23511yescharacteristic
282C2×C3C5×D3×D411yescharacteristic
283C2×C3C5×#23711yescharacteristic
284C2×C3C2×C3×C5×D411yescharacteristic
285C2×C3C3×C5×#4111yescharacteristic
286C2×C3C3×C5×#4111yescharacteristic
287D3—22no—
288C5C3×#88311yescharacteristic, direct product factor, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—32no—
293C22—61no—
294C22—61no—
295C22C22×C3×C5×D311yescharacteristic
296C4—21no—
297C4—21no—
298C4—61no—
299C4—61no—
300C4—61no—
301C3—21no—
302C3C5×#88311yescharacteristic, direct product factor
303C3C3×C5×#12311yescharacteristic, complemented
304C2—41no—
305C2—62no—
306C2C3×C5×#23511yescharacteristic
307C2C3×C5×D3×D411yescharacteristic
308C2C3×C5×#23711yescharacteristic
subgroupG/H
centerC22×C3×C5C22×D3
commutator, cosocleC22×C3C23×C3×C5
FrattiniC22C22×C3×C5×D3
FittingC32×C5×C22⋊C4C2
socleC22×C32×C5C23
prime radicalC2×C3×C5×#67C2
hypercenterC3×C5×C22⋊C4D3
hypocenterC3C3×C5×#123
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C5 ◁ C3×C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×D3C2semidirect
3. C32×C5×C22⋊C4C2semidirect
4. C3×C5×#217C2semidirect
5. C3×C5×(C2×D3)⋊C4C2semidirect
6. C3×C5×C4⋊Dic3C2semidirect
7. C3×C5×Dic3⋊C4C2semidirect
9. C5×#883C3direct (N × H)
19. C3×#883C5direct (N × H)
23. C2×C4×C5×D3C2×C3semidirect
29. C3×C5×C22⋊C4D3semidirect
30. C3×C5×C22⋊C4C2×C3semidirect
31. C5×#217C2×C3semidirect
32. C5×(C2×D3)⋊C4C2×C3semidirect
33. C5×C4⋊Dic3C2×C3semidirect
34. C5×Dic3⋊C4C2×C3semidirect
49. C2×C3×C4×D3C2×C5semidirect
50. C32×C22⋊C4C2×C5semidirect
51. C3×#217C2×C5semidirect
52. C3×(C2×D3)⋊C4C2×C5semidirect
53. C3×C4⋊Dic3C2×C5semidirect
54. C3×Dic3⋊C4C2×C5semidirect
77. #883C3×C5direct (N × H)
89. C5×C22⋊C4C3×D3semidirect
132. C2×C4×D3C2×C3×C5semidirect
138. C3×C22⋊C4C5×D3semidirect
139. C3×C22⋊C4C2×C3×C5semidirect
140. #217C2×C3×C5semidirect
141. (C2×D3)⋊C4C2×C3×C5semidirect
142. C4⋊Dic3C2×C3×C5semidirect
143. Dic3⋊C4C2×C3×C5semidirect
144. C32×C5#123semidirect
225. C22⋊C4C3×C5×D3semidirect
227. C3×C5#883direct (N × H)
228. C3×C5C3×#123semidirect
264. C32C5×#123semidirect
288. C5C3×#883direct (N × H)
302. C3C5×#883direct (N × H)
303. C3C3×C5×#123semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#672
C2×C3×C4×C5×D32
C32×C5×C22⋊C42
C3×C5×#2172
C3×C5×(C2×D3)⋊C42
C3×C5×C4⋊Dic32
C3×C5×Dic3⋊C42
C3×C5×#1233
C5×#8833
C3×#8835