Finite Group Databasebeta

← C3×C5×#884 #13087896

Subgroups — C3×C5×#884 #13087896

Order 1440.

410 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#67C211yesmaximal, characteristic, complemented
2C2×C3×C5×#67C211yesmaximal, characteristic, complemented
3C2×C3×C5×D12C211yesmaximal, characteristic, complemented
4C2×C3×C4×C5×D3C211yesmaximal, characteristic, complemented
5C32×C5×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×C5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7C3×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
8C3×C5×#121—31nomaximal
9C5×#884C311yesmaximal, characteristic, direct product factor
10C23×C32×C5C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C22×C3×C5×D3C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C32×C4×C5C2211yescharacteristic
15C3×C5×#67—22no—
16C3×C5×#67—22no—
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19C3×C5×D12—22no—
20C3×C4×C5×D3—22no—
21C3×#884C511yesmaximal, characteristic, direct product factor, Hall
22C2×C3×C5×D4—31no—
23C2×C3×C5×D4—31no—
24C2×C3×C5×D4—31no—
25C22×C3×C4×C5—31no—
26C2×C5×#67C2×C311yescharacteristic, complemented
27C2×C5×#67C2×C311yescharacteristic, complemented
28C2×C5×D12C2×C311yescharacteristic, complemented
29C2×C4×C5×D3C2×C311yescharacteristic, complemented
30C3×C5×C4⋊C4—31no—
31C3×C5×C22⋊C4—21no—
32C3×C5×C22⋊C4—31no—
33C3×C5×C22⋊C4D311yescharacteristic, complemented
34C3×C5×C22⋊C4C2×C311yescharacteristic, complemented
35C5×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
36C5×Dic3⋊C4C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C22×C32×C5—21no—
39C22×C32×C5—21no—
40C22×C32×C5C2311yescharacteristic
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3—21no—
43C2×C3×C5×D3—21no—
44C2×C3×C5×D3—21no—
45C2×C3×C5×D3—21no—
46C2×C3×C5×D3D412yescomplemented
47C32×C4×C5—21no—
48C32×C4×C5—21no—
49C3×C5×Dic3—21no—
50C3×C5×Dic3D412yescomplemented
51C5×#121—31noHall
52C2×C3×#67C2×C511yescharacteristic, complemented
53C2×C3×#67C2×C511yescharacteristic, complemented
54C2×C3×D12C2×C511yescharacteristic, complemented
55C2×C3×C4×D3C2×C511yescharacteristic, complemented
56C32×C22⋊C4C2×C511yescharacteristic, complemented
57C3×(C2×D3)⋊C4C2×C511yescharacteristic, complemented
58C3×Dic3⋊C4C2×C511yescharacteristic, complemented
59C23×C3×C5—21no—
60C23×C3×C5—31no—
61C23×C3×C5—31no—
62C23×C3×C5C2×D311yescharacteristic
63C23×C3×C5C22×C311yescharacteristic
64C22×C5×D3C22×C311yescharacteristic
65C22×C5×D3C22×C311yescharacteristic
66C3×C5×D4—62no—
67C3×C5×D4—62no—
68C3×C5×D4—62no—
69C2×C3×C4×C5—21no—
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—31no—
72C2×C3×C4×C5—31no—
73C2×C3×C4×C5—62no—
74C2×C3×C4×C5C2×D311yescharacteristic
75C2×C3×C4×C5C2×D311yescharacteristic
76C2×C3×C4×C5C22×C311yescharacteristic
77C2×C3×C4×C5C22×C311yescharacteristic
78C5×#67—22no—
79C5×#67—22no—
80C2×C5×Dic3C22×C311yescharacteristic
81C2×C5×Dic3C22×C311yescharacteristic
82C5×D12—22no—
83C4×C5×D3—22no—
84C3×#121—31no—
85#884C3×C511yescharacteristic, direct product factor
86C2×C32×C5—41no—
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5#4111yescharacteristic
90C3×C5×D3—22no—
91C3×C5×D3—41no—
92C2×C5×D4—31no—
93C2×C5×D4—31no—
94C2×C5×D4—31no—
95C22×C4×C5—31no—
96C5×C4⋊C4—31no—
97C5×C22⋊C4—31no—
98C5×C22⋊C4C3×D311yescharacteristic, complemented
99C23×C32C22×C511yescharacteristic
100C22×C3×D3C22×C511yescharacteristic
101C22×C3×D3C22×C511yescharacteristic
102C2×C32×C4C22×C511yescharacteristic
103C2×C32×C4C22×C511yescharacteristic
104C3×#67—22no—
105C3×#67—22no—
106C2×C3×Dic3C22×C511yescharacteristic
107C2×C3×Dic3C22×C511yescharacteristic
108C3×D12—22no—
109C3×C4×D3—22no—
110C22×C3×C5—22no—
111C22×C3×C5—21no—
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—32no—
119C22×C3×C5—41no—
120C22×C3×C5—41no—
121C22×C3×C5—61no—
122C22×C3×C5—61no—
123C22×C3×C5—61no—
124C22×C3×C5—61no—
125C22×C3×C5—61no—
126C22×C3×C5C22×D311yescharacteristic
127C22×C3×C5C23×C311yescharacteristic
128C2×C5×D3—21no—
129C2×C5×D3—21no—
130C2×C5×D3—21no—
131C2×C5×D3—21no—
132C2×C5×D3—21no—
133C2×C5×D3C3×D412yescomplemented
134C3×C4×C5—21no—
135C3×C4×C5—21no—
136C3×C4×C5—21no—
137C3×C4×C5—21no—
138C3×C4×C5—32no—
139C3×C4×C5—41no—
140C3×C4×C5—41no—
141C3×C4×C5—61no—
142C5×Dic3—21no—
143C5×Dic3C3×D412yescomplemented
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—31no—
147C22×C3×C4—31no—
148C2×#67C2×C3×C511yescharacteristic, complemented
149C2×#67C2×C3×C511yescharacteristic, complemented
150C2×D12C2×C3×C511yescharacteristic, complemented
151C2×C4×D3C2×C3×C511yescharacteristic, complemented
152C3×C4⋊C4—31no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—31no—
155C3×C22⋊C4C5×D311yescharacteristic, complemented
156C3×C22⋊C4C2×C3×C511yescharacteristic, complemented
157(C2×D3)⋊C4C2×C3×C511yescharacteristic, complemented
158Dic3⋊C4C2×C3×C511yescharacteristic, complemented
159C32×C5#12111yescharacteristic, complemented, Hall
160C23×C5—31no—
161C23×C5—31no—
162C23×C5C2×C3×D311yescharacteristic
163C5×D4—62no—
164C5×D4—62no—
165C5×D4—62no—
166C2×C4×C5—31no—
167C2×C4×C5—31no—
168C2×C4×C5—62no—
169C2×C4×C5C2×C3×D311yescharacteristic
170C2×C4×C5C2×C3×D311yescharacteristic
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C23×C511yescharacteristic
175C2×C3×D3—21no—
176C2×C3×D3—21no—
177C2×C3×D3—21no—
178C2×C3×D3—21no—
179C2×C3×D3—21no—
180C2×C3×D3C5×D412yescomplemented
181C32×C4—21no—
182C32×C4—21no—
183C3×Dic3—21no—
184C3×Dic3C5×D412yescomplemented
185#121—31noSylow
186C2×C3×C5—21no—
187C2×C3×C5—21no—
188C2×C3×C5—21no—
189C2×C3×C5—42no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192C2×C3×C5—62no—
193C2×C3×C5—121no—
194C2×C3×C5#23511yescharacteristic
195C2×C3×C5D3×D411yescharacteristic
196C2×C3×C5D3×D411yescharacteristic
197C2×C3×C5C2×C3×D411yescharacteristic
198C2×C3×C5C2×C3×D411yescharacteristic
199C2×C3×C5C3×#4111yescharacteristic
200C5×D3—22no—
201C5×D3—41no—
202C23×C3—21no—
203C23×C3—31no—
204C23×C3—31no—
205C23×C3C2×C5×D311yescharacteristic
206C23×C3C22×C3×C511yescharacteristic
207C22×D3C22×C3×C511yescharacteristic
208C22×D3C22×C3×C511yescharacteristic
209C3×D4—62no—
210C3×D4—62no—
211C3×D4—62no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—62no—
217C2×C3×C4C2×C5×D311yescharacteristic
218C2×C3×C4C2×C5×D311yescharacteristic
219C2×C3×C4C22×C3×C511yescharacteristic
220C2×C3×C4C22×C3×C511yescharacteristic
221#67—22no—
222#67—22no—
223C2×Dic3C22×C3×C511yescharacteristic
224C2×Dic3C22×C3×C511yescharacteristic
225D12—22no—
226C4×D3—22no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—32no—
231C22×C5—61no—
232C22×C5—61no—
233C22×C5—61no—
234C22×C5—61no—
235C22×C5—61no—
236C22×C5C22×C3×D311yescharacteristic
237C4×C5—21no—
238C4×C5—21no—
239C4×C5—32no—
240C4×C5—61no—
241C2×C32—41no—
242C2×C32C2×C5×D411yescharacteristic
243C2×C32C2×C5×D411yescharacteristic
244C2×C32C5×#4111yescharacteristic
245C3×D3—22no—
246C3×D3—41no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—31no—
250C22×C4—31no—
251C4⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4C3×C5×D311yescharacteristic, complemented
254C3×C5—21no—
255C3×C5#88411yescharacteristic, direct product factor
256C3×C5C3×#12111yescharacteristic, complemented
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—32no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3C22×C5×D311yescharacteristic
274C22×C3C23×C3×C511yescharacteristic
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3C3×C5×D412yescomplemented
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—32no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—61no—
289Dic3—21no—
290Dic3C3×C5×D412yescomplemented
291C2×C5—41no—
292C2×C5—62no—
293C2×C5—121no—
294C2×C5C3×#23511yescharacteristic
295C2×C5C3×D3×D411yescharacteristic
296C2×C5C3×D3×D411yescharacteristic
297C32C5×#12111yescharacteristic, complemented, Sylow
298C23—31no—
299C23—31no—
300C23C2×C3×C5×D311yescharacteristic
301D4—62no—
302D4—62no—
303D4—62no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—62no—
307C2×C4C2×C3×C5×D311yescharacteristic
308C2×C4C2×C3×C5×D311yescharacteristic
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—62no—
316C2×C3—121no—
317C2×C3C5×#23511yescharacteristic
318C2×C3C5×D3×D411yescharacteristic
319C2×C3C5×D3×D411yescharacteristic
320C2×C3C2×C3×C5×D411yescharacteristic
321C2×C3C2×C3×C5×D411yescharacteristic
322C2×C3C3×C5×#4111yescharacteristic
323D3—22no—
324D3—41no—
325C5C3×#88411yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—32no—
330C22—61no—
331C22—61no—
332C22—61no—
333C22—61no—
334C22—61no—
335C22C22×C3×C5×D311yescharacteristic
336C4—21no—
337C4—21no—
338C4—32no—
339C4—61no—
340C3—21no—
341C3C5×#88411yescharacteristic, direct product factor
342C3C3×C5×#12111yescharacteristic, complemented
343C2—41no—
344C2—62no—
345C2—121no—
346C2C3×C5×#23511yescharacteristic
347C2C3×C5×D3×D411yescharacteristic
348C2C3×C5×D3×D411yescharacteristic
subgroupG/H
centerC22×C3×C5C22×D3
commutator, cosocleC22×C3C23×C3×C5
FrattiniC22C22×C3×C5×D3
FittingC32×C5×C22⋊C4C2
socleC22×C32×C5C23
hypercenterC3×C5×C22⋊C4D3
hypocenterC3C3×C5×#121
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
1. C2×C3×C5×#67C2semidirect
2. C2×C3×C5×#67C2semidirect
3. C2×C3×C5×D12C2semidirect
4. C2×C3×C4×C5×D3C2semidirect
5. C32×C5×C22⋊C4C2semidirect
6. C3×C5×(C2×D3)⋊C4C2semidirect
7. C3×C5×Dic3⋊C4C2semidirect
9. C5×#884C3direct (N × H)
21. C3×#884C5direct (N × H)
26. C2×C5×#67C2×C3semidirect
27. C2×C5×#67C2×C3semidirect
28. C2×C5×D12C2×C3semidirect
29. C2×C4×C5×D3C2×C3semidirect
33. C3×C5×C22⋊C4D3semidirect
34. C3×C5×C22⋊C4C2×C3semidirect
35. C5×(C2×D3)⋊C4C2×C3semidirect
36. C5×Dic3⋊C4C2×C3semidirect
46. C2×C3×C5×D3D4semidirect
50. C3×C5×Dic3D4semidirect
52. C2×C3×#67C2×C5semidirect
53. C2×C3×#67C2×C5semidirect
54. C2×C3×D12C2×C5semidirect
55. C2×C3×C4×D3C2×C5semidirect
56. C32×C22⋊C4C2×C5semidirect
57. C3×(C2×D3)⋊C4C2×C5semidirect
58. C3×Dic3⋊C4C2×C5semidirect
85. #884C3×C5direct (N × H)
98. C5×C22⋊C4C3×D3semidirect
133. C2×C5×D3C3×D4semidirect
143. C5×Dic3C3×D4semidirect
148. C2×#67C2×C3×C5semidirect
149. C2×#67C2×C3×C5semidirect
150. C2×D12C2×C3×C5semidirect
151. C2×C4×D3C2×C3×C5semidirect
155. C3×C22⋊C4C5×D3semidirect
156. C3×C22⋊C4C2×C3×C5semidirect
157. (C2×D3)⋊C4C2×C3×C5semidirect
158. Dic3⋊C4C2×C3×C5semidirect
159. C32×C5#121semidirect
180. C2×C3×D3C5×D4semidirect
184. C3×Dic3C5×D4semidirect
253. C22⋊C4C3×C5×D3semidirect
255. C3×C5#884direct (N × H)
256. C3×C5C3×#121semidirect
280. C2×D3C3×C5×D4semidirect
290. Dic3C3×C5×D4semidirect
297. C32C5×#121semidirect
325. C5C3×#884direct (N × H)
341. C3C5×#884direct (N × H)
342. C3C3×C5×#121semidirect

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×C5×D122
C2×C3×C4×C5×D32
C32×C5×C22⋊C42
C3×C5×(C2×D3)⋊C42
C3×C5×Dic3⋊C42
C3×C5×#1213
C5×#8843
C3×#8845