Subgroups — C₃×C₅×#1440 #14274474

Order 1920.

1472 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#95	C₂	1	4	yes	maximal, complemented
4	C₅×#1440	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
7	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
8	C₂×C₃×C₅×C₂²⋊C₄	—	2	4	no	—
9	C₂×C₃×C₅×C₂²⋊C₄	—	2	4	no	—
10	C₂×C₃×C₅×C₂²⋊C₄	—	2	8	no	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	4	yes	complemented
12	C₃×C₅×#95	—	2	8	no	—
13	C₃×#1440	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	—
15	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
16	C₂×C₅×#95	C₂×C₃	1	4	yes	complemented
17	C₂⁴×C₃×C₅	—	2	4	no	—
18	C₂⁴×C₃×C₅	—	2	4	no	—
19	C₂⁴×C₃×C₅	—	2	4	no	—
20	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
21	C₂⁴×C₃×C₅	D₄	1	4	yes	—
22	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	4	no	—
26	C₂²×C₃×C₄×C₅	—	2	4	no	—
27	C₂²×C₃×C₄×C₅	—	2	4	no	—
28	C₂²×C₃×C₄×C₅	—	2	4	no	—
29	C₂²×C₃×C₄×C₅	—	2	4	no	—
30	C₂²×C₃×C₄×C₅	—	2	4	no	—
31	C₂²×C₃×C₄×C₅	—	2	4	no	—
32	C₂²×C₃×C₄×C₅	—	2	4	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
35	C₃×C₅×C₂²⋊C₄	—	2	16	no	—
36	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
37	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	2	yes	—
38	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
39	C₂×C₃×#95	C₂×C₅	1	4	yes	complemented
40	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
42	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
43	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
46	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	4	yes	complemented
47	C₅×#95	—	2	8	no	—
48	#1440	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×C₃×C₅	—	2	4	no	—
50	C₂³×C₃×C₅	—	2	8	no	—
51	C₂³×C₃×C₅	—	2	4	no	—
52	C₂³×C₃×C₅	—	2	8	no	—
53	C₂³×C₃×C₅	—	2	8	no	—
54	C₂³×C₃×C₅	—	2	4	no	—
55	C₂³×C₃×C₅	—	4	4	no	—
56	C₂³×C₃×C₅	—	4	4	no	—
57	C₂³×C₃×C₅	—	4	4	no	—
58	C₂³×C₃×C₅	—	4	4	no	—
59	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
62	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
63	C₂³×C₃×C₅	#41	1	4	yes	—
64	C₂³×C₃×C₅	#41	1	4	yes	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	2	8	no	—
67	C₂×C₃×C₄×C₅	—	2	8	no	—
68	C₂×C₃×C₄×C₅	—	2	8	no	—
69	C₂×C₃×C₄×C₅	—	2	8	no	—
70	C₂×C₃×C₄×C₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	4	4	no	—
72	C₂×C₃×C₄×C₅	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	—	4	4	no	—
77	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
78	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
79	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
80	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
81	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
82	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
83	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	4	yes	complemented
84	C₃×#95	—	2	8	no	—
85	C₂⁴×C₅	—	2	4	no	—
86	C₂⁴×C₅	—	2	4	no	—
87	C₂⁴×C₅	—	2	4	no	—
88	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
89	C₂⁴×C₅	C₃×D₄	1	4	yes	—
90	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	4	no	—
94	C₂²×C₄×C₅	—	2	4	no	—
95	C₂²×C₄×C₅	—	2	4	no	—
96	C₂²×C₄×C₅	—	2	4	no	—
97	C₂²×C₄×C₅	—	2	4	no	—
98	C₂²×C₄×C₅	—	2	4	no	—
99	C₂²×C₄×C₅	—	2	4	no	—
100	C₂²×C₄×C₅	—	2	4	no	—
101	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
102	C₂²×C₄×C₅	C₃×D₄	1	8	yes	—
103	C₅×C₂²⋊C₄	—	2	16	no	—
104	C₅×C₂²⋊C₄	—	2	8	no	—
105	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	—
106	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂×#95	C₂×C₃×C₅	1	4	yes	complemented
108	C₂²×C₃×C₅	—	2	4	no	—
109	C₂²×C₃×C₅	—	2	8	no	—
110	C₂²×C₃×C₅	—	4	8	no	—
111	C₂²×C₃×C₅	—	4	8	no	—
112	C₂²×C₃×C₅	—	4	8	no	—
113	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₃×C₅	#117	1	2	yes	—
115	C₂²×C₃×C₅	C₄×D₄	1	4	yes	—
116	C₂²×C₃×C₅	#120	1	2	yes	—
117	C₂²×C₃×C₅	#121	1	4	yes	—
118	C₂²×C₃×C₅	#121	1	4	yes	—
119	C₂²×C₃×C₅	#123	1	4	yes	—
120	C₂²×C₃×C₅	#123	1	2	yes	—
121	C₂²×C₃×C₅	#123	1	4	yes	—
122	C₂²×C₃×C₅	#124	1	4	yes	—
123	C₂²×C₃×C₅	#126	1	4	yes	—
124	C₃×C₄×C₅	—	4	4	no	—
125	C₃×C₄×C₅	—	4	8	no	—
126	C₂⁴×C₃	—	2	4	no	—
127	C₂⁴×C₃	—	2	4	no	—
128	C₂⁴×C₃	—	2	4	no	—
129	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
130	C₂⁴×C₃	C₅×D₄	1	4	yes	—
131	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	—	2	4	no	—
133	C₂²×C₃×C₄	—	2	4	no	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	2	4	no	—
137	C₂²×C₃×C₄	—	2	4	no	—
138	C₂²×C₃×C₄	—	2	4	no	—
139	C₂²×C₃×C₄	—	2	4	no	—
140	C₂²×C₃×C₄	—	2	4	no	—
141	C₂²×C₃×C₄	—	2	4	no	—
142	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
143	C₂²×C₃×C₄	C₅×D₄	1	8	yes	—
144	C₃×C₂²⋊C₄	—	2	16	no	—
145	C₃×C₂²⋊C₄	—	2	8	no	—
146	C₂³×C₅	—	2	4	no	—
147	C₂³×C₅	—	2	8	no	—
148	C₂³×C₅	—	2	4	no	—
149	C₂³×C₅	—	2	8	no	—
150	C₂³×C₅	—	2	8	no	—
151	C₂³×C₅	—	2	4	no	—
152	C₂³×C₅	—	4	4	no	—
153	C₂³×C₅	—	4	4	no	—
154	C₂³×C₅	—	4	4	no	—
155	C₂³×C₅	—	4	4	no	—
156	C₂³×C₅	C₃×C₂²⋊C₄	1	4	yes	—
157	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
159	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
160	C₂³×C₅	C₃×#41	1	4	yes	—
161	C₂³×C₅	C₃×#41	1	4	yes	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	8	no	—
164	C₂×C₄×C₅	—	2	8	no	—
165	C₂×C₄×C₅	—	2	8	no	—
166	C₂×C₄×C₅	—	2	8	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	4	no	—
169	C₂×C₄×C₅	—	4	4	no	—
170	C₂×C₄×C₅	—	4	4	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	4	no	—
173	C₂×C₄×C₅	—	4	4	no	—
174	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
176	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₂²⋊C₄	—	2	8	no	—
180	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	4	yes	complemented
181	#95	—	2	8	no	—
182	C₂×C₃×C₅	—	4	4	no	—
183	C₂×C₃×C₅	#381	1	1	yes	characteristic
184	C₂×C₃×C₅	#386	1	2	yes	—
185	C₂×C₃×C₅	#388	1	4	yes	—
186	C₂×C₃×C₅	#394	1	4	yes	—
187	C₂×C₃×C₅	#397	1	4	yes	—
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	2	8	no	—
190	C₂³×C₃	—	2	4	no	—
191	C₂³×C₃	—	2	8	no	—
192	C₂³×C₃	—	2	8	no	—
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	—	4	4	no	—
195	C₂³×C₃	—	4	4	no	—
196	C₂³×C₃	—	4	4	no	—
197	C₂³×C₃	—	4	4	no	—
198	C₂³×C₃	C₅×C₂²⋊C₄	1	4	yes	—
199	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
200	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
201	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
202	C₂³×C₃	C₅×#41	1	4	yes	—
203	C₂³×C₃	C₅×#41	1	4	yes	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	8	no	—
206	C₂×C₃×C₄	—	2	8	no	—
207	C₂×C₃×C₄	—	2	8	no	—
208	C₂×C₃×C₄	—	2	8	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	4	4	no	—
211	C₂×C₃×C₄	—	4	4	no	—
212	C₂×C₃×C₄	—	4	4	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	—	4	4	no	—
216	C₂²×C₅	—	2	4	no	—
217	C₂²×C₅	—	2	8	no	—
218	C₂²×C₅	—	4	8	no	—
219	C₂²×C₅	—	4	8	no	—
220	C₂²×C₅	—	4	8	no	—
221	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	C₃×#117	1	2	yes	—
223	C₂²×C₅	C₃×C₄×D₄	1	4	yes	—
224	C₂²×C₅	C₃×#120	1	2	yes	—
225	C₂²×C₅	C₃×#121	1	4	yes	—
226	C₂²×C₅	C₃×#121	1	4	yes	—
227	C₂²×C₅	C₃×#123	1	4	yes	—
228	C₂²×C₅	C₃×#123	1	2	yes	—
229	C₂²×C₅	C₃×#123	1	4	yes	—
230	C₂²×C₅	C₃×#124	1	4	yes	—
231	C₂²×C₅	C₃×#126	1	4	yes	—
232	C₄×C₅	—	4	4	no	—
233	C₄×C₅	—	4	8	no	—
234	C₂⁴	—	2	4	no	—
235	C₂⁴	—	2	4	no	—
236	C₂⁴	—	2	4	no	—
237	C₂⁴	C₂×C₃×C₄×C₅	1	2	yes	—
238	C₂⁴	C₃×C₅×D₄	1	4	yes	—
239	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
251	C₂²×C₄	C₃×C₅×D₄	1	8	yes	—
252	C₂²⋊C₄	—	2	16	no	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₃×C₅	#1440	1	1	yes	characteristic, direct product factor, Hall
255	C₂²×C₃	—	2	4	no	—
256	C₂²×C₃	—	2	8	no	—
257	C₂²×C₃	—	4	8	no	—
258	C₂²×C₃	—	4	8	no	—
259	C₂²×C₃	—	4	8	no	—
260	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂²×C₃	C₅×#117	1	2	yes	—
262	C₂²×C₃	C₄×C₅×D₄	1	4	yes	—
263	C₂²×C₃	C₅×#120	1	2	yes	—
264	C₂²×C₃	C₅×#121	1	4	yes	—
265	C₂²×C₃	C₅×#121	1	4	yes	—
266	C₂²×C₃	C₅×#123	1	4	yes	—
267	C₂²×C₃	C₅×#123	1	2	yes	—
268	C₂²×C₃	C₅×#123	1	4	yes	—
269	C₂²×C₃	C₅×#124	1	4	yes	—
270	C₂²×C₃	C₅×#126	1	4	yes	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	8	no	—
273	C₂×C₅	—	4	4	no	—
274	C₂×C₅	C₃×#381	1	1	yes	characteristic
275	C₂×C₅	C₃×#386	1	2	yes	—
276	C₂×C₅	C₃×#388	1	4	yes	—
277	C₂×C₅	C₃×#394	1	4	yes	—
278	C₂×C₅	C₃×#397	1	4	yes	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	8	no	—
281	C₂³	—	2	4	no	—
282	C₂³	—	2	8	no	—
283	C₂³	—	2	8	no	—
284	C₂³	—	2	4	no	—
285	C₂³	—	4	4	no	—
286	C₂³	—	4	4	no	—
287	C₂³	—	4	4	no	—
288	C₂³	—	4	4	no	—
289	C₂³	C₃×C₅×C₂²⋊C₄	1	4	yes	—
290	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
291	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
292	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
293	C₂³	C₃×C₅×#41	1	4	yes	—
294	C₂³	C₃×C₅×#41	1	4	yes	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₃	—	4	4	no	—
308	C₂×C₃	C₅×#381	1	1	yes	characteristic
309	C₂×C₃	C₅×#386	1	2	yes	—
310	C₂×C₃	C₅×#388	1	4	yes	—
311	C₂×C₃	C₅×#394	1	4	yes	—
312	C₂×C₃	C₅×#397	1	4	yes	—
313	C₅	C₃×#1440	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	4	no	—
315	C₂²	—	2	8	no	—
316	C₂²	—	4	8	no	—
317	C₂²	—	4	8	no	—
318	C₂²	—	4	8	no	—
319	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
320	C₂²	C₃×C₅×#117	1	2	yes	—
321	C₂²	C₃×C₄×C₅×D₄	1	4	yes	—
322	C₂²	C₃×C₅×#120	1	2	yes	—
323	C₂²	C₃×C₅×#121	1	4	yes	—
324	C₂²	C₃×C₅×#121	1	4	yes	—
325	C₂²	C₃×C₅×#123	1	4	yes	—
326	C₂²	C₃×C₅×#123	1	2	yes	—
327	C₂²	C₃×C₅×#123	1	4	yes	—
328	C₂²	C₃×C₅×#124	1	4	yes	—
329	C₂²	C₃×C₅×#126	1	4	yes	—
330	C₄	—	4	4	no	—
331	C₄	—	4	8	no	—
332	C₃	C₅×#1440	1	1	yes	characteristic, direct product factor, Sylow
333	C₂	—	4	4	no	—
334	C₂	C₃×C₅×#381	1	1	yes	characteristic
335	C₂	C₃×C₅×#386	1	2	yes	—
336	C₂	C₃×C₅×#388	1	4	yes	—
337	C₂	C₃×C₅×#394	1	4	yes	—
338	C₂	C₃×C₅×#397	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃×C₅	C₂³
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂⁴	C₂³×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₅×#95	C₂	semidirect
4. C₅×#1440	C₃	direct (N × H)
11. C₂×C₃×C₅×C₂²⋊C₄	C₄	semidirect
13. C₃×#1440	C₅	direct (N × H)
16. C₂×C₅×#95	C₂×C₃	semidirect
39. C₂×C₃×#95	C₂×C₅	semidirect
46. C₂×C₅×C₂²⋊C₄	C₃×C₄	semidirect
48. #1440	C₃×C₅	direct (N × H)
83. C₂×C₃×C₂²⋊C₄	C₄×C₅	semidirect
107. C₂×#95	C₂×C₃×C₅	semidirect
180. C₂×C₂²⋊C₄	C₃×C₄×C₅	semidirect
254. C₃×C₅	#1440	direct (N × H)
313. C₅	C₃×#1440	direct (N × H)
332. C₃	C₅×#1440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₂²⋊C₄	2
C₂×C₃×C₅×#95	2
C₅×#1440	3
C₃×#1440	5

Subgroups — C3×C5×#1440 #14274474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1440 #14274474