Subgroups — C₃×C₅×#2016 #14275050

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#103	C₂	1	2	yes	maximal
4	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₅×#2016	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
10	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
11	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
12	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
14	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×#103	—	2	4	no	—
18	C₃×C₅×#95	—	2	2	no	—
19	C₃×#2016	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₅×Q₁₆	C₂×C₃	1	2	yes	—
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×#103	C₂×C₃	1	2	yes	—
23	C₅×#391	C₂×C₃	1	1	yes	characteristic
24	C₅×#336	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	—	2	2	no	—
33	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
34	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	—	2	1	no	—
37	C₂²×C₃×C₄×C₅	—	2	1	no	—
38	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×Q₁₆	—	2	16	no	—
41	C₂×C₃×C₅×C₈	—	2	2	no	—
42	C₂×C₃×C₅×C₈	—	2	2	no	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₃×C₅×C₄⋊C₄	—	2	8	no	—
45	C₃×C₅×C₄⋊C₄	—	2	2	no	—
46	C₃×C₄²×C₅	—	2	2	no	—
47	C₂²×C₃×Q₁₆	C₂×C₅	1	2	yes	—
48	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
49	C₂×C₃×#103	C₂×C₅	1	2	yes	—
50	C₃×#391	C₂×C₅	1	1	yes	characteristic
51	C₃×#336	C₂×C₅	1	1	yes	characteristic
52	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
54	C₂×C₅×Q₁₆	—	2	4	no	—
55	C₂×C₅×Q₁₆	—	2	4	no	—
56	C₂×C₅×Q₁₆	—	2	4	no	—
57	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
58	C₅×C₄⋊Q₈	—	2	4	no	—
59	C₂×C₅×C₄⋊C₄	—	2	2	no	—
60	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	C₅×#103	—	2	4	no	—
63	C₅×#95	—	2	2	no	—
64	#2016	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×Q₈	—	2	4	no	—
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	2	4	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	complemented
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
90	C₂×C₃×Q₁₆	—	2	4	no	—
91	C₂×C₃×Q₁₆	—	2	4	no	—
92	C₂×C₃×Q₁₆	—	2	4	no	—
93	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
94	C₃×C₄⋊Q₈	—	2	4	no	—
95	C₂×C₃×C₄⋊C₄	—	2	2	no	—
96	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	C₃×#103	—	2	4	no	—
99	C₃×#95	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	—	2	1	no	—
102	C₂×C₅×Q₈	—	2	2	no	—
103	C₂×C₅×Q₈	—	2	2	no	—
104	C₂×C₅×Q₈	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	2	no	—
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
109	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
110	C₂²×C₄×C₅	—	2	2	no	—
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₅×Q₁₆	—	2	16	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
119	C₅×C₄⋊C₄	—	2	8	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₂²×Q₁₆	C₂×C₃×C₅	1	2	yes	—
123	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
124	C₂×#103	C₂×C₃×C₅	1	2	yes	—
125	#391	C₂×C₃×C₅	1	1	yes	characteristic
126	#336	C₂×C₃×C₅	1	1	yes	characteristic
127	C₂²×C₃×C₅	#120	1	1	yes	characteristic
128	C₂²×C₃×C₅	#121	1	1	yes	characteristic
129	C₂²×C₃×C₅	#127	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
131	C₂²×C₃×C₅	#137	1	2	yes	—
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	4	4	no	—
134	C₃×C₄×C₅	—	4	2	no	—
135	C₃×C₄×C₅	—	8	1	no	—
136	C₃×C₄×C₅	#120	1	2	yes	—
137	C₃×C₄×C₅	#121	1	2	yes	—
138	C₂×C₃×Q₈	—	2	2	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
147	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
148	C₂²×C₃×C₄	—	2	1	no	—
149	C₂²×C₃×C₄	—	2	2	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
152	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₃×Q₁₆	—	2	16	no	—
154	C₂×C₃×C₈	—	2	2	no	—
155	C₂×C₃×C₈	—	2	2	no	—
156	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
157	C₃×C₄⋊C₄	—	2	8	no	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	C₃×C₄²	—	2	2	no	—
160	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
161	C₅×Q₈	—	2	4	no	—
162	C₅×Q₈	—	2	4	no	—
163	C₅×Q₈	—	2	4	no	—
164	C₅×Q₈	—	4	2	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	complemented
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
182	C₅×C₈	—	2	4	no	—
183	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
184	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
192	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
193	#103	—	2	4	no	—
194	#95	—	2	2	no	—
195	C₂×C₃×C₅	#392	1	1	yes	characteristic
196	C₂×C₃×C₅	#451	1	2	yes	—
197	C₂×C₃×C₅	#462	1	2	yes	—
198	C₂×C₃×C₅	#494	1	1	yes	characteristic
199	C₂×C₃×C₅	#497	1	1	yes	characteristic
200	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
201	C₃×Q₈	—	2	4	no	—
202	C₃×Q₈	—	2	4	no	—
203	C₃×Q₈	—	2	4	no	—
204	C₃×Q₈	—	4	2	no	—
205	C₃×Q₈	—	4	2	no	—
206	C₃×Q₈	—	4	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	complemented
218	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
222	C₃×C₈	—	2	4	no	—
223	C₂²×C₅	C₃×#120	1	1	yes	characteristic
224	C₂²×C₅	C₃×#121	1	1	yes	characteristic
225	C₂²×C₅	C₃×#127	1	1	yes	characteristic
226	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
227	C₂²×C₅	C₃×#137	1	2	yes	—
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	8	1	no	—
232	C₄×C₅	C₃×#120	1	2	yes	—
233	C₄×C₅	C₃×#121	1	2	yes	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₃×C₅×D₄	1	4	yes	—
243	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
248	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
249	Q₁₆	—	2	16	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₃×C₅	#2016	1	1	yes	characteristic, direct product factor, Hall
257	C₂²×C₃	C₅×#120	1	1	yes	characteristic
258	C₂²×C₃	C₅×#121	1	1	yes	characteristic
259	C₂²×C₃	C₅×#127	1	1	yes	characteristic
260	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
261	C₂²×C₃	C₅×#137	1	2	yes	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	4	4	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	—	8	1	no	—
266	C₃×C₄	C₅×#120	1	2	yes	—
267	C₃×C₄	C₅×#121	1	2	yes	—
268	C₂×C₅	C₃×#392	1	1	yes	characteristic
269	C₂×C₅	C₃×#451	1	2	yes	—
270	C₂×C₅	C₃×#462	1	2	yes	—
271	C₂×C₅	C₃×#494	1	1	yes	characteristic
272	C₂×C₅	C₃×#497	1	1	yes	characteristic
273	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	Q₈	—	2	4	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	2	4	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	complemented
291	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
293	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
294	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₂×C₃	C₅×#392	1	1	yes	characteristic
297	C₂×C₃	C₅×#451	1	2	yes	—
298	C₂×C₃	C₅×#462	1	2	yes	—
299	C₂×C₃	C₅×#494	1	1	yes	characteristic
300	C₂×C₃	C₅×#497	1	1	yes	characteristic
301	C₅	C₃×#2016	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	C₃×C₅×#120	1	1	yes	characteristic
303	C₂²	C₃×C₅×#121	1	1	yes	characteristic
304	C₂²	C₃×C₅×#127	1	1	yes	characteristic
305	C₂²	C₂×C₃×C₅×Q₁₆	1	2	yes	—
306	C₂²	C₃×C₅×#137	1	2	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	4	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	1	no	—
311	C₄	C₃×C₅×#120	1	2	yes	—
312	C₄	C₃×C₅×#121	1	2	yes	—
313	C₃	C₅×#2016	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	C₃×C₅×#392	1	1	yes	characteristic
315	C₂	C₃×C₅×#451	1	2	yes	—
316	C₂	C₃×C₅×#462	1	2	yes	—
317	C₂	C₃×C₅×#494	1	1	yes	characteristic
318	C₂	C₃×C₅×#497	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2016	C₃	direct (N × H)
19. C₃×#2016	C₅	direct (N × H)
64. #2016	C₃×C₅	direct (N × H)
82. C₂×C₃×C₄×C₅	Q₁₆	semidirect
177. C₂×C₄×C₅	C₃×Q₁₆	semidirect
217. C₂×C₃×C₄	C₅×Q₁₆	semidirect
256. C₃×C₅	#2016	direct (N × H)
290. C₂×C₄	C₃×C₅×Q₁₆	semidirect
301. C₅	C₃×#2016	direct (N × H)
313. C₃	C₅×#2016	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×Q₁₆	2
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×C₃×C₅×#103	2
C₃×C₅×#391	2
C₃×C₅×#336	2
C₅×#2016	3
C₃×#2016	5

Subgroups — C3×C5×#2016 #14275050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2016 #14275050