Subgroups — C₃×C₅×#1971 #14275005

Order 1920.

608 conjugacy classes of subgroups.

The subgroup lattice has 300 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	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#103	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#391	C₂	1	2	yes	maximal
6	C₅×#1971	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
9	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
10	C₂×C₃×C₅×Q₁₆	C₄	1	4	yes	complemented
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#106	—	2	2	no	—
16	C₃×C₅×#103	—	2	4	no	—
17	C₃×C₅×#97	—	2	2	no	—
18	C₃×C₅×#95	—	2	4	no	—
19	C₃×#1971	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×#103	C₂×C₃	1	2	yes	—
23	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
24	C₅×#391	C₂×C₃	1	2	yes	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	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	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×Q₁₆	—	2	8	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₄²×C₅	—	2	1	no	—
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₂²×C₃×Q₁₆	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
48	C₂×C₃×#103	C₂×C₅	1	2	yes	—
49	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
50	C₃×#391	C₂×C₅	1	2	yes	—
51	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
52	C₂×C₅×Q₁₆	—	2	2	no	—
53	C₂×C₅×Q₁₆	—	2	2	no	—
54	C₂×C₅×Q₁₆	C₃×C₄	1	4	yes	complemented
55	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×#106	—	2	2	no	—
60	C₅×#103	—	2	4	no	—
61	C₅×#97	—	2	2	no	—
62	C₅×#95	—	2	4	no	—
63	#1971	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	4	no	—
66	C₃×C₅×Q₈	—	2	4	no	—
67	C₃×C₅×Q₈	—	4	2	no	—
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
85	C₂×C₃×Q₁₆	—	2	2	no	—
86	C₂×C₃×Q₁₆	—	2	2	no	—
87	C₂×C₃×Q₁₆	C₄×C₅	1	4	yes	complemented
88	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₃×#106	—	2	2	no	—
93	C₃×#103	—	2	4	no	—
94	C₃×#97	—	2	2	no	—
95	C₃×#95	—	2	4	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	2	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
105	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₅×Q₁₆	—	2	8	no	—
107	C₂×C₅×C₈	—	2	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
118	C₂×#106	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂×#103	C₂×C₃×C₅	1	2	yes	—
120	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
121	#391	C₂×C₃×C₅	1	2	yes	—
122	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₃×C₅	#121	1	1	yes	characteristic
124	C₂²×C₃×C₅	#127	1	1	yes	characteristic
125	C₂²×C₃×C₅	#137	1	4	yes	—
126	C₃×C₄×C₅	—	4	4	no	—
127	C₃×C₄×C₅	—	4	2	no	—
128	C₃×C₄×C₅	—	8	2	no	—
129	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₃×C₄×C₅	#121	1	1	yes	characteristic
132	C₃×C₄×C₅	#124	1	1	yes	characteristic
133	C₂×C₃×Q₈	—	2	2	no	—
134	C₂×C₃×Q₈	—	2	2	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	2	2	no	—
137	C₂×C₃×Q₈	—	2	2	no	—
138	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
139	C₂²×C₃×C₄	—	2	2	no	—
140	C₂²×C₃×C₄	—	2	2	no	—
141	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₃×Q₁₆	—	2	8	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
147	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
148	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
149	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₅×Q₈	—	2	4	no	—
156	C₅×Q₈	—	2	4	no	—
157	C₅×Q₈	—	4	2	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	4	no	—
166	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
171	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	C₃×C₂²⋊C₄	1	4	yes	—
174	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	C₃×C₄×C₅	1	4	yes	complemented
178	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
181	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
182	#106	—	2	2	no	—
183	#103	—	2	4	no	—
184	#97	—	2	2	no	—
185	#95	—	2	4	no	—
186	C₂×C₃×C₅	#390	1	1	yes	characteristic
187	C₂×C₃×C₅	#441	1	2	yes	—
188	C₂×C₃×C₅	#470	1	2	yes	—
189	C₂×C₃×C₅	#497	1	2	yes	—
190	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
191	C₃×Q₈	—	2	4	no	—
192	C₃×Q₈	—	2	4	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₃×Q₈	—	4	2	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	—	4	2	no	—
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	—	4	4	no	—
202	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
207	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
208	C₃×C₈	—	2	2	no	—
209	C₃×C₈	C₅×C₂²⋊C₄	1	4	yes	—
210	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₅	C₃×#121	1	1	yes	characteristic
212	C₂²×C₅	C₃×#127	1	1	yes	characteristic
213	C₂²×C₅	C₃×#137	1	4	yes	—
214	C₄×C₅	—	4	4	no	—
215	C₄×C₅	—	4	2	no	—
216	C₄×C₅	—	8	2	no	—
217	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
219	C₄×C₅	C₃×#121	1	1	yes	characteristic
220	C₄×C₅	C₃×#124	1	1	yes	characteristic
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
230	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
231	Q₁₆	—	2	8	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
235	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
236	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
237	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₃×C₅	#1971	1	1	yes	characteristic, direct product factor, Hall
243	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
244	C₂²×C₃	C₅×#121	1	1	yes	characteristic
245	C₂²×C₃	C₅×#127	1	1	yes	characteristic
246	C₂²×C₃	C₅×#137	1	4	yes	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	—	8	2	no	—
250	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
252	C₃×C₄	C₅×#121	1	1	yes	characteristic
253	C₃×C₄	C₅×#124	1	1	yes	characteristic
254	C₂×C₅	C₃×#390	1	1	yes	characteristic
255	C₂×C₅	C₃×#441	1	2	yes	—
256	C₂×C₅	C₃×#470	1	2	yes	—
257	C₂×C₅	C₃×#497	1	2	yes	—
258	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
275	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	C₃×C₅×C₂²⋊C₄	1	4	yes	—
278	C₂×C₃	C₅×#390	1	1	yes	characteristic
279	C₂×C₃	C₅×#441	1	2	yes	—
280	C₂×C₃	C₅×#470	1	2	yes	—
281	C₂×C₃	C₅×#497	1	2	yes	—
282	C₅	C₃×#1971	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
284	C₂²	C₃×C₅×#121	1	1	yes	characteristic
285	C₂²	C₃×C₅×#127	1	1	yes	characteristic
286	C₂²	C₃×C₅×#137	1	4	yes	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
291	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
292	C₄	C₃×C₅×#121	1	1	yes	characteristic
293	C₄	C₃×C₅×#124	1	1	yes	characteristic
294	C₃	C₅×#1971	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	C₃×C₅×#390	1	1	yes	characteristic
296	C₂	C₃×C₅×#441	1	2	yes	—
297	C₂	C₃×C₅×#470	1	2	yes	—
298	C₂	C₃×C₅×#497	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×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×#1971	C₃	direct (N × H)
10. C₂×C₃×C₅×Q₁₆	C₄	semidirect
19. C₃×#1971	C₅	direct (N × H)
54. C₂×C₅×Q₁₆	C₃×C₄	semidirect
63. #1971	C₃×C₅	direct (N × H)
87. C₂×C₃×Q₁₆	C₄×C₅	semidirect
177. C₂×Q₁₆	C₃×C₄×C₅	semidirect
242. C₃×C₅	#1971	direct (N × H)
282. C₅	C₃×#1971	direct (N × H)
294. C₃	C₅×#1971	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×#106	2
C₂×C₃×C₅×#103	2
C₂×C₃×C₅×#97	2
C₃×C₅×#391	2
C₅×#1971	3
C₃×#1971	5

Subgroups — C3×C5×#1971 #14275005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1971 #14275005