Subgroups — C₃×C₅×#1969 #14275003

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#103	C₂	1	2	yes	maximal
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#391	C₂	1	2	yes	maximal
6	C₅×#1969	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×#107	—	2	2	no	—
16	C₃×C₅×#103	—	2	4	no	—
17	C₃×C₄×C₅×C₈	—	2	2	no	—
18	C₃×C₅×#95	—	2	4	no	—
19	C₃×#1969	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×#103	C₂×C₃	1	2	yes	—
23	C₂×C₄×C₅×C₈	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×#107	C₂×C₅	1	1	yes	characteristic
48	C₂×C₃×#103	C₂×C₅	1	2	yes	—
49	C₂×C₃×C₄×C₈	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×#107	—	2	2	no	—
60	C₅×#103	—	2	4	no	—
61	C₄×C₅×C₈	—	2	2	no	—
62	C₅×#95	—	2	4	no	—
63	#1969	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	2	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	Q₁₆	1	4	yes	complemented
77	C₂×C₃×C₄×C₅	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
83	C₃×C₅×C₈	—	2	2	no	—
84	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
86	C₂×C₃×Q₁₆	—	2	2	no	—
87	C₂×C₃×Q₁₆	—	2	2	no	—
88	C₂×C₃×Q₁₆	C₄×C₅	1	4	yes	complemented
89	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₃×#107	—	2	2	no	—
94	C₃×#103	—	2	4	no	—
95	C₃×C₄×C₈	—	2	2	no	—
96	C₃×#95	—	2	4	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	—	2	2	no	—
102	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₅×Q₁₆	—	2	8	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
113	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₄²×C₅	—	2	1	no	—
118	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
120	C₂×#103	C₂×C₃×C₅	1	2	yes	—
121	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
122	#391	C₂×C₃×C₅	1	2	yes	—
123	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	#121	1	1	yes	characteristic
125	C₂²×C₃×C₅	#127	1	1	yes	characteristic
126	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
127	C₂²×C₃×C₅	#135	1	2	yes	—
128	C₃×C₄×C₅	—	2	4	no	—
129	C₃×C₄×C₅	—	4	4	no	—
130	C₃×C₄×C₅	—	8	2	no	—
131	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₃×C₄×C₅	#121	1	1	yes	characteristic
134	C₃×C₄×C₅	#124	1	1	yes	characteristic
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	—	2	2	no	—
139	C₂×C₃×Q₈	—	2	2	no	—
140	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₃×Q₁₆	—	2	8	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
150	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
151	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₅×Q₈	—	2	4	no	—
158	C₅×Q₈	—	2	4	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	4	no	—
168	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	complemented
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
174	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
175	C₅×C₈	—	2	2	no	—
176	C₅×C₈	C₃×C₂²⋊C₄	1	4	yes	—
177	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	C₃×C₄×C₅	1	4	yes	complemented
181	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
182	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
184	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
185	#107	—	2	2	no	—
186	#103	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	#95	—	2	4	no	—
189	C₂×C₃×C₅	#390	1	1	yes	characteristic
190	C₂×C₃×C₅	C₄×Q₁₆	1	2	yes	—
191	C₂×C₃×C₅	#467	1	2	yes	—
192	C₂×C₃×C₅	#494	1	1	yes	characteristic
193	C₂×C₃×C₅	#495	1	1	yes	characteristic
194	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
195	C₃×Q₈	—	2	4	no	—
196	C₃×Q₈	—	2	4	no	—
197	C₃×Q₈	—	4	2	no	—
198	C₃×Q₈	—	4	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	4	no	—
206	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	complemented
207	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
212	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	C₅×C₂²⋊C₄	1	4	yes	—
215	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
216	C₂²×C₅	C₃×#121	1	1	yes	characteristic
217	C₂²×C₅	C₃×#127	1	1	yes	characteristic
218	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
219	C₂²×C₅	C₃×#135	1	2	yes	—
220	C₄×C₅	—	2	4	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅	C₃×#121	1	1	yes	characteristic
226	C₄×C₅	C₃×#124	1	1	yes	characteristic
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
236	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
237	Q₁₆	—	2	8	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
241	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
242	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
243	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₃×C₅	#1969	1	1	yes	characteristic, direct product factor, Hall
249	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
250	C₂²×C₃	C₅×#121	1	1	yes	characteristic
251	C₂²×C₃	C₅×#127	1	1	yes	characteristic
252	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
253	C₂²×C₃	C₅×#135	1	2	yes	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	4	4	no	—
256	C₃×C₄	—	8	2	no	—
257	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
259	C₃×C₄	C₅×#121	1	1	yes	characteristic
260	C₃×C₄	C₅×#124	1	1	yes	characteristic
261	C₂×C₅	C₃×#390	1	1	yes	characteristic
262	C₂×C₅	C₃×C₄×Q₁₆	1	2	yes	—
263	C₂×C₅	C₃×#467	1	2	yes	—
264	C₂×C₅	C₃×#494	1	1	yes	characteristic
265	C₂×C₅	C₃×#495	1	1	yes	characteristic
266	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	2	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	complemented
279	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
283	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
284	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	C₃×C₅×C₂²⋊C₄	1	4	yes	—
287	C₂×C₃	C₅×#390	1	1	yes	characteristic
288	C₂×C₃	C₄×C₅×Q₁₆	1	2	yes	—
289	C₂×C₃	C₅×#467	1	2	yes	—
290	C₂×C₃	C₅×#494	1	1	yes	characteristic
291	C₂×C₃	C₅×#495	1	1	yes	characteristic
292	C₅	C₃×#1969	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
294	C₂²	C₃×C₅×#121	1	1	yes	characteristic
295	C₂²	C₃×C₅×#127	1	1	yes	characteristic
296	C₂²	C₂×C₃×C₅×Q₁₆	1	2	yes	—
297	C₂²	C₃×C₅×#135	1	2	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	2	no	—
301	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
302	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
303	C₄	C₃×C₅×#121	1	1	yes	characteristic
304	C₄	C₃×C₅×#124	1	1	yes	characteristic
305	C₃	C₅×#1969	1	1	yes	characteristic, direct product factor, Sylow
306	C₂	C₃×C₅×#390	1	1	yes	characteristic
307	C₂	C₃×C₄×C₅×Q₁₆	1	2	yes	—
308	C₂	C₃×C₅×#467	1	2	yes	—
309	C₂	C₃×C₅×#494	1	1	yes	characteristic
310	C₂	C₃×C₅×#495	1	1	yes	characteristic

	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×#1969	C₃	direct (N × H)
10. C₂×C₃×C₅×Q₁₆	C₄	semidirect
19. C₃×#1969	C₅	direct (N × H)
54. C₂×C₅×Q₁₆	C₃×C₄	semidirect
63. #1969	C₃×C₅	direct (N × H)
76. C₂×C₃×C₄×C₅	Q₁₆	semidirect
88. C₂×C₃×Q₁₆	C₄×C₅	semidirect
168. C₂×C₄×C₅	C₃×Q₁₆	semidirect
180. C₂×Q₁₆	C₃×C₄×C₅	semidirect
206. C₂×C₃×C₄	C₅×Q₁₆	semidirect
248. C₃×C₅	#1969	direct (N × H)
278. C₂×C₄	C₃×C₅×Q₁₆	semidirect
292. C₅	C₃×#1969	direct (N × H)
305. C₃	C₅×#1969	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×#107	2
C₂×C₃×C₅×#103	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#391	2
C₅×#1969	3
C₃×#1969	5

Subgroups — C3×C5×#1969 #14275003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1969 #14275003