Subgroups — C₃×C₅×#1979 #14275013

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#108	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#419	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#101	C₂	1	2	yes	maximal
5	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
6	C₅×#1979	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	1	no	—
9	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
10	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
11	C₂×C₃×C₅×Q₁₆	C₄	1	4	yes	complemented
12	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
13	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#117	C₂²	1	1	yes	characteristic
16	C₃×C₅×#108	—	2	2	no	—
17	C₃×C₅×#103	—	2	4	no	—
18	C₃×C₅×#101	—	2	4	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×C₄×C₅×C₈	—	2	1	no	—
21	C₃×#1979	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
24	C₅×#419	C₂×C₃	1	2	yes	—
25	C₂×C₅×#101	C₂×C₃	1	2	yes	—
26	C₅×#405	C₂×C₃	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×Q₁₆	—	2	4	no	—
36	C₃×C₅×Q₁₆	—	2	4	no	—
37	C₃×C₅×M₁₆	—	2	1	no	—
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₃×C₅×M₁₆	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	2	no	—
43	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
47	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
48	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
49	C₃×C₄²×C₅	—	2	1	no	—
50	C₂²×C₃×Q₁₆	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
52	C₃×#419	C₂×C₅	1	2	yes	—
53	C₂×C₃×#101	C₂×C₅	1	2	yes	—
54	C₃×#405	C₂×C₅	1	1	yes	characteristic
55	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₂×C₅×Q₁₆	—	2	1	no	—
57	C₂×C₅×Q₁₆	—	2	1	no	—
58	C₂×C₅×Q₁₆	—	2	2	no	—
59	C₂×C₅×Q₁₆	C₃×C₄	1	4	yes	complemented
60	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
61	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₅×#117	C₂²×C₃	1	1	yes	characteristic
64	C₅×#108	—	2	2	no	—
65	C₅×#103	—	2	4	no	—
66	C₅×#101	—	2	4	no	—
67	C₅×#97	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	#1979	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	#41	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	2	4	no	—
73	C₃×C₅×Q₈	—	4	2	no	—
74	C₃×C₅×Q₈	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	2	no	—
86	C₃×C₅×C₈	—	4	2	no	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
90	C₂×C₃×Q₁₆	—	2	1	no	—
91	C₂×C₃×Q₁₆	—	2	1	no	—
92	C₂×C₃×Q₁₆	—	2	2	no	—
93	C₂×C₃×Q₁₆	C₄×C₅	1	4	yes	complemented
94	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
95	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₃×#117	C₂²×C₅	1	1	yes	characteristic
98	C₃×#108	—	2	2	no	—
99	C₃×#103	—	2	4	no	—
100	C₃×#101	—	2	4	no	—
101	C₃×#97	—	2	1	no	—
102	C₃×C₄×C₈	—	2	1	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	C₂×C₃×C₄	1	4	yes	—
109	C₂²×C₄×C₅	—	2	2	no	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₅×Q₁₆	—	2	4	no	—
112	C₅×Q₁₆	—	2	4	no	—
113	C₅×M₁₆	—	2	1	no	—
114	C₅×M₁₆	—	2	2	no	—
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	C₃×D₄	1	2	yes	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
120	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
123	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
124	C₅×C₂²⋊C₄	—	2	1	no	—
125	C₄²×C₅	—	2	1	no	—
126	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
127	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
128	#419	C₂×C₃×C₅	1	2	yes	—
129	C₂×#101	C₂×C₃×C₅	1	2	yes	—
130	#405	C₂×C₃×C₅	1	1	yes	characteristic
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#121	1	1	yes	characteristic
134	C₂²×C₃×C₅	#124	1	1	yes	characteristic
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	#121	1	1	yes	characteristic
140	C₃×C₄×C₅	#127	1	1	yes	characteristic
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	C₂×C₄×C₅	1	4	yes	—
147	C₂²×C₃×C₄	—	2	2	no	—
148	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₃×Q₁₆	—	2	4	no	—
150	C₃×Q₁₆	—	2	4	no	—
151	C₃×M₁₆	—	2	1	no	—
152	C₃×M₁₆	—	2	2	no	—
153	C₃×M₁₆	—	2	2	no	—
154	C₃×M₁₆	C₅×D₄	1	2	yes	—
155	C₂×C₃×C₈	—	2	1	no	—
156	C₂×C₃×C₈	—	2	2	no	—
157	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
158	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
161	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
162	C₃×C₂²⋊C₄	—	2	1	no	—
163	C₃×C₄²	—	2	1	no	—
164	C₂³×C₅	C₃×#41	1	1	yes	characteristic
165	C₅×Q₈	—	2	4	no	—
166	C₅×Q₈	—	2	4	no	—
167	C₅×Q₈	—	4	2	no	—
168	C₅×Q₈	—	4	2	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	—	4	2	no	—
181	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
182	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
183	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	C₃×C₄×C₅	1	4	yes	complemented
188	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
189	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
190	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
191	#117	C₂²×C₃×C₅	1	1	yes	characteristic
192	#108	—	2	2	no	—
193	#103	—	2	4	no	—
194	#101	—	2	4	no	—
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₂×C₃×C₅	—	2	1	no	—
198	C₂×C₃×C₅	—	2	1	no	—
199	C₂×C₃×C₅	#390	1	1	yes	characteristic
200	C₂×C₃×C₅	#473	1	2	yes	—
201	C₂³×C₃	C₅×#41	1	1	yes	characteristic
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	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
216	C₃×C₈	—	2	2	no	—
217	C₃×C₈	—	4	2	no	—
218	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
219	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₅	C₃×#121	1	1	yes	characteristic
223	C₂²×C₅	C₃×#124	1	1	yes	characteristic
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
228	C₄×C₅	C₃×#121	1	1	yes	characteristic
229	C₄×C₅	C₃×#127	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
238	Q₁₆	—	2	4	no	—
239	Q₁₆	—	2	4	no	—
240	M₁₆	—	2	1	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	C₃×C₅×D₄	1	2	yes	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
247	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
248	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
249	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
250	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₃×C₅	#1979	1	1	yes	characteristic, direct product factor, Hall
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
256	C₂²×C₃	C₅×#121	1	1	yes	characteristic
257	C₂²×C₃	C₅×#124	1	1	yes	characteristic
258	C₃×C₄	—	4	4	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
262	C₃×C₄	C₅×#121	1	1	yes	characteristic
263	C₃×C₄	C₅×#127	1	1	yes	characteristic
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	C₃×#390	1	1	yes	characteristic
267	C₂×C₅	C₃×#473	1	2	yes	—
268	C₂³	C₃×C₅×#41	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃×C₅×D₄	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	C₃×C₅×#41	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
286	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	C₅×#390	1	1	yes	characteristic
290	C₂×C₃	C₅×#473	1	2	yes	—
291	C₅	C₃×#1979	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
294	C₂²	C₃×C₅×#121	1	1	yes	characteristic
295	C₂²	C₃×C₅×#124	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	4	4	no	—
298	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
299	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
300	C₄	C₃×C₅×#121	1	1	yes	characteristic
301	C₄	C₃×C₅×#127	1	1	yes	characteristic
302	C₃	C₅×#1979	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	C₃×C₅×#390	1	1	yes	characteristic
306	C₂	C₃×C₅×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1979	C₃	direct (N × H)
11. C₂×C₃×C₅×Q₁₆	C₄	semidirect
21. C₃×#1979	C₅	direct (N × H)
59. C₂×C₅×Q₁₆	C₃×C₄	semidirect
69. #1979	C₃×C₅	direct (N × H)
93. C₂×C₃×Q₁₆	C₄×C₅	semidirect
187. C₂×Q₁₆	C₃×C₄×C₅	semidirect
253. C₃×C₅	#1979	direct (N × H)
291. C₅	C₃×#1979	direct (N × H)
302. C₃	C₅×#1979	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×#108	2
C₃×C₅×#419	2
C₂×C₃×C₅×#101	2
C₃×C₅×#405	2
C₅×#1979	3
C₃×#1979	5

Subgroups — C3×C5×#1979 #14275013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1979 #14275013