Subgroups — C₃×C₅×#1967 #14275001

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#101	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₅×#100	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1967	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₃×C₅×#124	—	2	1	no	—
12	C₃×C₅×#124	—	2	1	no	—
13	C₃×C₅×#124	—	2	2	no	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	—	2	2	no	—
17	C₃×C₅×#101	—	2	4	no	—
18	C₃×C₅×#100	—	2	4	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×C₄×C₅×C₈	—	2	1	no	—
21	C₃×#1967	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
23	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₅×#101	C₂×C₃	1	2	yes	complemented
25	C₂×C₅×#100	C₂×C₃	1	2	yes	—
26	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	—	2	1	no	—
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	2	no	—
35	C₂×C₃×C₅×D₄	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₃×C₅×M₁₆	—	2	2	no	—
41	C₃×C₅×M₁₆	—	2	2	no	—
42	C₃×C₅×M₁₆	D₄	1	4	yes	—
43	C₂×C₃×C₅×C₈	—	2	2	no	—
44	C₂×C₃×C₅×C₈	—	2	2	no	—
45	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
46	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
47	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
51	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×C₃×#101	C₂×C₅	1	2	yes	complemented
53	C₂×C₃×#100	C₂×C₅	1	2	yes	—
54	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	C₅×#124	—	2	1	no	—
60	C₅×#124	—	2	1	no	—
61	C₅×#124	—	2	2	no	—
62	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	4	yes	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×#101	—	2	4	no	—
66	C₅×#100	—	2	4	no	—
67	C₅×#97	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	#1967	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	2	4	no	—
72	C₂³×C₃×C₅	—	4	4	no	—
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₅×Q₈	—	4	2	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×D₄	—	4	2	no	—
78	C₃×C₅×D₄	—	4	1	no	—
79	C₃×C₅×D₄	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	4	2	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	2	yes	—
89	C₃×C₅×C₈	—	2	4	no	—
90	C₃×C₅×C₈	—	4	2	no	—
91	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
94	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
95	C₃×#124	—	2	1	no	—
96	C₃×#124	—	2	1	no	—
97	C₃×#124	—	2	2	no	—
98	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	4	yes	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄⋊C₈	—	2	2	no	—
101	C₃×#101	—	2	4	no	—
102	C₃×#100	—	2	4	no	—
103	C₃×#97	—	2	1	no	—
104	C₃×C₄×C₈	—	2	1	no	—
105	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
106	C₂×C₅×Q₈	—	2	1	no	—
107	C₂×C₅×Q₈	—	2	1	no	—
108	C₂×C₅×Q₈	—	2	2	no	—
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	2	1	no	—
112	C₂×C₅×D₄	—	2	2	no	—
113	C₂×C₅×D₄	—	2	2	no	—
114	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
115	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₅×M₁₆	—	2	2	no	—
118	C₅×M₁₆	—	2	2	no	—
119	C₅×M₁₆	—	2	2	no	—
120	C₅×M₁₆	C₃×D₄	1	4	yes	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	2	4	no	—
124	C₅×C₂²⋊C₄	—	2	4	no	—
125	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
126	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
127	C₄²×C₅	C₃×D₄	1	2	yes	—
128	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic
129	#423	C₂×C₃×C₅	1	1	yes	characteristic, complemented
130	C₂×#101	C₂×C₃×C₅	1	2	yes	complemented
131	C₂×#100	C₂×C₃×C₅	1	2	yes	—
132	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	2	no	—
135	C₂²×C₃×C₅	—	4	4	no	—
136	C₂²×C₃×C₅	—	8	4	no	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	4	2	no	—
142	C₃×C₄×C₅	#121	1	2	yes	—
143	C₃×C₄×C₅	#124	1	1	yes	characteristic
144	C₃×C₄×C₅	#127	1	1	yes	characteristic
145	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
146	C₂×C₃×Q₈	—	2	2	no	—
147	C₂×C₃×Q₈	—	2	1	no	—
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	—	2	1	no	—
153	C₂×C₃×D₄	—	2	2	no	—
154	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
155	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	—	2	2	no	—
159	C₃×M₁₆	—	2	2	no	—
160	C₃×M₁₆	C₅×D₄	1	4	yes	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₂×C₃×C₈	—	2	2	no	—
163	C₃×C₂²⋊C₄	—	2	4	no	—
164	C₃×C₂²⋊C₄	—	2	4	no	—
165	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
166	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
167	C₃×C₄²	C₅×D₄	1	2	yes	—
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	—	2	4	no	—
170	C₂³×C₅	—	4	4	no	—
171	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₅×Q₈	—	4	2	no	—
173	C₅×Q₈	—	4	1	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	1	no	—
177	C₅×D₄	—	4	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	4	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
185	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₃×#41	1	2	yes	—
187	C₅×C₈	—	2	4	no	—
188	C₅×C₈	—	4	2	no	—
189	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
190	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
191	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
192	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
193	#124	—	2	1	no	—
194	#124	—	2	1	no	—
195	#124	—	2	2	no	—
196	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	4	yes	—
197	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	#101	—	2	4	no	—
200	#100	—	2	4	no	—
201	#97	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₂×C₃×C₅	—	2	2	no	—
204	C₂×C₃×C₅	—	8	2	no	—
205	C₂×C₃×C₅	#390	1	1	yes	characteristic
206	C₂×C₃×C₅	#413	1	2	yes	—
207	C₂³×C₃	—	2	2	no	—
208	C₂³×C₃	—	2	4	no	—
209	C₂³×C₃	—	4	4	no	—
210	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₃×Q₈	—	4	2	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×Q₈	—	4	1	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	1	no	—
216	C₃×D₄	—	4	1	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
221	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₅×#41	1	2	yes	—
226	C₃×C₈	—	2	4	no	—
227	C₃×C₈	—	4	2	no	—
228	C₂²×C₅	—	2	2	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	—	4	4	no	—
231	C₂²×C₅	—	8	4	no	—
232	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
233	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	C₃×#121	1	2	yes	—
238	C₄×C₅	C₃×#124	1	1	yes	characteristic
239	C₄×C₅	C₃×#127	1	1	yes	characteristic
240	C₂⁴	C₂×C₃×C₄×C₅	1	2	yes	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
250	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
251	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	C₃×C₅×D₄	1	4	yes	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
261	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
262	C₄²	C₃×C₅×D₄	1	2	yes	—
263	C₃×C₅	#1967	1	1	yes	characteristic, direct product factor, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	4	4	no	—
267	C₂²×C₃	—	8	4	no	—
268	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₅×#121	1	2	yes	—
274	C₃×C₄	C₅×#124	1	1	yes	characteristic
275	C₃×C₄	C₅×#127	1	1	yes	characteristic
276	C₂×C₅	—	2	2	no	—
277	C₂×C₅	—	8	2	no	—
278	C₂×C₅	C₃×#390	1	1	yes	characteristic
279	C₂×C₅	C₃×#413	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	4	no	—
282	C₂³	—	4	4	no	—
283	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	D₄	—	4	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
294	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
295	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
297	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₃×C₅×#41	1	2	yes	—
299	C₈	—	2	4	no	—
300	C₈	—	4	2	no	—
301	C₂×C₃	—	2	2	no	—
302	C₂×C₃	—	8	2	no	—
303	C₂×C₃	C₅×#390	1	1	yes	characteristic
304	C₂×C₃	C₅×#413	1	2	yes	—
305	C₅	C₃×#1967	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₂²	C₃×C₄×C₅×D₄	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	2	no	—
315	C₄	C₃×C₅×#121	1	2	yes	—
316	C₄	C₃×C₅×#124	1	1	yes	characteristic
317	C₄	C₃×C₅×#127	1	1	yes	characteristic
318	C₃	C₅×#1967	1	1	yes	characteristic, direct product factor, Sylow
319	C₂	—	2	2	no	—
320	C₂	—	8	2	no	—
321	C₂	C₃×C₅×#390	1	1	yes	characteristic
322	C₂	C₃×C₅×#413	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₂×C₃×C₅×#101	C₂	semidirect
5. C₃×C₄×C₅×M₁₆	C₂	semidirect
6. C₅×#1967	C₃	direct (N × H)
21. C₃×#1967	C₅	direct (N × H)
23. C₅×#423	C₂×C₃	semidirect
24. C₂×C₅×#101	C₂×C₃	semidirect
26. C₄×C₅×M₁₆	C₂×C₃	semidirect
51. C₃×#423	C₂×C₅	semidirect
52. C₂×C₃×#101	C₂×C₅	semidirect
54. C₃×C₄×M₁₆	C₂×C₅	semidirect
69. #1967	C₃×C₅	direct (N × H)
129. #423	C₂×C₃×C₅	semidirect
130. C₂×#101	C₂×C₃×C₅	semidirect
132. C₄×M₁₆	C₂×C₃×C₅	semidirect
263. C₃×C₅	#1967	direct (N × H)
305. C₅	C₃×#1967	direct (N × H)
318. C₃	C₅×#1967	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#124	2
C₃×C₅×#423	2
C₂×C₃×C₅×#101	2
C₂×C₃×C₅×#100	2
C₃×C₄×C₅×M₁₆	2
C₅×#1967	3
C₃×#1967	5

Subgroups — C3×C5×#1967 #14275001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1967 #14275001