Subgroups — C₃×C₅×#3142 #14276176

Order 1920.

784 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	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#491	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#490	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#488	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#417	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3142	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×C₅×#125	—	2	2	no	—
14	C₃×C₅×#125	C₂²	1	2	yes	—
15	C₃×C₅×#125	C₂²	1	2	yes	—
16	C₃×C₅×#124	C₂²	1	2	yes	—
17	C₃×C₅×#121	—	2	2	no	—
18	C₃×C₄×C₅×D₄	—	2	2	no	—
19	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₅×#103	C₂²	1	8	yes	—
23	C₃×C₅×#102	C₂²	1	8	yes	—
24	C₃×C₅×#97	C₂²	1	2	yes	—
25	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
26	C₃×#3142	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#532	C₂×C₃	1	1	yes	characteristic
28	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#491	C₂×C₃	1	2	yes	complemented
30	C₅×#490	C₂×C₃	1	2	yes	complemented
31	C₅×#488	C₂×C₃	1	4	yes	complemented
32	C₅×#417	C₂×C₃	1	4	yes	complemented
33	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₅×#41	—	2	8	no	—
35	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
36	C₂×C₃×C₅×D₄	—	2	2	no	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄×C₅	—	2	2	no	—
40	C₂²×C₃×C₄×C₅	—	2	2	no	—
41	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×M₁₆	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	—	2	4	no	—
47	C₃×C₅×C₄⋊C₄	—	2	2	no	—
48	C₃×C₅×C₄⋊C₄	—	2	2	no	—
49	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
51	C₃×C₄²×C₅	D₄	1	2	yes	—
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₃×#532	C₂×C₅	1	1	yes	characteristic
55	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#491	C₂×C₅	1	2	yes	complemented
57	C₃×#490	C₂×C₅	1	2	yes	complemented
58	C₃×#488	C₂×C₅	1	4	yes	complemented
59	C₃×#417	C₂×C₅	1	4	yes	complemented
60	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
63	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
64	C₅×#127	C₂²×C₃	1	1	yes	characteristic
65	C₅×#125	—	2	2	no	—
66	C₅×#125	C₂²×C₃	1	2	yes	—
67	C₅×#125	C₂²×C₃	1	2	yes	—
68	C₅×#124	C₂²×C₃	1	2	yes	—
69	C₅×#121	—	2	2	no	—
70	C₄×C₅×D₄	—	2	2	no	—
71	C₂×C₅×C₄⋊C₄	—	2	2	no	—
72	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
74	C₅×#103	C₂²×C₃	1	8	yes	—
75	C₅×#102	C₂²×C₃	1	8	yes	—
76	C₅×#97	C₂²×C₃	1	2	yes	—
77	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
78	#3142	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
79	C₂³×C₃×C₅	—	2	2	no	—
80	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×Q₈	—	2	4	no	—
82	C₃×C₅×D₄	—	2	4	no	—
83	C₃×C₅×D₄	—	4	4	no	—
84	C₃×C₅×D₄	—	4	4	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	—	4	2	no	—
88	C₂×C₃×C₄×C₅	—	4	4	no	—
89	C₂×C₃×C₄×C₅	—	4	4	no	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	#41	1	4	yes	—
94	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
95	C₃×C₅×C₈	—	2	4	no	—
96	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
97	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
98	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×#127	C₂²×C₅	1	1	yes	characteristic
100	C₃×#125	—	2	2	no	—
101	C₃×#125	C₂²×C₅	1	2	yes	—
102	C₃×#125	C₂²×C₅	1	2	yes	—
103	C₃×#124	C₂²×C₅	1	2	yes	—
104	C₃×#121	—	2	2	no	—
105	C₃×C₄×D₄	—	2	2	no	—
106	C₂×C₃×C₄⋊C₄	—	2	2	no	—
107	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
108	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
109	C₃×#103	C₂²×C₅	1	8	yes	—
110	C₃×#102	C₂²×C₅	1	8	yes	—
111	C₃×#97	C₂²×C₅	1	2	yes	—
112	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
113	C₅×#41	—	2	8	no	—
114	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
115	C₂×C₅×D₄	—	2	2	no	—
116	C₂×C₅×D₄	—	2	2	no	—
117	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
118	C₂²×C₄×C₅	—	2	2	no	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
121	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
122	C₅×M₁₆	—	2	4	no	—
123	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
124	C₅×C₄⋊C₄	—	2	4	no	—
125	C₅×C₄⋊C₄	—	2	4	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
129	C₅×C₂²⋊C₄	—	2	4	no	—
130	C₄²×C₅	C₃×D₄	1	2	yes	—
131	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
132	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
133	#532	C₂×C₃×C₅	1	1	yes	characteristic
134	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic, complemented
135	#491	C₂×C₃×C₅	1	2	yes	complemented
136	#490	C₂×C₃×C₅	1	2	yes	complemented
137	#488	C₂×C₃×C₅	1	4	yes	complemented
138	#417	C₂×C₃×C₅	1	4	yes	complemented
139	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
140	C₂²×C₃×C₅	—	2	2	no	—
141	C₂²×C₃×C₅	—	4	2	no	—
142	C₂²×C₃×C₅	—	4	4	no	—
143	C₂²×C₃×C₅	#124	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	—	4	2	no	—
148	C₃×C₄×C₅	—	4	4	no	—
149	C₃×C₄×C₅	#124	1	2	yes	—
150	C₃×C₄×C₅	C₂×#41	1	2	yes	—
151	C₃×#41	—	2	8	no	—
152	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
153	C₂×C₃×D₄	—	2	2	no	—
154	C₂×C₃×D₄	—	2	2	no	—
155	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
156	C₂²×C₃×C₄	—	2	2	no	—
157	C₂²×C₃×C₄	—	2	2	no	—
158	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
159	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₃×M₁₆	—	2	4	no	—
161	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₄⋊C₄	—	2	4	no	—
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
167	C₃×C₂²⋊C₄	—	2	4	no	—
168	C₃×C₄²	C₅×D₄	1	2	yes	—
169	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
171	C₂³×C₅	—	2	2	no	—
172	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₅×Q₈	—	2	4	no	—
174	C₅×D₄	—	2	4	no	—
175	C₅×D₄	—	4	4	no	—
176	C₅×D₄	—	4	4	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	4	4	no	—
181	C₂×C₄×C₅	—	4	4	no	—
182	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×#41	1	4	yes	—
186	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
187	C₅×C₈	—	2	4	no	—
188	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
189	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
190	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
191	#127	C₂²×C₃×C₅	1	1	yes	characteristic
192	#125	—	2	2	no	—
193	#125	C₂²×C₃×C₅	1	2	yes	—
194	#125	C₂²×C₃×C₅	1	2	yes	—
195	#124	C₂²×C₃×C₅	1	2	yes	—
196	#121	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
200	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
201	#103	C₂²×C₃×C₅	1	8	yes	—
202	#102	C₂²×C₃×C₅	1	8	yes	—
203	#97	C₂²×C₃×C₅	1	2	yes	—
204	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	—	8	2	no	—
207	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
208	C₂×C₃×C₅	#575	1	2	yes	—
209	C₂³×C₃	—	2	2	no	—
210	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
211	C₃×Q₈	—	2	4	no	—
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	4	4	no	—
214	C₃×D₄	—	4	4	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	4	4	no	—
219	C₂×C₃×C₄	—	4	4	no	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₅×#41	1	4	yes	—
224	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
225	C₃×C₈	—	2	4	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	—	4	4	no	—
228	C₂²×C₅	—	4	2	no	—
229	C₂²×C₅	C₃×#124	1	2	yes	—
230	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	4	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	C₃×#124	1	2	yes	—
236	C₄×C₅	C₂×C₃×#41	1	2	yes	—
237	#41	—	2	8	no	—
238	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
245	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
246	M₁₆	—	2	4	no	—
247	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	C₃×C₅×D₄	1	2	yes	—
255	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
256	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
257	C₃×C₅	#3142	1	1	yes	characteristic, direct product factor, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	4	4	no	—
261	C₂²×C₃	C₅×#124	1	2	yes	—
262	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	4	4	no	—
267	C₃×C₄	C₅×#124	1	2	yes	—
268	C₃×C₄	C₂×C₅×#41	1	2	yes	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	8	2	no	—
271	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
272	C₂×C₅	C₃×#575	1	2	yes	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
285	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
286	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₃×C₅×#41	1	4	yes	—
288	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	8	2	no	—
292	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
293	C₂×C₃	C₅×#575	1	2	yes	—
294	C₅	C₃×#3142	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₃×C₅×#124	1	2	yes	—
299	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	4	no	—
304	C₄	C₃×C₅×#124	1	2	yes	—
305	C₄	C₂×C₃×C₅×#41	1	2	yes	—
306	C₃	C₅×#3142	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	8	2	no	—
309	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
310	C₂	C₃×C₅×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₃×C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ 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. C₂×C₃×C₅×#125	C₂	semidirect
3. C₃×C₅×#491	C₂	semidirect
4. C₃×C₅×#490	C₂	semidirect
5. C₃×C₅×#488	C₂	semidirect
6. C₃×C₅×#417	C₂	semidirect
7. C₃×C₄×C₅×M₁₆	C₂	semidirect
8. C₅×#3142	C₃	direct (N × H)
26. C₃×#3142	C₅	direct (N × H)
28. C₂×C₅×#125	C₂×C₃	semidirect
29. C₅×#491	C₂×C₃	semidirect
30. C₅×#490	C₂×C₃	semidirect
31. C₅×#488	C₂×C₃	semidirect
32. C₅×#417	C₂×C₃	semidirect
33. C₄×C₅×M₁₆	C₂×C₃	semidirect
55. C₂×C₃×#125	C₂×C₅	semidirect
56. C₃×#491	C₂×C₅	semidirect
57. C₃×#490	C₂×C₅	semidirect
58. C₃×#488	C₂×C₅	semidirect
59. C₃×#417	C₂×C₅	semidirect
60. C₃×C₄×M₁₆	C₂×C₅	semidirect
78. #3142	C₃×C₅	direct (N × H)
134. C₂×#125	C₂×C₃×C₅	semidirect
135. #491	C₂×C₃×C₅	semidirect
136. #490	C₂×C₃×C₅	semidirect
137. #488	C₂×C₃×C₅	semidirect
138. #417	C₂×C₃×C₅	semidirect
139. C₄×M₁₆	C₂×C₃×C₅	semidirect
257. C₃×C₅	#3142	direct (N × H)
294. C₅	C₃×#3142	direct (N × H)
306. C₃	C₅×#3142	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₂×C₃×C₅×#125	2
C₃×C₅×#491	2
C₃×C₅×#490	2
C₃×C₅×#488	2
C₃×C₅×#417	2
C₃×C₄×C₅×M₁₆	2
C₅×#3142	3
C₃×#3142	5

Subgroups — C3×C5×#3142 #14276176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3142 #14276176