Subgroups — C₃×C₅×#3163 #14276197

Order 1920.

680 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#544	C₂	1	2	yes	maximal
2	C₃×C₅×#501	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#499	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#498	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#428	C₂	1	2	yes	maximal
6	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
8	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
9	C₅×#3163	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
14	C₃×C₅×#125	—	2	2	no	—
15	C₃×C₅×#125	C₂²	1	4	yes	—
16	C₃×C₅×C₂²⋊Q₈	—	2	4	no	—
17	C₃×C₅×#117	C₂²	1	2	yes	—
18	C₃×C₅×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×#107	C₂²	1	4	yes	—
21	C₃×C₅×#107	C₂²	1	2	yes	—
22	C₃×C₅×#106	C₂²	1	1	yes	characteristic
23	C₃×C₅×#106	C₂²	1	2	yes	—
24	C₃×C₅×#106	C₂²	1	4	yes	—
25	C₃×C₅×#106	C₂²	1	1	yes	characteristic
26	C₃×C₅×#106	C₂²	1	2	yes	—
27	C₃×C₅×#97	C₂²	1	2	yes	—
28	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
29	C₃×#3163	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₅×#544	C₂×C₃	1	2	yes	—
31	C₅×#501	C₂×C₃	1	4	yes	complemented
32	C₅×#499	C₂×C₃	1	2	yes	complemented
33	C₅×#498	C₂×C₃	1	2	yes	complemented
34	C₅×#428	C₂×C₃	1	2	yes	—
35	C₅×#427	C₂×C₃	1	1	yes	characteristic
36	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic
37	C₅×#405	C₂×C₃	1	1	yes	characteristic
38	C₂×C₃×C₅×Q₈	—	2	4	no	—
39	C₂²×C₃×C₄×C₅	—	2	2	no	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₃×C₅×M₁₆	Q₈	1	4	yes	complemented
42	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	2	2	no	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	4	no	—
49	C₃×C₅×C₄⋊C₄	—	2	4	no	—
50	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
51	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
52	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
53	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
54	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
55	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
56	C₃×C₄²×C₅	—	2	2	no	—
57	C₃×C₄²×C₅	C₂³	1	2	yes	—
58	C₃×#544	C₂×C₅	1	2	yes	—
59	C₃×#501	C₂×C₅	1	4	yes	complemented
60	C₃×#499	C₂×C₅	1	2	yes	complemented
61	C₃×#498	C₂×C₅	1	2	yes	complemented
62	C₃×#428	C₂×C₅	1	2	yes	—
63	C₃×#427	C₂×C₅	1	1	yes	characteristic
64	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
65	C₃×#405	C₂×C₅	1	1	yes	characteristic
66	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₅×C₄⋊Q₈	—	2	2	no	—
69	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
70	C₅×#125	—	2	2	no	—
71	C₅×#125	C₂²×C₃	1	4	yes	—
72	C₅×C₂²⋊Q₈	—	2	4	no	—
73	C₅×#117	C₂²×C₃	1	2	yes	—
74	C₅×#117	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₅×#107	C₂²×C₃	1	2	yes	—
77	C₅×#107	C₂²×C₃	1	4	yes	—
78	C₅×#106	C₂²×C₃	1	2	yes	—
79	C₅×#106	C₂²×C₃	1	2	yes	—
80	C₅×#106	C₂²×C₃	1	1	yes	characteristic
81	C₅×#106	C₂²×C₃	1	1	yes	characteristic
82	C₅×#106	C₂²×C₃	1	4	yes	—
83	C₅×#97	C₂²×C₃	1	2	yes	—
84	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
85	#3163	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×Q₈	—	8	4	no	—
88	C₂×C₃×C₄×C₅	—	2	2	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	—	8	2	no	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
98	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
99	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
100	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
101	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄⋊Q₈	—	2	2	no	—
103	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
104	C₃×#125	—	2	2	no	—
105	C₃×#125	C₂²×C₅	1	4	yes	—
106	C₃×C₂²⋊Q₈	—	2	4	no	—
107	C₃×#117	C₂²×C₅	1	2	yes	—
108	C₃×#117	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
110	C₃×#107	C₂²×C₅	1	2	yes	—
111	C₃×#107	C₂²×C₅	1	4	yes	—
112	C₃×#106	C₂²×C₅	1	2	yes	—
113	C₃×#106	C₂²×C₅	1	2	yes	—
114	C₃×#106	C₂²×C₅	1	1	yes	characteristic
115	C₃×#106	C₂²×C₅	1	1	yes	characteristic
116	C₃×#106	C₂²×C₅	1	4	yes	—
117	C₃×#97	C₂²×C₅	1	2	yes	—
118	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
119	C₂×C₅×Q₈	—	2	4	no	—
120	C₂²×C₄×C₅	—	2	2	no	—
121	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
122	C₅×M₁₆	C₃×Q₈	1	4	yes	complemented
123	C₂×C₅×C₈	C₃×Q₈	1	4	yes	complemented
124	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	2	4	no	—
129	C₅×C₄⋊C₄	—	2	4	no	—
130	C₅×C₄⋊C₄	—	2	4	no	—
131	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
132	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
133	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
134	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₅×C₂²⋊C₄	—	2	2	no	—
136	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
137	C₄²×C₅	—	2	2	no	—
138	C₄²×C₅	C₂³×C₃	1	2	yes	—
139	#544	C₂×C₃×C₅	1	2	yes	—
140	#501	C₂×C₃×C₅	1	4	yes	complemented
141	#499	C₂×C₃×C₅	1	2	yes	complemented
142	#498	C₂×C₃×C₅	1	2	yes	complemented
143	#428	C₂×C₃×C₅	1	2	yes	—
144	#427	C₂×C₃×C₅	1	1	yes	characteristic
145	C₂×#106	C₂×C₃×C₅	1	1	yes	characteristic
146	#405	C₂×C₃×C₅	1	1	yes	characteristic
147	C₂²×C₃×C₅	—	2	2	no	—
148	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
149	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₃×C₄×C₅	—	2	4	no	—
151	C₃×C₄×C₅	—	4	2	no	—
152	C₃×C₄×C₅	—	4	4	no	—
153	C₃×C₄×C₅	—	4	2	no	—
154	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
155	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₃×Q₈	—	2	4	no	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₃×M₁₆	C₅×Q₈	1	4	yes	complemented
161	C₂×C₃×C₈	C₅×Q₈	1	4	yes	complemented
162	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
164	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	—	2	4	no	—
167	C₃×C₄⋊C₄	—	2	4	no	—
168	C₃×C₄⋊C₄	—	2	4	no	—
169	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
170	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
171	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
172	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
173	C₃×C₂²⋊C₄	—	2	2	no	—
174	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
175	C₃×C₄²	—	2	2	no	—
176	C₃×C₄²	C₂³×C₅	1	2	yes	—
177	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₅×Q₈	—	8	4	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	4	no	—
182	C₂×C₄×C₅	—	8	2	no	—
183	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
184	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
187	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
188	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
189	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
190	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
191	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
192	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
195	#125	—	2	2	no	—
196	#125	C₂²×C₃×C₅	1	4	yes	—
197	C₂²⋊Q₈	—	2	4	no	—
198	#117	C₂²×C₃×C₅	1	2	yes	—
199	#117	C₂²×C₃×C₅	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
201	#107	C₂²×C₃×C₅	1	2	yes	—
202	#107	C₂²×C₃×C₅	1	4	yes	—
203	#106	C₂²×C₃×C₅	1	2	yes	—
204	#106	C₂²×C₃×C₅	1	2	yes	—
205	#106	C₂²×C₃×C₅	1	1	yes	characteristic
206	#106	C₂²×C₃×C₅	1	1	yes	characteristic
207	#106	C₂²×C₃×C₅	1	4	yes	—
208	#97	C₂²×C₃×C₅	1	2	yes	—
209	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
212	C₂×C₃×C₅	#577	1	2	yes	—
213	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
214	C₃×Q₈	—	8	4	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	8	2	no	—
219	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
222	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
223	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
224	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
225	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
226	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
229	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	—	4	4	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
235	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
236	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
237	C₂×Q₈	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
240	M₁₆	C₃×C₅×Q₈	1	4	yes	complemented
241	C₂×C₈	C₃×C₅×Q₈	1	4	yes	complemented
242	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
243	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
244	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
250	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
251	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
252	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂³×C₃×C₅	1	2	yes	—
257	C₃×C₅	#3163	1	1	yes	characteristic, direct product factor, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
260	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	—	4	4	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
266	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
267	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
270	C₂×C₅	C₃×#577	1	2	yes	—
271	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
272	Q₈	—	8	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
278	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
280	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
281	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
282	C₈	C₂×C₃×C₅×Q₈	1	2	yes	—
283	C₈	C₂×C₃×C₅×Q₈	1	2	yes	—
284	C₈	C₂×C₃×C₅×Q₈	1	4	yes	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
287	C₂×C₃	C₅×#577	1	2	yes	—
288	C₅	C₃×#3163	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
291	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
297	C₄	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
298	C₄	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
299	C₃	C₅×#3163	1	1	yes	characteristic, direct product factor, Sylow
300	C₂	—	2	2	no	—
301	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
302	C₂	C₃×C₅×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#501	C₂	semidirect
3. C₃×C₅×#499	C₂	semidirect
4. C₃×C₅×#498	C₂	semidirect
9. C₅×#3163	C₃	direct (N × H)
29. C₃×#3163	C₅	direct (N × H)
31. C₅×#501	C₂×C₃	semidirect
32. C₅×#499	C₂×C₃	semidirect
33. C₅×#498	C₂×C₃	semidirect
41. C₃×C₅×M₁₆	Q₈	semidirect
42. C₂×C₃×C₅×C₈	Q₈	semidirect
59. C₃×#501	C₂×C₅	semidirect
60. C₃×#499	C₂×C₅	semidirect
61. C₃×#498	C₂×C₅	semidirect
85. #3163	C₃×C₅	direct (N × H)
122. C₅×M₁₆	C₃×Q₈	semidirect
123. C₂×C₅×C₈	C₃×Q₈	semidirect
140. #501	C₂×C₃×C₅	semidirect
141. #499	C₂×C₃×C₅	semidirect
142. #498	C₂×C₃×C₅	semidirect
160. C₃×M₁₆	C₅×Q₈	semidirect
161. C₂×C₃×C₈	C₅×Q₈	semidirect
240. M₁₆	C₃×C₅×Q₈	semidirect
241. C₂×C₈	C₃×C₅×Q₈	semidirect
257. C₃×C₅	#3163	direct (N × H)
288. C₅	C₃×#3163	direct (N × H)
299. C₃	C₅×#3163	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×#544	2
C₃×C₅×#501	2
C₃×C₅×#499	2
C₃×C₅×#498	2
C₃×C₅×#428	2
C₃×C₅×#427	2
C₂×C₃×C₅×#106	2
C₃×C₅×#405	2
C₅×#3163	3
C₃×#3163	5

Subgroups — C3×C5×#3163 #14276197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3163 #14276197