Subgroups — C₃×C₅×#2660 #14275694

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#398	C₂	1	4	yes	maximal
4	C₃×C₅×#395	C₂	1	4	yes	maximal
5	C₃×C₅×#384	C₂	1	2	yes	maximal
6	C₃×C₅×#382	C₂	1	2	yes	maximal
7	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
8	C₅×#2660	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
11	C₃×C₅×#125	—	2	4	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
21	C₃×C₅×#95	C₂²	1	2	yes	—
22	C₃×C₅×#95	C₂²	1	2	yes	—
23	C₃×C₅×#95	C₂²	1	8	yes	—
24	C₃×#2660	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
27	C₅×#398	C₂×C₃	1	4	yes	—
28	C₅×#395	C₂×C₃	1	4	yes	—
29	C₅×#384	C₂×C₃	1	2	yes	—
30	C₅×#382	C₂×C₃	1	2	yes	—
31	C₅×#376	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	—	2	4	no	—
33	C₂×C₃×C₅×Q₈	—	4	4	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₅×C₄⋊C₄	—	2	2	no	—
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	—	2	4	no	—
47	C₃×C₅×C₄⋊C₄	—	4	8	no	—
48	C₃×C₄²×C₅	D₄	1	4	yes	—
49	C₃×C₄²×C₅	Q₈	1	8	yes	complemented
50	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
52	C₃×#398	C₂×C₅	1	4	yes	—
53	C₃×#395	C₂×C₅	1	4	yes	—
54	C₃×#384	C₂×C₅	1	2	yes	—
55	C₃×#382	C₂×C₅	1	2	yes	—
56	C₃×#376	C₂×C₅	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
58	C₅×C₄⋊Q₈	—	2	4	no	—
59	C₅×#125	—	2	4	no	—
60	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	—
67	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
69	C₅×#95	C₂²×C₃	1	2	yes	—
70	C₅×#95	C₂²×C₃	1	2	yes	—
71	C₅×#95	C₂²×C₃	1	8	yes	—
72	#2660	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
74	C₃×C₅×Q₈	—	4	8	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	2	8	no	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
89	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
90	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
91	C₃×C₄⋊Q₈	—	2	4	no	—
92	C₃×#125	—	2	4	no	—
93	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
97	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
99	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	8	yes	—
100	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
102	C₃×#95	C₂²×C₅	1	2	yes	—
103	C₃×#95	C₂²×C₅	1	2	yes	—
104	C₃×#95	C₂²×C₅	1	8	yes	—
105	C₂×C₅×Q₈	—	2	4	no	—
106	C₂×C₅×Q₈	—	4	4	no	—
107	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
110	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
113	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	4	no	—
115	C₅×C₄⋊C₄	—	2	4	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	4	no	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₄⋊C₄	—	4	8	no	—
121	C₄²×C₅	C₃×D₄	1	4	yes	—
122	C₄²×C₅	C₃×Q₈	1	8	yes	complemented
123	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
124	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
125	#398	C₂×C₃×C₅	1	4	yes	—
126	#395	C₂×C₃×C₅	1	4	yes	—
127	#384	C₂×C₃×C₅	1	2	yes	—
128	#382	C₂×C₃×C₅	1	2	yes	—
129	#376	C₂×C₃×C₅	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
132	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
133	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
134	C₃×C₄×C₅	—	2	2	no	—
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×C₄×C₅	—	2	8	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	—	4	4	no	—
139	C₃×C₄×C₅	—	4	2	no	—
140	C₃×C₄×C₅	C₄⋊Q₈	1	4	yes	—
141	C₂×C₃×Q₈	—	2	4	no	—
142	C₂×C₃×Q₈	—	4	4	no	—
143	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
145	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
149	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	—	2	4	no	—
155	C₃×C₄⋊C₄	—	2	4	no	—
156	C₃×C₄⋊C₄	—	4	8	no	—
157	C₃×C₄²	C₅×D₄	1	4	yes	—
158	C₃×C₄²	C₅×Q₈	1	8	yes	complemented
159	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
160	C₅×Q₈	—	4	8	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	8	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
175	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
176	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	#125	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
182	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
184	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	8	yes	—
186	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
188	#95	C₂²×C₃×C₅	1	2	yes	—
189	#95	C₂²×C₃×C₅	1	2	yes	—
190	#95	C₂²×C₃×C₅	1	8	yes	—
191	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
192	C₂×C₃×C₅	#535	1	1	yes	characteristic
193	C₂×C₃×C₅	#536	1	1	yes	characteristic
194	C₂×C₃×C₅	#544	1	4	yes	—
195	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
196	C₃×Q₈	—	4	8	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	8	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
211	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
212	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
214	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
215	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	4	2	no	—
220	C₄×C₅	—	4	4	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	C₃×C₄⋊Q₈	1	4	yes	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
226	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
227	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
228	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
231	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	8	no	—
239	C₄²	C₃×C₅×D₄	1	4	yes	—
240	C₄²	C₃×C₅×Q₈	1	8	yes	complemented
241	C₃×C₅	#2660	1	1	yes	characteristic, direct product factor, Hall
242	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
243	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
244	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
245	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	8	no	—
249	C₃×C₄	—	4	2	no	—
250	C₃×C₄	—	4	4	no	—
251	C₃×C₄	—	4	2	no	—
252	C₃×C₄	C₅×C₄⋊Q₈	1	4	yes	—
253	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
254	C₂×C₅	C₃×#535	1	1	yes	characteristic
255	C₂×C₅	C₃×#536	1	1	yes	characteristic
256	C₂×C₅	C₃×#544	1	4	yes	—
257	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
258	Q₈	—	4	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
272	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
273	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
274	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
275	C₂×C₃	C₅×#535	1	1	yes	characteristic
276	C₂×C₃	C₅×#536	1	1	yes	characteristic
277	C₂×C₃	C₅×#544	1	4	yes	—
278	C₅	C₃×#2660	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
280	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
281	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
282	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₃×C₅×C₄⋊Q₈	1	4	yes	—
290	C₃	C₅×#2660	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
292	C₂	C₃×C₅×#535	1	1	yes	characteristic
293	C₂	C₃×C₅×#536	1	1	yes	characteristic
294	C₂	C₃×C₅×#544	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#2660	C₃	direct (N × H)
24. C₃×#2660	C₅	direct (N × H)
49. C₃×C₄²×C₅	Q₈	semidirect
72. #2660	C₃×C₅	direct (N × H)
122. C₄²×C₅	C₃×Q₈	semidirect
158. C₃×C₄²	C₅×Q₈	semidirect
240. C₄²	C₃×C₅×Q₈	semidirect
241. C₃×C₅	#2660	direct (N × H)
278. C₅	C₃×#2660	direct (N × H)
290. C₃	C₅×#2660	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×C₄⋊Q₈	2
C₂×C₃×C₅×#125	2
C₃×C₅×#398	2
C₃×C₅×#395	2
C₃×C₅×#384	2
C₃×C₅×#382	2
C₃×C₅×#376	2
C₅×#2660	3
C₃×#2660	5

Subgroups — C3×C5×#2660 #14275694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2660 #14275694