Subgroups — C₃×C₅×#2072 #14275106

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#103	C₂	1	2	yes	maximal
2	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#340	C₂	1	2	yes	maximal
5	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₅×#2072	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×#103	—	2	4	no	—
15	C₃×C₅×#95	—	2	1	no	—
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×#2072	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#103	C₂×C₃	1	2	yes	—
19	C₅×#391	C₂×C₃	1	1	yes	characteristic
20	C₅×#384	C₂×C₃	1	1	yes	characteristic
21	C₅×#340	C₂×C₃	1	2	yes	—
22	C₅×#336	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	—	2	1	no	—
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	—	4	2	no	—
40	C₃×C₅×C₄⋊C₄	—	4	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₂×C₃×#103	C₂×C₅	1	2	yes	—
43	C₃×#391	C₂×C₅	1	1	yes	characteristic
44	C₃×#384	C₂×C₅	1	1	yes	characteristic
45	C₃×#340	C₂×C₅	1	2	yes	—
46	C₃×#336	C₂×C₅	1	1	yes	characteristic
47	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×#103	—	2	4	no	—
55	C₅×#95	—	2	1	no	—
56	C₅×#95	—	2	1	no	—
57	#2072	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×Q₈	—	2	4	no	—
60	C₃×C₅×Q₈	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	4	1	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
75	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₃×C₄×C₅	#41	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
80	C₃×C₅×C₈	—	4	2	no	—
81	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
82	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄⋊C₄	—	2	1	no	—
84	C₂×C₃×C₄⋊C₄	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
86	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₃×#103	—	2	4	no	—
89	C₃×#95	—	2	1	no	—
90	C₃×#95	—	2	1	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
95	C₂²×C₄×C₅	—	2	1	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	2	1	no	—
98	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₂×#103	C₂×C₃×C₅	1	2	yes	—
111	#391	C₂×C₃×C₅	1	1	yes	characteristic
112	#384	C₂×C₃×C₅	1	1	yes	characteristic
113	#340	C₂×C₃×C₅	1	2	yes	—
114	#336	C₂×C₃×C₅	1	1	yes	characteristic
115	C₂²×C₃×C₅	#121	1	1	yes	characteristic
116	C₂²×C₃×C₅	#123	1	1	yes	characteristic
117	C₂²×C₃×C₅	#124	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
120	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂²×C₃×C₅	#137	1	1	yes	characteristic
122	C₃×C₄×C₅	—	2	2	no	—
123	C₃×C₄×C₅	—	2	2	no	—
124	C₃×C₄×C₅	—	4	2	no	—
125	C₃×C₄×C₅	—	8	2	no	—
126	C₃×C₄×C₅	—	8	1	no	—
127	C₃×C₄×C₅	#123	1	2	yes	—
128	C₃×C₄×C₅	#126	1	2	yes	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	2	no	—
131	C₂×C₃×Q₈	—	2	2	no	—
132	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
133	C₂²×C₃×C₄	—	2	1	no	—
134	C₂²×C₃×C₄	—	2	1	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	4	2	no	—
146	C₃×C₄⋊C₄	—	4	2	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
149	C₅×Q₈	—	2	4	no	—
150	C₅×Q₈	—	4	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
165	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₃×#41	1	2	yes	—
168	C₂×C₄×C₅	C₃×#41	1	2	yes	—
169	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
170	C₅×C₈	—	4	2	no	—
171	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
172	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
177	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
178	#103	—	2	4	no	—
179	#95	—	2	1	no	—
180	#95	—	2	1	no	—
181	C₂×C₃×C₅	#397	1	1	yes	characteristic
182	C₂×C₃×C₅	#460	1	1	yes	characteristic
183	C₂×C₃×C₅	#462	1	1	yes	characteristic
184	C₂×C₃×C₅	#483	1	1	yes	characteristic
185	C₂×C₃×C₅	#484	1	1	yes	characteristic
186	C₂×C₃×C₅	#487	1	1	yes	characteristic
187	C₂×C₃×C₅	#489	1	1	yes	characteristic
188	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
189	C₃×Q₈	—	2	4	no	—
190	C₃×Q₈	—	4	2	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	—	4	1	no	—
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	—	4	2	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
205	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₅×#41	1	2	yes	—
208	C₂×C₃×C₄	C₅×#41	1	2	yes	—
209	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
210	C₃×C₈	—	4	2	no	—
211	C₂²×C₅	C₃×#121	1	1	yes	characteristic
212	C₂²×C₅	C₃×#123	1	1	yes	characteristic
213	C₂²×C₅	C₃×#124	1	1	yes	characteristic
214	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
215	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
216	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₅	C₃×#137	1	1	yes	characteristic
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	4	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	8	1	no	—
223	C₄×C₅	C₃×#123	1	2	yes	—
224	C₄×C₅	C₃×#126	1	2	yes	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄²	—	2	2	no	—
244	C₃×C₅	#2072	1	1	yes	characteristic, direct product factor, Hall
245	C₂²×C₃	C₅×#121	1	1	yes	characteristic
246	C₂²×C₃	C₅×#123	1	1	yes	characteristic
247	C₂²×C₃	C₅×#124	1	1	yes	characteristic
248	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
249	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
250	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂²×C₃	C₅×#137	1	1	yes	characteristic
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	8	2	no	—
256	C₃×C₄	—	8	1	no	—
257	C₃×C₄	C₅×#123	1	2	yes	—
258	C₃×C₄	C₅×#126	1	2	yes	—
259	C₂×C₅	C₃×#397	1	1	yes	characteristic
260	C₂×C₅	C₃×#460	1	1	yes	characteristic
261	C₂×C₅	C₃×#462	1	1	yes	characteristic
262	C₂×C₅	C₃×#483	1	1	yes	characteristic
263	C₂×C₅	C₃×#484	1	1	yes	characteristic
264	C₂×C₅	C₃×#487	1	1	yes	characteristic
265	C₂×C₅	C₃×#489	1	1	yes	characteristic
266	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
283	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
284	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₃×C₅×#41	1	2	yes	—
286	C₂×C₄	C₃×C₅×#41	1	2	yes	—
287	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
288	C₈	—	4	2	no	—
289	C₂×C₃	C₅×#397	1	1	yes	characteristic
290	C₂×C₃	C₅×#460	1	1	yes	characteristic
291	C₂×C₃	C₅×#462	1	1	yes	characteristic
292	C₂×C₃	C₅×#483	1	1	yes	characteristic
293	C₂×C₃	C₅×#484	1	1	yes	characteristic
294	C₂×C₃	C₅×#487	1	1	yes	characteristic
295	C₂×C₃	C₅×#489	1	1	yes	characteristic
296	C₅	C₃×#2072	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	C₃×C₅×#121	1	1	yes	characteristic
298	C₂²	C₃×C₅×#123	1	1	yes	characteristic
299	C₂²	C₃×C₅×#124	1	1	yes	characteristic
300	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
301	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
302	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂²	C₃×C₅×#137	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	C₃×C₅×#123	1	2	yes	—
310	C₄	C₃×C₅×#126	1	2	yes	—
311	C₃	C₅×#2072	1	1	yes	characteristic, direct product factor, Sylow
312	C₂	C₃×C₅×#397	1	1	yes	characteristic
313	C₂	C₃×C₅×#460	1	1	yes	characteristic
314	C₂	C₃×C₅×#462	1	1	yes	characteristic
315	C₂	C₃×C₅×#483	1	1	yes	characteristic
316	C₂	C₃×C₅×#484	1	1	yes	characteristic
317	C₂	C₃×C₅×#487	1	1	yes	characteristic
318	C₂	C₃×C₅×#489	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. C₅×#2072	C₃	direct (N × H)
17. C₃×#2072	C₅	direct (N × H)
57. #2072	C₃×C₅	direct (N × H)
244. C₃×C₅	#2072	direct (N × H)
296. C₅	C₃×#2072	direct (N × H)
311. C₃	C₅×#2072	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×#103	2
C₃×C₅×#391	2
C₃×C₅×#384	2
C₃×C₅×#340	2
C₃×C₅×#336	2
C₅×#2072	3
C₃×#2072	5

Subgroups — C3×C5×#2072 #14275106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2072 #14275106