Subgroups — C₃×C₅×#2076 #14275110

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#380	C₂	1	2	yes	maximal
4	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#340	C₂	1	2	yes	maximal, complemented
6	C₅×#2076	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
13	C₃×C₅×#107	—	2	2	no	—
14	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
15	C₃×C₅×#95	—	2	2	no	—
16	C₃×#2076	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₅×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₅×#380	C₂×C₃	1	2	yes	—
20	C₅×#340	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#340	C₂×C₃	1	2	yes	complemented
22	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	1	no	—
24	C₂²×C₃×C₄×C₅	—	2	1	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₃×C₅×C₄⋊C₄	—	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×C₂²⋊C₄	—	4	4	no	—
40	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
42	C₃×#380	C₂×C₅	1	2	yes	—
43	C₃×#340	C₂×C₅	1	2	yes	complemented
44	C₃×#340	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₅×#107	—	2	2	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×#95	—	2	2	no	—
54	#2076	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	2	1	no	—
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	—	2	1	no	—
58	C₂³×C₃×C₅	—	2	2	no	—
59	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	4	2	no	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	2	yes	—
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×C₅×C₈	—	4	1	no	—
78	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×#107	—	2	2	no	—
85	C₃×C₂²⋊C₈	—	2	2	no	—
86	C₃×#95	—	2	2	no	—
87	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	—	2	1	no	—
89	C₂²×C₄×C₅	—	2	1	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	C₂×C₂²⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
107	#380	C₂×C₃×C₅	1	2	yes	—
108	#340	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	#340	C₂×C₃×C₅	1	2	yes	complemented
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	4	no	—
112	C₂²×C₃×C₅	—	4	1	no	—
113	C₂²×C₃×C₅	—	4	1	no	—
114	C₂²×C₃×C₅	—	4	2	no	—
115	C₂²×C₃×C₅	#121	1	1	yes	characteristic
116	C₂²×C₃×C₅	#123	1	2	yes	—
117	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
118	C₂²×C₃×C₅	#135	1	1	yes	characteristic
119	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
120	C₃×C₄×C₅	—	4	1	no	—
121	C₃×C₄×C₅	—	8	2	no	—
122	C₃×C₄×C₅	—	8	2	no	—
123	C₃×C₄×C₅	#123	1	1	yes	characteristic
124	C₃×C₄×C₅	#124	1	1	yes	characteristic
125	C₃×C₄×C₅	#126	1	2	yes	—
126	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₂²⋊C₄	—	4	4	no	—
144	C₂³×C₅	—	2	1	no	—
145	C₂³×C₅	—	2	2	no	—
146	C₂³×C₅	—	2	1	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	C₃×Q₁₆	1	2	yes	—
149	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₃×#41	1	2	yes	—
163	C₂×C₄×C₅	C₃×#41	1	2	yes	—
164	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
165	C₅×C₈	—	2	2	no	—
166	C₅×C₈	—	4	1	no	—
167	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
168	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
169	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
171	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	#107	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	#95	—	2	2	no	—
176	C₂×C₃×C₅	—	4	2	no	—
177	C₂×C₃×C₅	#397	1	1	yes	characteristic
178	C₂×C₃×C₅	#467	1	1	yes	characteristic
179	C₂×C₃×C₅	#469	1	1	yes	characteristic
180	C₂×C₃×C₅	#482	1	2	yes	—
181	C₂×C₃×C₅	#484	1	2	yes	—
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	—	2	2	no	—
186	C₂³×C₃	C₅×Q₁₆	1	2	yes	—
187	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	4	2	no	—
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₅×#41	1	2	yes	—
201	C₂×C₃×C₄	C₅×#41	1	2	yes	—
202	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	1	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	2	4	no	—
207	C₂²×C₅	—	4	1	no	—
208	C₂²×C₅	—	4	1	no	—
209	C₂²×C₅	—	4	2	no	—
210	C₂²×C₅	C₃×#121	1	1	yes	characteristic
211	C₂²×C₅	C₃×#123	1	2	yes	—
212	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
213	C₂²×C₅	C₃×#135	1	1	yes	characteristic
214	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
215	C₄×C₅	—	4	1	no	—
216	C₄×C₅	—	8	2	no	—
217	C₄×C₅	—	8	2	no	—
218	C₄×C₅	C₃×#123	1	1	yes	characteristic
219	C₄×C₅	C₃×#124	1	1	yes	characteristic
220	C₄×C₅	C₃×#126	1	2	yes	—
221	C₂⁴	C₃×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₃×C₅	#2076	1	1	yes	characteristic, direct product factor, Hall
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	4	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	4	2	no	—
245	C₂²×C₃	C₅×#121	1	1	yes	characteristic
246	C₂²×C₃	C₅×#123	1	2	yes	—
247	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
248	C₂²×C₃	C₅×#135	1	1	yes	characteristic
249	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
250	C₃×C₄	—	4	1	no	—
251	C₃×C₄	—	8	2	no	—
252	C₃×C₄	—	8	2	no	—
253	C₃×C₄	C₅×#123	1	1	yes	characteristic
254	C₃×C₄	C₅×#124	1	1	yes	characteristic
255	C₃×C₄	C₅×#126	1	2	yes	—
256	C₂×C₅	—	4	2	no	—
257	C₂×C₅	C₃×#397	1	1	yes	characteristic
258	C₂×C₅	C₃×#467	1	1	yes	characteristic
259	C₂×C₅	C₃×#469	1	1	yes	characteristic
260	C₂×C₅	C₃×#482	1	2	yes	—
261	C₂×C₅	C₃×#484	1	2	yes	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	C₃×C₅×Q₁₆	1	2	yes	—
267	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
280	C₂×C₄	C₃×C₅×#41	1	2	yes	—
281	C₂×C₄	C₃×C₅×#41	1	2	yes	—
282	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	1	no	—
285	C₂×C₃	—	4	2	no	—
286	C₂×C₃	C₅×#397	1	1	yes	characteristic
287	C₂×C₃	C₅×#467	1	1	yes	characteristic
288	C₂×C₃	C₅×#469	1	1	yes	characteristic
289	C₂×C₃	C₅×#482	1	2	yes	—
290	C₂×C₃	C₅×#484	1	2	yes	—
291	C₅	C₃×#2076	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₃×C₅×#121	1	1	yes	characteristic
298	C₂²	C₃×C₅×#123	1	2	yes	—
299	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
300	C₂²	C₃×C₅×#135	1	1	yes	characteristic
301	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₃×C₅×#123	1	1	yes	characteristic
306	C₄	C₃×C₅×#124	1	1	yes	characteristic
307	C₄	C₃×C₅×#126	1	2	yes	—
308	C₃	C₅×#2076	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	4	2	no	—
310	C₂	C₃×C₅×#397	1	1	yes	characteristic
311	C₂	C₃×C₅×#467	1	1	yes	characteristic
312	C₂	C₃×C₅×#469	1	1	yes	characteristic
313	C₂	C₃×C₅×#482	1	2	yes	—
314	C₂	C₃×C₅×#484	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#107	C₂	semidirect
4. C₃×C₅×#340	C₂	semidirect
5. C₃×C₅×#340	C₂	semidirect
6. C₅×#2076	C₃	direct (N × H)
16. C₃×#2076	C₅	direct (N × H)
17. C₂×C₅×#107	C₂×C₃	semidirect
20. C₅×#340	C₂×C₃	semidirect
21. C₅×#340	C₂×C₃	semidirect
40. C₂×C₃×#107	C₂×C₅	semidirect
43. C₃×#340	C₂×C₅	semidirect
44. C₃×#340	C₂×C₅	semidirect
54. #2076	C₃×C₅	direct (N × H)
105. C₂×#107	C₂×C₃×C₅	semidirect
108. #340	C₂×C₃×C₅	semidirect
109. #340	C₂×C₃×C₅	semidirect
239. C₃×C₅	#2076	direct (N × H)
291. C₅	C₃×#2076	direct (N × H)
308. C₃	C₅×#2076	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×#107	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₃×C₅×#380	2
C₃×C₅×#340	2
C₅×#2076	3
C₃×#2076	5

Subgroups — C3×C5×#2076 #14275110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2076 #14275110