Subgroups — C₃×C₅×#2084 #14275118

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#106	C₂	1	2	yes	maximal
2	C₃×C₅×#389	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×#2084	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
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×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#106	—	2	4	no	—
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×#2084	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#106	C₂×C₃	1	2	yes	—
19	C₅×#389	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	C₂²×C₃×C₄×C₅	—	2	1	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	1	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	4	2	no	—
37	C₃×C₅×C₄⋊C₄	—	4	2	no	—
38	C₃×C₅×C₄⋊C₄	—	4	2	no	—
39	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
40	C₃×C₄²×C₅	—	2	2	no	—
41	C₂×C₃×#106	C₂×C₅	1	2	yes	—
42	C₃×#389	C₂×C₅	1	1	yes	characteristic
43	C₃×#384	C₂×C₅	1	1	yes	characteristic
44	C₃×#340	C₂×C₅	1	2	yes	—
45	C₃×#336	C₂×C₅	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×#106	—	2	4	no	—
55	C₅×#95	—	2	1	no	—
56	#2084	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
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	1	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	4	1	no	—
69	C₂×C₃×C₄×C₅	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
72	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂×C₃×C₄×C₅	#41	1	2	yes	—
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₃×C₅×C₈	—	2	4	no	—
78	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄⋊C₄	—	2	1	no	—
80	C₂×C₃×C₄⋊C₄	—	2	1	no	—
81	C₂×C₃×C₄⋊C₄	—	2	1	no	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₃×#106	—	2	4	no	—
87	C₃×#95	—	2	1	no	—
88	C₂²×C₄×C₅	—	2	1	no	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	1	no	—
91	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
105	C₄²×C₅	—	2	2	no	—
106	C₂×#106	C₂×C₃×C₅	1	2	yes	—
107	#389	C₂×C₃×C₅	1	1	yes	characteristic
108	#384	C₂×C₃×C₅	1	1	yes	characteristic
109	#340	C₂×C₃×C₅	1	2	yes	—
110	#336	C₂×C₃×C₅	1	1	yes	characteristic
111	C₂²×C₃×C₅	#121	1	1	yes	characteristic
112	C₂²×C₃×C₅	#123	1	1	yes	characteristic
113	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂²×C₃×C₅	#137	1	1	yes	characteristic
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	4	2	no	—
121	C₃×C₄×C₅	—	8	2	no	—
122	C₃×C₄×C₅	—	8	1	no	—
123	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
124	C₃×C₄×C₅	#125	1	2	yes	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	4	2	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄⋊C₄	—	4	2	no	—
141	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
142	C₃×C₄²	—	2	2	no	—
143	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
158	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
161	C₂×C₄×C₅	C₃×#41	1	2	yes	—
162	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
163	C₅×C₈	—	2	4	no	—
164	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
170	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
172	#106	—	2	4	no	—
173	#95	—	2	1	no	—
174	C₂×C₃×C₅	#398	1	1	yes	characteristic
175	C₂×C₃×C₅	#460	1	1	yes	characteristic
176	C₂×C₃×C₅	#461	1	1	yes	characteristic
177	C₂×C₃×C₅	#481	1	1	yes	characteristic
178	C₂×C₃×C₅	#483	1	1	yes	characteristic
179	C₂×C₃×C₅	#498	1	1	yes	characteristic
180	C₂×C₃×C₅	#501	1	1	yes	characteristic
181	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
196	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
199	C₂×C₃×C₄	C₅×#41	1	2	yes	—
200	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
201	C₃×C₈	—	2	4	no	—
202	C₂²×C₅	C₃×#121	1	1	yes	characteristic
203	C₂²×C₅	C₃×#123	1	1	yes	characteristic
204	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
205	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
206	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
207	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²×C₅	C₃×#137	1	1	yes	characteristic
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	—	8	2	no	—
213	C₄×C₅	—	8	1	no	—
214	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
215	C₄×C₅	C₃×#125	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₃×C₅×Q₈	1	4	yes	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₃×C₅	#2084	1	1	yes	characteristic, direct product factor, Hall
235	C₂²×C₃	C₅×#121	1	1	yes	characteristic
236	C₂²×C₃	C₅×#123	1	1	yes	characteristic
237	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
240	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₂²×C₃	C₅×#137	1	1	yes	characteristic
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	—	8	2	no	—
246	C₃×C₄	—	8	1	no	—
247	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
248	C₃×C₄	C₅×#125	1	2	yes	—
249	C₂×C₅	C₃×#398	1	1	yes	characteristic
250	C₂×C₅	C₃×#460	1	1	yes	characteristic
251	C₂×C₅	C₃×#461	1	1	yes	characteristic
252	C₂×C₅	C₃×#481	1	1	yes	characteristic
253	C₂×C₅	C₃×#483	1	1	yes	characteristic
254	C₂×C₅	C₃×#498	1	1	yes	characteristic
255	C₂×C₅	C₃×#501	1	1	yes	characteristic
256	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
271	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
272	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
274	C₂×C₄	C₃×C₅×#41	1	2	yes	—
275	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₂×C₃	C₅×#398	1	1	yes	characteristic
278	C₂×C₃	C₅×#460	1	1	yes	characteristic
279	C₂×C₃	C₅×#461	1	1	yes	characteristic
280	C₂×C₃	C₅×#481	1	1	yes	characteristic
281	C₂×C₃	C₅×#483	1	1	yes	characteristic
282	C₂×C₃	C₅×#498	1	1	yes	characteristic
283	C₂×C₃	C₅×#501	1	1	yes	characteristic
284	C₅	C₃×#2084	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	C₃×C₅×#121	1	1	yes	characteristic
286	C₂²	C₃×C₅×#123	1	1	yes	characteristic
287	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
288	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
289	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
290	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂²	C₃×C₅×#137	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	1	no	—
297	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
298	C₄	C₃×C₅×#125	1	2	yes	—
299	C₃	C₅×#2084	1	1	yes	characteristic, direct product factor, Sylow
300	C₂	C₃×C₅×#398	1	1	yes	characteristic
301	C₂	C₃×C₅×#460	1	1	yes	characteristic
302	C₂	C₃×C₅×#461	1	1	yes	characteristic
303	C₂	C₃×C₅×#481	1	1	yes	characteristic
304	C₂	C₃×C₅×#483	1	1	yes	characteristic
305	C₂	C₃×C₅×#498	1	1	yes	characteristic
306	C₂	C₃×C₅×#501	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×#2084	C₃	direct (N × H)
17. C₃×#2084	C₅	direct (N × H)
56. #2084	C₃×C₅	direct (N × H)
234. C₃×C₅	#2084	direct (N × H)
284. C₅	C₃×#2084	direct (N × H)
299. C₃	C₅×#2084	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×#106	2
C₃×C₅×#389	2
C₃×C₅×#384	2
C₃×C₅×#340	2
C₃×C₅×#336	2
C₅×#2084	3
C₃×#2084	5

Subgroups — C3×C5×#2084 #14275118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2084 #14275118