Subgroups — C₃×C₅×#1989 #14275023

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#340	C₂	1	2	yes	maximal
5	C₃×C₅×#340	C₂	1	2	yes	maximal
6	C₅×#1989	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×#125	—	2	2	no	—
9	C₃×C₅×#125	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	1	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×C₄²×C₅	—	2	1	no	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	—	2	2	no	—
18	C₃×C₅×#95	—	2	1	no	—
19	C₃×#1989	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₅×#340	C₂×C₃	1	2	yes	—
24	C₅×#340	C₂×C₃	1	2	yes	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
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	4	no	—
33	C₃×C₅×C₄⋊C₄	—	2	4	no	—
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₅×C₄⋊C₄	Q₈	1	4	yes	—
41	C₃×C₄²×C₅	—	4	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
45	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
46	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
47	C₃×#340	C₂×C₅	1	2	yes	—
48	C₃×#340	C₂×C₅	1	2	yes	—
49	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
50	C₅×#125	—	2	2	no	—
51	C₅×#125	C₃×C₄	1	4	yes	complemented
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₄²×C₅	—	2	1	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	2	2	no	—
60	C₅×#95	—	2	1	no	—
61	#1989	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	4	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	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₃×C₅×C₈	—	4	2	no	—
81	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
82	C₃×#125	—	2	2	no	—
83	C₃×#125	C₄×C₅	1	4	yes	complemented
84	C₂×C₃×C₄⋊C₄	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	—	2	1	no	—
86	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₂×C₃×C₄²	—	2	1	no	—
90	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄⋊C₈	—	2	2	no	—
92	C₃×#95	—	2	1	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	1	no	—
95	C₂²×C₄×C₅	—	2	1	no	—
96	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
107	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
108	C₅×C₄⋊C₄	C₃×Q₈	1	4	yes	—
109	C₄²×C₅	—	4	2	no	—
110	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
111	C₄²×C₅	C₃×D₄	1	2	yes	—
112	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
113	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
115	#340	C₂×C₃×C₅	1	2	yes	—
116	#340	C₂×C₃×C₅	1	2	yes	—
117	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₃×C₅	#135	1	2	yes	—
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	—	4	2	no	—
126	C₃×C₄×C₅	—	4	4	no	—
127	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
128	C₃×C₄×C₅	#124	1	1	yes	characteristic
129	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	2	1	no	—
133	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₃×C₄⋊C₄	—	2	4	no	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
145	C₃×C₄⋊C₄	C₅×Q₈	1	4	yes	—
146	C₃×C₄²	—	4	2	no	—
147	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
148	C₃×C₄²	C₅×D₄	1	2	yes	—
149	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	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	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
162	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
166	C₂×C₄×C₅	C₃×#41	1	2	yes	—
167	C₅×C₈	—	4	2	no	—
168	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
169	#125	—	2	2	no	—
170	#125	C₃×C₄×C₅	1	4	yes	complemented
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	#95	—	2	1	no	—
180	C₂×C₃×C₅	#391	1	1	yes	characteristic
181	C₂×C₃×C₅	#416	1	1	yes	characteristic
182	C₂×C₃×C₅	#417	1	1	yes	characteristic
183	C₂×C₃×C₅	#478	1	2	yes	—
184	C₂×C₃×C₅	#479	1	2	yes	—
185	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	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	1	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
198	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
199	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
202	C₂×C₃×C₄	C₅×#41	1	2	yes	—
203	C₃×C₈	—	4	2	no	—
204	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
206	C₂²×C₅	C₃×#135	1	2	yes	—
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	—	4	2	no	—
213	C₄×C₅	—	4	4	no	—
214	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
215	C₄×C₅	C₃×#124	1	1	yes	characteristic
216	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
221	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
231	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
232	C₄⋊C₄	C₃×C₅×Q₈	1	4	yes	—
233	C₄²	—	4	2	no	—
234	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
235	C₄²	C₃×C₅×D₄	1	2	yes	—
236	C₃×C₅	#1989	1	1	yes	characteristic, direct product factor, Hall
237	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
239	C₂²×C₃	C₅×#135	1	2	yes	—
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	—	4	2	no	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
248	C₃×C₄	C₅×#124	1	1	yes	characteristic
249	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
250	C₂×C₅	C₃×#391	1	1	yes	characteristic
251	C₂×C₅	C₃×#416	1	1	yes	characteristic
252	C₂×C₅	C₃×#417	1	1	yes	characteristic
253	C₂×C₅	C₃×#478	1	2	yes	—
254	C₂×C₅	C₃×#479	1	2	yes	—
255	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
270	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
272	C₂×C₄	C₃×C₅×#41	1	2	yes	—
273	C₈	—	4	2	no	—
274	C₂×C₃	C₅×#391	1	1	yes	characteristic
275	C₂×C₃	C₅×#416	1	1	yes	characteristic
276	C₂×C₃	C₅×#417	1	1	yes	characteristic
277	C₂×C₃	C₅×#478	1	2	yes	—
278	C₂×C₃	C₅×#479	1	2	yes	—
279	C₅	C₃×#1989	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
282	C₂²	C₃×C₅×#135	1	2	yes	—
283	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂²	C₃×C₅×#137	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
291	C₄	C₃×C₅×#124	1	1	yes	characteristic
292	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
293	C₃	C₅×#1989	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	C₃×C₅×#391	1	1	yes	characteristic
295	C₂	C₃×C₅×#416	1	1	yes	characteristic
296	C₂	C₃×C₅×#417	1	1	yes	characteristic
297	C₂	C₃×C₅×#478	1	2	yes	—
298	C₂	C₃×C₅×#479	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×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×#1989	C₃	direct (N × H)
9. C₃×C₅×#125	C₄	semidirect
19. C₃×#1989	C₅	direct (N × H)
51. C₅×#125	C₃×C₄	semidirect
61. #1989	C₃×C₅	direct (N × H)
83. C₃×#125	C₄×C₅	semidirect
170. #125	C₃×C₄×C₅	semidirect
236. C₃×C₅	#1989	direct (N × H)
279. C₅	C₃×#1989	direct (N × H)
293. C₃	C₅×#1989	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×#125	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#340	2
C₅×#1989	3
C₃×#1989	5

Subgroups — C3×C5×#1989 #14275023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1989 #14275023