Subgroups — C₃×C₅×#1950 #14274984

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	2	yes	maximal
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#341	C₂	1	2	yes	maximal
6	C₅×#1950	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	4	no	—
13	C₃×C₅×C₄⋊C₈	—	2	1	no	—
14	C₃×C₅×C₄⋊C₈	—	2	1	no	—
15	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
16	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
17	C₃×C₅×#97	—	2	1	no	—
18	C₃×C₄×C₅×C₈	—	2	1	no	—
19	C₃×#1950	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#108	C₂×C₃	1	2	yes	—
21	C₅×#423	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
24	C₅×#341	C₂×C₃	1	2	yes	—
25	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃×C₅×M₁₆	—	2	1	no	—
29	C₃×C₅×M₁₆	—	2	1	no	—
30	C₃×C₅×M₁₆	—	2	2	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	D₄	1	2	yes	—
33	C₃×C₅×M₁₆	Q₈	1	2	yes	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₃×#108	C₂×C₅	1	2	yes	—
45	C₃×#423	C₂×C₅	1	1	yes	characteristic
46	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
47	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
48	C₃×#341	C₂×C₅	1	2	yes	—
49	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
50	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×#108	—	2	4	no	—
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×C₄⋊C₈	—	2	1	no	—
57	C₅×C₄⋊C₈	C₃×C₄	1	2	yes	—
58	C₅×C₄⋊C₈	C₃×C₄	1	2	yes	—
59	C₅×#97	—	2	1	no	—
60	C₄×C₅×C₈	—	2	1	no	—
61	#1950	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
75	C₃×C₅×C₈	—	2	4	no	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃×C₅×C₈	—	4	2	no	—
79	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
80	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
82	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₃×#108	—	2	4	no	—
85	C₃×C₄⋊C₈	—	2	1	no	—
86	C₃×C₄⋊C₈	—	2	1	no	—
87	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
88	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
89	C₃×#97	—	2	1	no	—
90	C₃×C₄×C₈	—	2	1	no	—
91	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₅×M₁₆	—	2	1	no	—
95	C₅×M₁₆	—	2	1	no	—
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	C₃×D₄	1	2	yes	—
99	C₅×M₁₆	C₃×Q₈	1	2	yes	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
106	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
110	C₂×#108	C₂×C₃×C₅	1	2	yes	—
111	#423	C₂×C₃×C₅	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
113	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
114	#341	C₂×C₃×C₅	1	2	yes	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₃×C₅	#125	1	1	yes	characteristic
119	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	2	1	no	—
122	C₃×C₄×C₅	—	2	1	no	—
123	C₃×C₄×C₅	#108	1	2	yes	complemented
124	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
126	C₃×C₄×C₅	#121	1	1	yes	characteristic
127	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₃×M₁₆	—	2	1	no	—
132	C₃×M₁₆	—	2	1	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	2	no	—
135	C₃×M₁₆	C₅×D₄	1	2	yes	—
136	C₃×M₁₆	C₅×Q₈	1	2	yes	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
143	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
144	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
146	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
147	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
160	C₅×C₈	—	2	4	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	2	2	no	—
163	C₅×C₈	—	4	2	no	—
164	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
165	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
168	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
169	#108	—	2	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	C₃×C₄×C₅	1	2	yes	—
173	C₄⋊C₈	C₃×C₄×C₅	1	2	yes	—
174	#97	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₂×C₃×C₅	—	2	1	no	—
177	C₂×C₃×C₅	—	2	1	no	—
178	C₂×C₃×C₅	#389	1	1	yes	characteristic
179	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
180	C₂×C₃×C₅	#430	1	1	yes	characteristic
181	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
192	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
193	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
194	C₃×C₈	—	2	4	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	4	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₅	C₃×#125	1	1	yes	characteristic
202	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	C₃×#108	1	2	yes	complemented
207	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
208	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
209	C₄×C₅	C₃×#121	1	1	yes	characteristic
210	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
214	M₁₆	—	2	1	no	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	C₃×C₅×D₄	1	2	yes	—
219	M₁₆	C₃×C₅×Q₈	1	2	yes	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×C₃×C₄×C₅	1	4	yes	—
226	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
227	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
228	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
229	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
230	C₃×C₅	#1950	1	1	yes	characteristic, direct product factor, Hall
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂²×C₃	C₅×#125	1	1	yes	characteristic
235	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	C₅×#108	1	2	yes	complemented
240	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
241	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
242	C₃×C₄	C₅×#121	1	1	yes	characteristic
243	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	C₃×#389	1	1	yes	characteristic
247	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
248	C₂×C₅	C₃×#430	1	1	yes	characteristic
249	C₂³	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
260	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
261	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	2	no	—
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	C₅×#389	1	1	yes	characteristic
269	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
270	C₂×C₃	C₅×#430	1	1	yes	characteristic
271	C₅	C₃×#1950	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
275	C₂²	C₃×C₅×#125	1	1	yes	characteristic
276	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	C₃×C₅×#108	1	2	yes	complemented
281	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
282	C₄	C₃×C₄×C₅×Q₈	1	1	yes	characteristic
283	C₄	C₃×C₅×#121	1	1	yes	characteristic
284	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
285	C₃	C₅×#1950	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₃×C₅×#389	1	1	yes	characteristic
289	C₂	C₂×C₃×C₅×#108	1	1	yes	characteristic
290	C₂	C₃×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1950	C₃	direct (N × H)
19. C₃×#1950	C₅	direct (N × H)
61. #1950	C₃×C₅	direct (N × H)
123. C₃×C₄×C₅	#108	semidirect
206. C₄×C₅	C₃×#108	semidirect
230. C₃×C₅	#1950	direct (N × H)
239. C₃×C₄	C₅×#108	semidirect
271. C₅	C₃×#1950	direct (N × H)
280. C₄	C₃×C₅×#108	semidirect
285. C₃	C₅×#1950	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×#108	2
C₃×C₅×#423	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×#341	2
C₅×#1950	3
C₃×#1950	5

Subgroups — C3×C5×#1950 #14274984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1950 #14274984