Subgroups — C₃×C₅×#1958 #14274992

Order 1920.

672 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×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#103	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#102	C₂	1	2	yes	maximal
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1958	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₅×#124	—	2	2	no	—
11	C₃×C₅×#124	C₄	1	4	yes	complemented
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	4	no	—
16	C₃×C₅×#102	—	2	4	no	—
17	C₃×C₄×C₅×C₈	—	2	2	no	—
18	C₃×C₅×#95	—	2	2	no	—
19	C₃×#1958	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#103	C₂×C₃	1	2	yes	complemented
22	C₂×C₅×#102	C₂×C₃	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#382	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
44	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×#103	C₂×C₅	1	2	yes	complemented
48	C₂×C₃×#102	C₂×C₅	1	2	yes	—
49	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×#382	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
54	C₅×#124	—	2	2	no	—
55	C₅×#124	C₃×C₄	1	4	yes	complemented
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×#103	—	2	4	no	—
60	C₅×#102	—	2	4	no	—
61	C₄×C₅×C₈	—	2	2	no	—
62	C₅×#95	—	2	2	no	—
63	#1958	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	—	4	1	no	—
66	C₂³×C₃×C₅	—	4	1	no	—
67	C₂³×C₃×C₅	—	4	4	no	—
68	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
69	C₃×C₅×Q₈	—	2	4	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×D₄	—	2	4	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	4	no	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₃×C₅×C₈	—	2	4	no	—
85	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
88	C₃×#124	—	2	2	no	—
89	C₃×#124	C₄×C₅	1	4	yes	complemented
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₃×#103	—	2	4	no	—
94	C₃×#102	—	2	4	no	—
95	C₃×C₄×C₈	—	2	2	no	—
96	C₃×#95	—	2	2	no	—
97	C₂⁴×C₅	—	2	1	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	1	no	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
106	C₂²×C₄×C₅	—	2	1	no	—
107	C₂²×C₄×C₅	—	2	2	no	—
108	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	2	8	no	—
116	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
117	C₄²×C₅	C₃×D₄	1	2	yes	—
118	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂×#103	C₂×C₃×C₅	1	2	yes	complemented
120	C₂×#102	C₂×C₃×C₅	1	2	yes	—
121	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
122	#382	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	C₂²×C₃×C₅	—	4	2	no	—
124	C₂²×C₃×C₅	—	4	4	no	—
125	C₂²×C₃×C₅	—	8	2	no	—
126	C₂²×C₃×C₅	—	8	2	no	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#124	1	1	yes	characteristic
129	C₂²×C₃×C₅	#127	1	1	yes	characteristic
130	C₂²×C₃×C₅	#135	1	4	yes	—
131	C₃×C₄×C₅	—	2	4	no	—
132	C₃×C₄×C₅	—	4	2	no	—
133	C₃×C₄×C₅	—	8	2	no	—
134	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₃×C₄×C₅	#121	1	2	yes	—
136	C₂⁴×C₃	—	2	1	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	2	no	—
139	C₂×C₃×Q₈	—	2	2	no	—
140	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
141	C₂×C₃×D₄	—	2	2	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	—	2	2	no	—
144	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₃×C₄	—	2	1	no	—
146	C₂²×C₃×C₄	—	2	2	no	—
147	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
148	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	2	2	no	—
151	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	8	no	—
155	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₃×C₄²	C₅×D₄	1	2	yes	—
157	C₂³×C₅	—	2	2	no	—
158	C₂³×C₅	—	4	1	no	—
159	C₂³×C₅	—	4	1	no	—
160	C₂³×C₅	—	4	4	no	—
161	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₅×Q₈	—	2	4	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₅×D₄	—	2	4	no	—
165	C₅×D₄	—	4	2	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	2	no	—
171	C₂×C₄×C₅	—	4	4	no	—
172	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×#41	1	2	yes	—
177	C₅×C₈	—	2	4	no	—
178	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
179	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
180	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
181	#124	—	2	2	no	—
182	#124	C₃×C₄×C₅	1	4	yes	complemented
183	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
186	#103	—	2	4	no	—
187	#102	—	2	4	no	—
188	C₄×C₈	—	2	2	no	—
189	#95	—	2	2	no	—
190	C₂×C₃×C₅	—	8	2	no	—
191	C₂×C₃×C₅	#390	1	1	yes	characteristic
192	C₂×C₃×C₅	#416	1	2	yes	—
193	C₂×C₃×C₅	#488	1	2	yes	—
194	C₂×C₃×C₅	#495	1	2	yes	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	—	4	1	no	—
197	C₂³×C₃	—	4	1	no	—
198	C₂³×C₃	—	4	4	no	—
199	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
200	C₃×Q₈	—	2	4	no	—
201	C₃×Q₈	—	4	2	no	—
202	C₃×D₄	—	2	4	no	—
203	C₃×D₄	—	4	2	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	4	no	—
210	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
211	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
213	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₅×#41	1	2	yes	—
215	C₃×C₈	—	2	4	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	—	4	4	no	—
218	C₂²×C₅	—	8	2	no	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₅	C₃×#124	1	1	yes	characteristic
222	C₂²×C₅	C₃×#127	1	1	yes	characteristic
223	C₂²×C₅	C₃×#135	1	4	yes	—
224	C₄×C₅	—	2	4	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	—	8	2	no	—
227	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
228	C₄×C₅	C₃×#121	1	2	yes	—
229	C₂⁴	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
241	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	8	no	—
248	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
249	C₄²	C₃×C₅×D₄	1	2	yes	—
250	C₃×C₅	#1958	1	1	yes	characteristic, direct product factor, Hall
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	—	4	4	no	—
253	C₂²×C₃	—	8	2	no	—
254	C₂²×C₃	—	8	2	no	—
255	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₃	C₅×#124	1	1	yes	characteristic
257	C₂²×C₃	C₅×#127	1	1	yes	characteristic
258	C₂²×C₃	C₅×#135	1	4	yes	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	8	2	no	—
262	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
263	C₃×C₄	C₅×#121	1	2	yes	—
264	C₂×C₅	—	8	2	no	—
265	C₂×C₅	C₃×#390	1	1	yes	characteristic
266	C₂×C₅	C₃×#416	1	2	yes	—
267	C₂×C₅	C₃×#488	1	2	yes	—
268	C₂×C₅	C₃×#495	1	2	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	4	no	—
273	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	Q₈	—	2	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	4	yes	—
285	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
286	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
287	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
288	C₂×C₄	C₃×C₅×#41	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₂×C₃	—	8	2	no	—
291	C₂×C₃	C₅×#390	1	1	yes	characteristic
292	C₂×C₃	C₅×#416	1	2	yes	—
293	C₂×C₃	C₅×#488	1	2	yes	—
294	C₂×C₃	C₅×#495	1	2	yes	—
295	C₅	C₃×#1958	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	4	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
301	C₂²	C₃×C₅×#124	1	1	yes	characteristic
302	C₂²	C₃×C₅×#127	1	1	yes	characteristic
303	C₂²	C₃×C₅×#135	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
308	C₄	C₃×C₅×#121	1	2	yes	—
309	C₃	C₅×#1958	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	8	2	no	—
311	C₂	C₃×C₅×#390	1	1	yes	characteristic
312	C₂	C₃×C₅×#416	1	2	yes	—
313	C₂	C₃×C₅×#488	1	2	yes	—
314	C₂	C₃×C₅×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
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
2. C₂×C₃×C₅×#103	C₂	semidirect
4. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
5. C₃×C₅×#382	C₂	semidirect
6. C₅×#1958	C₃	direct (N × H)
11. C₃×C₅×#124	C₄	semidirect
19. C₃×#1958	C₅	direct (N × H)
21. C₂×C₅×#103	C₂×C₃	semidirect
23. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
24. C₅×#382	C₂×C₃	semidirect
47. C₂×C₃×#103	C₂×C₅	semidirect
49. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
50. C₃×#382	C₂×C₅	semidirect
55. C₅×#124	C₃×C₄	semidirect
63. #1958	C₃×C₅	direct (N × H)
89. C₃×#124	C₄×C₅	semidirect
119. C₂×#103	C₂×C₃×C₅	semidirect
121. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
122. #382	C₂×C₃×C₅	semidirect
182. #124	C₃×C₄×C₅	semidirect
250. C₃×C₅	#1958	direct (N × H)
295. C₅	C₃×#1958	direct (N × H)
309. C₃	C₅×#1958	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×#124	2
C₂×C₃×C₅×#103	2
C₂×C₃×C₅×#102	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#382	2
C₅×#1958	3
C₃×#1958	5

Subgroups — C3×C5×#1958 #14274992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1958 #14274992