Subgroups — C₃×C₅×#2634 #14275668

Order 1920.

832 conjugacy classes of subgroups.

The subgroup lattice has 332 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×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#400	C₂	1	2	yes	maximal
4	C₃×C₅×#400	C₂	1	2	yes	maximal
5	C₃×C₅×#398	C₂	1	2	yes	maximal
6	C₃×C₅×#395	C₂	1	2	yes	maximal
7	C₃×C₅×#387	C₂	1	4	yes	maximal
8	C₃×C₅×#383	C₂	1	1	yes	maximal, characteristic
9	C₅×#2634	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#125	—	2	4	no	—
12	C₃×C₄×C₅×Q₈	—	2	4	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
21	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₅×#95	C₂²	1	2	yes	—
23	C₃×C₅×#95	C₂²	1	2	yes	—
24	C₃×C₅×#95	C₂²	1	2	yes	—
25	C₃×C₅×#95	C₂²	1	4	yes	—
26	C₃×C₅×#95	C₂²	1	4	yes	—
27	C₃×C₅×#95	C₂²	1	4	yes	—
28	C₃×#2634	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
30	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
31	C₅×#400	C₂×C₃	1	2	yes	—
32	C₅×#400	C₂×C₃	1	2	yes	—
33	C₅×#398	C₂×C₃	1	2	yes	—
34	C₅×#395	C₂×C₃	1	2	yes	—
35	C₅×#387	C₂×C₃	1	4	yes	—
36	C₅×#383	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×C₅×Q₈	—	2	2	no	—
38	C₂×C₃×C₅×Q₈	—	2	4	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
43	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	2	no	—
49	C₃×C₅×C₄⋊C₄	—	2	4	no	—
50	C₃×C₅×C₄⋊C₄	—	2	2	no	—
51	C₃×C₅×C₄⋊C₄	—	2	2	no	—
52	C₃×C₅×C₄⋊C₄	—	2	4	no	—
53	C₃×C₅×C₄⋊C₄	—	2	8	no	—
54	C₃×C₄²×C₅	—	2	4	no	—
55	C₃×C₄²×C₅	D₄	1	4	yes	—
56	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
57	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
58	C₃×#400	C₂×C₅	1	2	yes	—
59	C₃×#400	C₂×C₅	1	2	yes	—
60	C₃×#398	C₂×C₅	1	2	yes	—
61	C₃×#395	C₂×C₅	1	2	yes	—
62	C₃×#387	C₂×C₅	1	4	yes	—
63	C₃×#383	C₂×C₅	1	1	yes	characteristic
64	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₅×#125	—	2	4	no	—
66	C₄×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
75	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
76	C₅×#95	C₂²×C₃	1	2	yes	—
77	C₅×#95	C₂²×C₃	1	2	yes	—
78	C₅×#95	C₂²×C₃	1	2	yes	—
79	C₅×#95	C₂²×C₃	1	4	yes	—
80	C₅×#95	C₂²×C₃	1	4	yes	—
81	C₅×#95	C₂²×C₃	1	4	yes	—
82	#2634	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
83	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
84	C₃×C₅×Q₈	—	2	8	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	—	2	2	no	—
88	C₂×C₃×C₄×C₅	—	2	2	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	—	2	2	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	—	2	2	no	—
93	C₂×C₃×C₄×C₅	—	2	4	no	—
94	C₂×C₃×C₄×C₅	—	2	4	no	—
95	C₂×C₃×C₄×C₅	—	2	4	no	—
96	C₂×C₃×C₄×C₅	—	2	4	no	—
97	C₂×C₃×C₄×C₅	—	2	4	no	—
98	C₂×C₃×C₄×C₅	—	2	4	no	—
99	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
100	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
101	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₄×C₅	#41	1	4	yes	—
103	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×#125	—	2	4	no	—
105	C₃×C₄×Q₈	—	2	4	no	—
106	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
107	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
108	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
109	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
111	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
114	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
115	C₃×#95	C₂²×C₅	1	2	yes	—
116	C₃×#95	C₂²×C₅	1	2	yes	—
117	C₃×#95	C₂²×C₅	1	2	yes	—
118	C₃×#95	C₂²×C₅	1	4	yes	—
119	C₃×#95	C₂²×C₅	1	4	yes	—
120	C₃×#95	C₂²×C₅	1	4	yes	—
121	C₂×C₅×Q₈	—	2	2	no	—
122	C₂×C₅×Q₈	—	2	4	no	—
123	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
124	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
125	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
126	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
127	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
129	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
131	C₅×C₄⋊C₄	—	2	4	no	—
132	C₅×C₄⋊C₄	—	2	2	no	—
133	C₅×C₄⋊C₄	—	2	4	no	—
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	—	2	2	no	—
136	C₅×C₄⋊C₄	—	2	4	no	—
137	C₅×C₄⋊C₄	—	2	8	no	—
138	C₄²×C₅	—	2	4	no	—
139	C₄²×C₅	C₃×D₄	1	4	yes	—
140	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
142	#400	C₂×C₃×C₅	1	2	yes	—
143	#400	C₂×C₃×C₅	1	2	yes	—
144	#398	C₂×C₃×C₅	1	2	yes	—
145	#395	C₂×C₃×C₅	1	2	yes	—
146	#387	C₂×C₃×C₅	1	4	yes	—
147	#383	C₂×C₃×C₅	1	1	yes	characteristic
148	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
150	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
151	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
152	C₃×C₄×C₅	—	2	2	no	—
153	C₃×C₄×C₅	—	2	4	no	—
154	C₃×C₄×C₅	—	2	2	no	—
155	C₃×C₄×C₅	—	2	2	no	—
156	C₃×C₄×C₅	—	4	2	no	—
157	C₃×C₄×C₅	—	4	2	no	—
158	C₃×C₄×C₅	—	4	2	no	—
159	C₃×C₄×C₅	—	4	4	no	—
160	C₂×C₃×Q₈	—	2	2	no	—
161	C₂×C₃×Q₈	—	2	4	no	—
162	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
163	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
164	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
165	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
166	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
169	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
170	C₃×C₄⋊C₄	—	2	4	no	—
171	C₃×C₄⋊C₄	—	2	2	no	—
172	C₃×C₄⋊C₄	—	2	4	no	—
173	C₃×C₄⋊C₄	—	2	2	no	—
174	C₃×C₄⋊C₄	—	2	2	no	—
175	C₃×C₄⋊C₄	—	2	4	no	—
176	C₃×C₄⋊C₄	—	2	8	no	—
177	C₃×C₄²	—	2	4	no	—
178	C₃×C₄²	C₅×D₄	1	4	yes	—
179	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
180	C₅×Q₈	—	2	8	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	2	4	no	—
191	C₂×C₄×C₅	—	2	4	no	—
192	C₂×C₄×C₅	—	2	4	no	—
193	C₂×C₄×C₅	—	2	4	no	—
194	C₂×C₄×C₅	—	2	4	no	—
195	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
196	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₄×C₅	C₃×#41	1	4	yes	—
199	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
200	#125	—	2	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
203	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
205	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
206	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
209	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
210	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
211	#95	C₂²×C₃×C₅	1	2	yes	—
212	#95	C₂²×C₃×C₅	1	2	yes	—
213	#95	C₂²×C₃×C₅	1	2	yes	—
214	#95	C₂²×C₃×C₅	1	4	yes	—
215	#95	C₂²×C₃×C₅	1	4	yes	—
216	#95	C₂²×C₃×C₅	1	4	yes	—
217	C₂×C₃×C₅	#525	1	1	yes	characteristic
218	C₂×C₃×C₅	#536	1	1	yes	characteristic
219	C₂×C₃×C₅	#536	1	1	yes	characteristic
220	C₂×C₃×C₅	#541	1	4	yes	—
221	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
222	C₃×Q₈	—	2	8	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	4	no	—
232	C₂×C₃×C₄	—	2	4	no	—
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	2	4	no	—
235	C₂×C₃×C₄	—	2	4	no	—
236	C₂×C₃×C₄	—	2	4	no	—
237	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
238	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
240	C₂×C₃×C₄	C₅×#41	1	4	yes	—
241	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
243	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
244	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
245	C₄×C₅	—	2	2	no	—
246	C₄×C₅	—	2	4	no	—
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	—	2	2	no	—
249	C₄×C₅	—	4	2	no	—
250	C₄×C₅	—	4	2	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	4	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
256	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
257	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
258	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
259	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
260	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
261	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
262	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄²	—	2	4	no	—
271	C₄²	C₃×C₅×D₄	1	4	yes	—
272	C₃×C₅	#2634	1	1	yes	characteristic, direct product factor, Hall
273	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
274	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
275	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
276	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	2	4	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	2	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	—	4	4	no	—
285	C₂×C₅	C₃×#525	1	1	yes	characteristic
286	C₂×C₅	C₃×#536	1	1	yes	characteristic
287	C₂×C₅	C₃×#536	1	1	yes	characteristic
288	C₂×C₅	C₃×#541	1	4	yes	—
289	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
290	Q₈	—	2	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
306	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
307	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
308	C₂×C₄	C₃×C₅×#41	1	4	yes	—
309	C₂×C₃	C₅×#525	1	1	yes	characteristic
310	C₂×C₃	C₅×#536	1	1	yes	characteristic
311	C₂×C₃	C₅×#536	1	1	yes	characteristic
312	C₂×C₃	C₅×#541	1	4	yes	—
313	C₅	C₃×#2634	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
315	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
316	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
317	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	4	no	—
326	C₃	C₅×#2634	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	C₃×C₅×#525	1	1	yes	characteristic
328	C₂	C₃×C₅×#536	1	1	yes	characteristic
329	C₂	C₃×C₅×#536	1	1	yes	characteristic
330	C₂	C₃×C₅×#541	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₅×#2634	C₃	direct (N × H)
28. C₃×#2634	C₅	direct (N × H)
82. #2634	C₃×C₅	direct (N × H)
272. C₃×C₅	#2634	direct (N × H)
313. C₅	C₃×#2634	direct (N × H)
326. C₃	C₅×#2634	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×C₄×C₅×Q₈	2
C₃×C₅×#400	2
C₃×C₅×#398	2
C₃×C₅×#395	2
C₃×C₅×#387	2
C₃×C₅×#383	2
C₅×#2634	3
C₃×#2634	5

Subgroups — C3×C5×#2634 #14275668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2634 #14275668