Subgroups — C₃×C₅×#2647 #14275681

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#400	C₂	1	4	yes	maximal
2	C₃×C₅×#398	C₂	1	4	yes	maximal
3	C₃×C₅×#389	C₂	1	2	yes	maximal
4	C₃×C₅×#389	C₂	1	2	yes	maximal
5	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
8	C₅×#2647	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×#95	C₂²	1	2	yes	—
18	C₃×C₅×#95	C₂²	1	2	yes	—
19	C₃×C₅×#95	C₂²	1	4	yes	—
20	C₃×C₅×#95	C₂²	1	4	yes	—
21	C₃×C₅×#95	C₂²	1	4	yes	—
22	C₃×#2647	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#400	C₂×C₃	1	4	yes	—
24	C₅×#398	C₂×C₃	1	4	yes	—
25	C₅×#389	C₂×C₃	1	2	yes	—
26	C₅×#389	C₂×C₃	1	2	yes	—
27	C₅×#384	C₂×C₃	1	1	yes	characteristic
28	C₅×#382	C₂×C₃	1	1	yes	characteristic
29	C₅×#376	C₂×C₃	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	4	4	no	—
40	C₃×C₅×C₄⋊C₄	—	4	4	no	—
41	C₃×C₅×C₄⋊C₄	—	4	4	no	—
42	C₃×C₄²×C₅	—	2	2	no	—
43	C₃×C₄²×C₅	—	2	2	no	—
44	C₃×C₄²×C₅	Q₈	1	4	yes	—
45	C₃×#400	C₂×C₅	1	4	yes	—
46	C₃×#398	C₂×C₅	1	4	yes	—
47	C₃×#389	C₂×C₅	1	2	yes	—
48	C₃×#389	C₂×C₅	1	2	yes	—
49	C₃×#384	C₂×C₅	1	1	yes	characteristic
50	C₃×#382	C₂×C₅	1	1	yes	characteristic
51	C₃×#376	C₂×C₅	1	1	yes	characteristic
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
56	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₅×#95	C₂²×C₃	1	2	yes	—
61	C₅×#95	C₂²×C₃	1	2	yes	—
62	C₅×#95	C₂²×C₃	1	4	yes	—
63	C₅×#95	C₂²×C₃	1	4	yes	—
64	C₅×#95	C₂²×C₃	1	4	yes	—
65	#2647	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
87	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
89	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
94	C₃×#95	C₂²×C₅	1	2	yes	—
95	C₃×#95	C₂²×C₅	1	2	yes	—
96	C₃×#95	C₂²×C₅	1	4	yes	—
97	C₃×#95	C₂²×C₅	1	4	yes	—
98	C₃×#95	C₂²×C₅	1	4	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
103	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
105	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	4	4	no	—
109	C₅×C₄⋊C₄	—	4	4	no	—
110	C₅×C₄⋊C₄	—	4	4	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	C₃×Q₈	1	4	yes	—
114	#400	C₂×C₃×C₅	1	4	yes	—
115	#398	C₂×C₃×C₅	1	4	yes	—
116	#389	C₂×C₃×C₅	1	2	yes	—
117	#389	C₂×C₃×C₅	1	2	yes	—
118	#384	C₂×C₃×C₅	1	1	yes	characteristic
119	#382	C₂×C₃×C₅	1	1	yes	characteristic
120	#376	C₂×C₃×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
124	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
125	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
126	C₃×C₄×C₅	—	2	4	no	—
127	C₃×C₄×C₅	—	4	1	no	—
128	C₃×C₄×C₅	—	4	4	no	—
129	C₃×C₄×C₅	—	4	1	no	—
130	C₃×C₄×C₅	—	4	2	no	—
131	C₃×C₄×C₅	—	4	4	no	—
132	C₃×C₄×C₅	#125	1	4	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
136	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
138	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	4	4	no	—
143	C₃×C₄⋊C₄	—	4	4	no	—
144	C₃×C₄⋊C₄	—	4	4	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	2	no	—
147	C₃×C₄²	C₅×Q₈	1	4	yes	—
148	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
149	C₂×C₄×C₅	—	2	2	no	—
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	4	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
165	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
166	C₂×C₄×C₅	C₃×#41	1	2	yes	—
167	C₂×C₄×C₅	C₃×#41	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
170	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
171	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
172	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
173	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
176	#95	C₂²×C₃×C₅	1	2	yes	—
177	#95	C₂²×C₃×C₅	1	2	yes	—
178	#95	C₂²×C₃×C₅	1	4	yes	—
179	#95	C₂²×C₃×C₅	1	4	yes	—
180	#95	C₂²×C₃×C₅	1	4	yes	—
181	C₂×C₃×C₅	C₂×#125	1	1	yes	characteristic
182	C₂×C₃×C₅	#540	1	1	yes	characteristic
183	C₂×C₃×C₅	#541	1	1	yes	characteristic
184	C₂×C₃×C₅	#542	1	2	yes	—
185	C₂×C₃×C₅	#544	1	2	yes	—
186	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
203	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
204	C₂×C₃×C₄	C₅×#41	1	2	yes	—
205	C₂×C₃×C₄	C₅×#41	1	2	yes	—
206	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
207	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
208	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
209	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
210	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
211	C₄×C₅	—	2	4	no	—
212	C₄×C₅	—	4	1	no	—
213	C₄×C₅	—	4	4	no	—
214	C₄×C₅	—	4	1	no	—
215	C₄×C₅	—	4	2	no	—
216	C₄×C₅	—	4	4	no	—
217	C₄×C₅	C₃×#125	1	4	yes	—
218	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
220	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
221	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
223	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
224	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	C₃×C₅×Q₈	1	4	yes	—
233	C₃×C₅	#2647	1	1	yes	characteristic, direct product factor, Hall
234	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
235	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
237	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
238	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	4	4	no	—
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	4	4	no	—
245	C₃×C₄	C₅×#125	1	4	yes	—
246	C₂×C₅	C₂×C₃×#125	1	1	yes	characteristic
247	C₂×C₅	C₃×#540	1	1	yes	characteristic
248	C₂×C₅	C₃×#541	1	1	yes	characteristic
249	C₂×C₅	C₃×#542	1	2	yes	—
250	C₂×C₅	C₃×#544	1	2	yes	—
251	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
268	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
269	C₂×C₄	C₃×C₅×#41	1	2	yes	—
270	C₂×C₄	C₃×C₅×#41	1	2	yes	—
271	C₂×C₃	C₂×C₅×#125	1	1	yes	characteristic
272	C₂×C₃	C₅×#540	1	1	yes	characteristic
273	C₂×C₃	C₅×#541	1	1	yes	characteristic
274	C₂×C₃	C₅×#542	1	2	yes	—
275	C₂×C₃	C₅×#544	1	2	yes	—
276	C₅	C₃×#2647	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
278	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
279	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
280	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
281	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
282	C₄	—	2	4	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	4	no	—
288	C₄	C₃×C₅×#125	1	4	yes	—
289	C₃	C₅×#2647	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	C₂×C₃×C₅×#125	1	1	yes	characteristic
291	C₂	C₃×C₅×#540	1	1	yes	characteristic
292	C₂	C₃×C₅×#541	1	1	yes	characteristic
293	C₂	C₃×C₅×#542	1	2	yes	—
294	C₂	C₃×C₅×#544	1	2	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
8. C₅×#2647	C₃	direct (N × H)
22. C₃×#2647	C₅	direct (N × H)
65. #2647	C₃×C₅	direct (N × H)
233. C₃×C₅	#2647	direct (N × H)
276. C₅	C₃×#2647	direct (N × H)
289. C₃	C₅×#2647	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#400	2
C₃×C₅×#398	2
C₃×C₅×#389	2
C₃×C₅×#384	2
C₃×C₅×#382	2
C₃×C₅×#376	2
C₅×#2647	3
C₃×#2647	5

Subgroups — C3×C5×#2647 #14275681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2647 #14275681