Subgroups — C₃×C₅×#1484 #14274518

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₅×#1484	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	—	2	1	no	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₃×C₅×#127	C₄	1	1	yes	characteristic
10	C₃×C₅×#124	—	2	1	no	—
11	C₃×C₅×#121	—	2	1	no	—
12	C₃×C₅×#121	C₄	1	2	yes	—
13	C₃×C₄×C₅×D₄	—	2	1	no	—
14	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×#97	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₃×#1484	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	C₂×C₃	1	1	yes	characteristic
23	C₅×#431	C₂×C₃	1	1	yes	characteristic, complemented
24	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×D₄⋊C₈	C₂×C₃	1	4	yes	complemented
26	C₃×C₅×#41	—	4	4	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	2	1	no	—
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₃×C₅×M₁₆	—	2	4	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×#532	C₂×C₅	1	1	yes	characteristic
48	C₃×#431	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×D₄⋊C₈	C₂×C₅	1	4	yes	complemented
51	C₂×C₅×#41	—	2	1	no	—
52	C₂×C₅×M₁₆	—	2	1	no	—
53	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
54	C₅×#127	C₃×C₄	1	1	yes	characteristic
55	C₅×#124	—	2	1	no	—
56	C₅×#121	—	2	1	no	—
57	C₅×#121	C₃×C₄	1	2	yes	—
58	C₄×C₅×D₄	—	2	1	no	—
59	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×#97	—	2	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
66	#1484	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×D₄	—	2	4	no	—
71	C₃×C₅×D₄	—	4	2	no	—
72	C₃×C₅×D₄	—	4	4	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	2	2	no	—
78	C₂×C₃×C₄×C₅	—	4	4	no	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	4	no	—
86	C₃×C₅×C₈	—	4	2	no	—
87	C₂×C₃×#41	—	2	1	no	—
88	C₂×C₃×M₁₆	—	2	1	no	—
89	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
90	C₃×#127	C₄×C₅	1	1	yes	characteristic
91	C₃×#124	—	2	1	no	—
92	C₃×#121	—	2	1	no	—
93	C₃×#121	C₄×C₅	1	2	yes	—
94	C₃×C₄×D₄	—	2	1	no	—
95	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
98	C₃×C₂²⋊C₈	—	2	1	no	—
99	C₃×#97	—	2	1	no	—
100	C₃×C₄×C₈	—	2	1	no	—
101	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
102	C₅×#41	—	4	4	no	—
103	C₂×C₅×Q₈	—	2	1	no	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	1	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
108	C₂²×C₄×C₅	—	2	2	no	—
109	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
112	C₅×M₁₆	—	2	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
117	C₅×C₂²⋊C₄	—	2	2	no	—
118	C₅×C₂²⋊C₄	—	2	2	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	#532	C₂×C₃×C₅	1	1	yes	characteristic
124	#431	C₂×C₃×C₅	1	1	yes	characteristic, complemented
125	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	D₄⋊C₈	C₂×C₃×C₅	1	4	yes	complemented
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	—	4	2	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	#104	1	2	yes	—
139	C₃×C₄×C₅	#131	1	2	yes	—
140	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
141	C₃×#41	—	4	4	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×D₄	—	2	2	no	—
144	C₂×C₃×D₄	—	2	1	no	—
145	C₂×C₃×D₄	—	2	1	no	—
146	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
147	C₂²×C₃×C₄	—	2	2	no	—
148	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
151	C₃×M₁₆	—	2	4	no	—
152	C₂×C₃×C₈	—	2	2	no	—
153	C₂×C₃×C₈	—	2	2	no	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₂²⋊C₄	—	2	2	no	—
158	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂³×C₅	—	2	2	no	—
163	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₅×Q₈	—	4	2	no	—
165	C₅×D₄	—	2	4	no	—
166	C₅×D₄	—	4	2	no	—
167	C₅×D₄	—	4	4	no	—
168	C₅×D₄	—	4	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	4	no	—
174	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₅×C₈	—	2	4	no	—
181	C₅×C₈	—	4	2	no	—
182	C₂×#41	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
185	#127	C₃×C₄×C₅	1	1	yes	characteristic
186	#124	—	2	1	no	—
187	#121	—	2	1	no	—
188	#121	C₃×C₄×C₅	1	2	yes	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
191	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
192	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
193	C₂²⋊C₈	—	2	1	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
197	C₂×C₃×C₅	—	4	1	no	—
198	C₂×C₃×C₅	—	8	2	no	—
199	C₂×C₃×C₅	#408	1	1	yes	characteristic
200	C₂×C₃×C₅	#418	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₃×Q₈	—	4	2	no	—
205	C₃×D₄	—	2	4	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	4	no	—
208	C₃×D₄	—	4	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	4	no	—
214	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₃×C₈	—	2	4	no	—
221	C₃×C₈	—	4	2	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	4	2	no	—
226	C₂²×C₅	—	4	2	no	—
227	C₂²×C₅	—	4	2	no	—
228	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	C₃×#104	1	2	yes	—
234	C₄×C₅	C₃×#131	1	2	yes	—
235	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
236	#41	—	4	4	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
246	M₁₆	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
254	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
255	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
256	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
257	C₃×C₅	#1484	1	1	yes	characteristic, direct product factor, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	C₅×#104	1	2	yes	—
270	C₃×C₄	C₅×#131	1	2	yes	—
271	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	—	8	2	no	—
274	C₂×C₅	C₃×#408	1	1	yes	characteristic
275	C₂×C₅	C₃×#418	1	1	yes	characteristic
276	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	—	8	2	no	—
299	C₂×C₃	C₅×#408	1	1	yes	characteristic
300	C₂×C₃	C₅×#418	1	1	yes	characteristic
301	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
302	C₅	C₃×#1484	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	C₃×C₅×#104	1	2	yes	—
315	C₄	C₃×C₅×#131	1	2	yes	—
316	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
317	C₃	C₅×#1484	1	1	yes	characteristic, direct product factor, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₃×C₅×#408	1	1	yes	characteristic
321	C₂	C₃×C₅×#418	1	1	yes	characteristic
322	C₂	C₂×C₃×C₅×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×C₅×#532	C₂
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
2. C₃×C₅×#431	C₂	semidirect
3. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. C₃×C₅×D₄⋊C₈	C₂	semidirect
5. C₅×#1484	C₃	direct (N × H)
21. C₃×#1484	C₅	direct (N × H)
23. C₅×#431	C₂×C₃	semidirect
24. C₄×C₅×M₁₆	C₂×C₃	semidirect
25. C₅×D₄⋊C₈	C₂×C₃	semidirect
48. C₃×#431	C₂×C₅	semidirect
49. C₃×C₄×M₁₆	C₂×C₅	semidirect
50. C₃×D₄⋊C₈	C₂×C₅	semidirect
66. #1484	C₃×C₅	direct (N × H)
124. #431	C₂×C₃×C₅	semidirect
125. C₄×M₁₆	C₂×C₃×C₅	semidirect
126. D₄⋊C₈	C₂×C₃×C₅	semidirect
257. C₃×C₅	#1484	direct (N × H)
302. C₅	C₃×#1484	direct (N × H)
317. C₃	C₅×#1484	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×#532	2
C₃×C₅×#431	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×D₄⋊C₈	2
C₅×#1484	3
C₃×#1484	5

Subgroups — C3×C5×#1484 #14274518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1484 #14274518