Subgroups — C₃×C₅×#1496 #14274530

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₅×#1496	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×C₅×#127	C₄	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	1	no	—
10	C₃×C₅×#121	—	2	1	no	—
11	C₃×C₅×#121	C₄	1	2	yes	—
12	C₃×C₄×C₅×D₄	—	2	1	no	—
13	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₃×#1496	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	C₂×C₃	1	1	yes	characteristic
23	C₅×#432	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#431	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	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₃×#532	C₂×C₅	1	1	yes	characteristic
47	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#431	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×D₄⋊C₈	C₂×C₅	1	4	yes	complemented
50	C₂×C₅×#41	—	2	1	no	—
51	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
52	C₅×#127	C₃×C₄	1	1	yes	characteristic
53	C₅×#124	—	2	1	no	—
54	C₅×#121	—	2	1	no	—
55	C₅×#121	C₃×C₄	1	2	yes	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	2	1	no	—
60	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×#97	—	2	1	no	—
64	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
65	#1496	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	—	2	2	no	—
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₃×C₅×D₄	—	2	4	no	—
70	C₃×C₅×D₄	—	4	2	no	—
71	C₃×C₅×D₄	—	4	2	no	—
72	C₃×C₅×D₄	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	4	no	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	—	4	2	no	—
85	C₃×C₅×C₈	—	4	2	no	—
86	C₂×C₃×#41	—	2	1	no	—
87	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
88	C₃×#127	C₄×C₅	1	1	yes	characteristic
89	C₃×#124	—	2	1	no	—
90	C₃×#121	—	2	1	no	—
91	C₃×#121	C₄×C₅	1	2	yes	—
92	C₃×C₄×D₄	—	2	1	no	—
93	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄⋊C₈	—	2	1	no	—
96	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
97	C₃×C₂²⋊C₈	—	2	1	no	—
98	C₃×C₂²⋊C₈	—	2	1	no	—
99	C₃×#97	—	2	1	no	—
100	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
101	C₅×#41	—	4	4	no	—
102	C₂×C₅×Q₈	—	2	1	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	2	1	no	—
105	C₂×C₅×D₄	—	2	1	no	—
106	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₄×C₅	—	2	2	no	—
108	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
118	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
121	#532	C₂×C₃×C₅	1	1	yes	characteristic
122	#432	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	#431	C₂×C₃×C₅	1	1	yes	characteristic, complemented
124	D₄⋊C₈	C₂×C₃×C₅	1	4	yes	complemented
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	4	2	no	—
129	C₂²×C₃×C₅	—	4	2	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	#131	1	2	yes	—
138	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
139	C₃×#41	—	4	4	no	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×D₄	—	2	2	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	—	2	1	no	—
144	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₃×C₄	—	2	2	no	—
146	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
156	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
157	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₂³×C₅	—	2	2	no	—
160	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₅×Q₈	—	4	2	no	—
162	C₅×D₄	—	2	4	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	4	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	4	no	—
171	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	—	4	2	no	—
179	C₂×#41	—	2	1	no	—
180	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
181	#127	C₃×C₄×C₅	1	1	yes	characteristic
182	#124	—	2	1	no	—
183	#121	—	2	1	no	—
184	#121	C₃×C₄×C₅	1	2	yes	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
187	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
194	C₂×C₃×C₅	—	4	1	no	—
195	C₂×C₃×C₅	—	8	2	no	—
196	C₂×C₃×C₅	#408	1	1	yes	characteristic
197	C₂×C₃×C₅	#418	1	1	yes	characteristic
198	C₂×C₃×C₅	#421	1	1	yes	characteristic
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₃×Q₈	—	4	2	no	—
202	C₃×D₄	—	2	4	no	—
203	C₃×D₄	—	4	2	no	—
204	C₃×D₄	—	4	2	no	—
205	C₃×D₄	—	4	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	4	1	no	—
210	C₂×C₃×C₄	—	4	4	no	—
211	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₃×C₈	—	4	2	no	—
218	C₃×C₈	—	4	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	4	2	no	—
223	C₂²×C₅	—	4	2	no	—
224	C₂²×C₅	—	4	2	no	—
225	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	C₃×#131	1	2	yes	—
232	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
233	#41	—	4	4	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
242	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
250	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
251	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
252	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
253	C₃×C₅	#1496	1	1	yes	characteristic, direct product factor, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	4	2	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	C₅×#131	1	2	yes	—
267	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
268	C₂×C₅	—	4	1	no	—
269	C₂×C₅	—	8	2	no	—
270	C₂×C₅	C₃×#408	1	1	yes	characteristic
271	C₂×C₅	C₃×#418	1	1	yes	characteristic
272	C₂×C₅	C₃×#421	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
289	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₂×C₃	—	4	1	no	—
294	C₂×C₃	—	8	2	no	—
295	C₂×C₃	C₅×#408	1	1	yes	characteristic
296	C₂×C₃	C₅×#418	1	1	yes	characteristic
297	C₂×C₃	C₅×#421	1	1	yes	characteristic
298	C₅	C₃×#1496	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	C₃×C₅×#131	1	2	yes	—
312	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
313	C₃	C₅×#1496	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	4	1	no	—
315	C₂	—	8	2	no	—
316	C₂	C₃×C₅×#408	1	1	yes	characteristic
317	C₂	C₃×C₅×#418	1	1	yes	characteristic
318	C₂	C₃×C₅×#421	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×#432	C₂	semidirect
3. C₃×C₅×#431	C₂	semidirect
4. C₃×C₅×D₄⋊C₈	C₂	semidirect
5. C₅×#1496	C₃	direct (N × H)
21. C₃×#1496	C₅	direct (N × H)
23. C₅×#432	C₂×C₃	semidirect
24. C₅×#431	C₂×C₃	semidirect
25. C₅×D₄⋊C₈	C₂×C₃	semidirect
47. C₃×#432	C₂×C₅	semidirect
48. C₃×#431	C₂×C₅	semidirect
49. C₃×D₄⋊C₈	C₂×C₅	semidirect
65. #1496	C₃×C₅	direct (N × H)
122. #432	C₂×C₃×C₅	semidirect
123. #431	C₂×C₃×C₅	semidirect
124. D₄⋊C₈	C₂×C₃×C₅	semidirect
253. C₃×C₅	#1496	direct (N × H)
298. C₅	C₃×#1496	direct (N × H)
313. C₃	C₅×#1496	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×#432	2
C₃×C₅×#431	2
C₃×C₅×D₄⋊C₈	2
C₅×#1496	3
C₃×#1496	5

Subgroups — C3×C5×#1496 #14274530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1496 #14274530