Subgroups — C₃×C₅×#1559 #14274593

Order 1920.

404 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#333	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#332	C₂	1	2	yes	maximal, complemented
6	C₅×#1559	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	1	no	—
11	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
12	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₃×C₄×C₅×Q₈	—	2	1	no	—
14	C₃×C₄×C₅×Q₈	C₄	1	2	yes	—
15	C₃×C₅×#117	—	2	1	no	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	—	2	1	no	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
21	C₃×C₅×#97	—	2	1	no	—
22	C₃×#1559	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#533	C₂×C₃	1	1	yes	characteristic
24	C₅×#432	C₂×C₃	1	1	yes	characteristic
25	C₅×#423	C₂×C₃	1	1	yes	characteristic
26	C₅×#333	C₂×C₃	1	2	yes	complemented
27	C₅×#332	C₂×C₃	1	2	yes	complemented
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×M₁₆	—	4	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
43	C₃×C₄²×C₅	—	2	2	no	—
44	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#533	C₂×C₅	1	1	yes	characteristic
49	C₃×#432	C₂×C₅	1	1	yes	characteristic
50	C₃×#423	C₂×C₅	1	1	yes	characteristic
51	C₃×#333	C₂×C₅	1	2	yes	complemented
52	C₃×#332	C₂×C₅	1	2	yes	complemented
53	C₂×C₅×M₁₆	—	2	1	no	—
54	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₅×#125	—	2	1	no	—
57	C₅×C₂²⋊Q₈	—	2	1	no	—
58	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×Q₈	C₃×C₄	1	2	yes	—
61	C₅×#117	—	2	1	no	—
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	—	2	1	no	—
64	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
65	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
66	C₅×C₂²⋊C₈	—	2	1	no	—
67	C₅×#97	—	2	1	no	—
68	#1559	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	1	no	—
72	C₃×C₅×Q₈	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	—	2	1	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	1	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	—	4	2	no	—
86	C₃×C₅×C₈	—	4	2	no	—
87	C₂×C₃×M₁₆	—	2	1	no	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
89	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
90	C₃×#125	—	2	1	no	—
91	C₃×C₂²⋊Q₈	—	2	1	no	—
92	C₃×C₂²⋊Q₈	C₄×C₅	1	2	yes	—
93	C₃×C₄×Q₈	—	2	1	no	—
94	C₃×C₄×Q₈	C₄×C₅	1	2	yes	—
95	C₃×#117	—	2	1	no	—
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	2	1	no	—
98	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
99	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
100	C₃×C₂²⋊C₈	—	2	1	no	—
101	C₃×#97	—	2	1	no	—
102	C₂×C₅×Q₈	—	2	1	no	—
103	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₅×M₁₆	—	4	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
119	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
122	#533	C₂×C₃×C₅	1	1	yes	characteristic
123	#432	C₂×C₃×C₅	1	1	yes	characteristic
124	#423	C₂×C₃×C₅	1	1	yes	characteristic
125	#333	C₂×C₃×C₅	1	2	yes	complemented
126	#332	C₂×C₃×C₅	1	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₃×C₄×C₅	—	2	1	no	—
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	—	4	2	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₃×C₄×C₅	#137	1	1	yes	characteristic
139	C₃×C₄×C₅	#137	1	2	yes	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
142	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
145	C₃×M₁₆	—	4	2	no	—
146	C₂×C₃×C₈	—	2	2	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₄²	—	2	2	no	—
156	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
157	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
160	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	4	1	no	—
163	C₅×Q₈	—	4	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	1	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	C₃×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₅×C₈	—	4	2	no	—
177	C₅×C₈	—	4	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
180	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
181	#125	—	2	1	no	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	C₃×C₄×C₅	1	2	yes	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₃×C₄×C₅	1	2	yes	—
186	#117	—	2	1	no	—
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	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
191	C₂²⋊C₈	—	2	1	no	—
192	#97	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	#413	1	1	yes	characteristic
195	C₂×C₃×C₅	#417	1	1	yes	characteristic
196	C₂×C₃×C₅	#419	1	1	yes	characteristic
197	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₃×Q₈	—	4	2	no	—
199	C₃×Q₈	—	4	1	no	—
200	C₃×Q₈	—	4	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₃×C₈	—	4	2	no	—
214	C₃×C₈	—	4	2	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	2	1	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₅	C₃×#137	1	1	yes	characteristic
227	C₄×C₅	C₃×#137	1	2	yes	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
230	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
245	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
246	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
247	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
248	C₃×C₅	#1559	1	1	yes	characteristic, direct product factor, Hall
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	4	2	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₃×C₄	C₅×#137	1	1	yes	characteristic
261	C₃×C₄	C₅×#137	1	2	yes	—
262	C₂×C₅	—	4	1	no	—
263	C₂×C₅	C₃×#413	1	1	yes	characteristic
264	C₂×C₅	C₃×#417	1	1	yes	characteristic
265	C₂×C₅	C₃×#419	1	1	yes	characteristic
266	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	C₅×#413	1	1	yes	characteristic
286	C₂×C₃	C₅×#417	1	1	yes	characteristic
287	C₂×C₃	C₅×#419	1	1	yes	characteristic
288	C₅	C₃×#1559	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₄	C₃×C₅×#137	1	1	yes	characteristic
301	C₄	C₃×C₅×#137	1	2	yes	—
302	C₃	C₅×#1559	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	4	1	no	—
304	C₂	C₃×C₅×#413	1	1	yes	characteristic
305	C₂	C₃×C₅×#417	1	1	yes	characteristic
306	C₂	C₃×C₅×#419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#333	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1559	C₃	direct (N × H)
22. C₃×#1559	C₅	direct (N × H)
26. C₅×#333	C₂×C₃	semidirect
27. C₅×#332	C₂×C₃	semidirect
51. C₃×#333	C₂×C₅	semidirect
52. C₃×#332	C₂×C₅	semidirect
68. #1559	C₃×C₅	direct (N × H)
125. #333	C₂×C₃×C₅	semidirect
126. #332	C₂×C₃×C₅	semidirect
248. C₃×C₅	#1559	direct (N × H)
288. C₅	C₃×#1559	direct (N × H)
302. C₃	C₅×#1559	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×#533	2
C₃×C₅×#432	2
C₃×C₅×#423	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1559	3
C₃×#1559	5

Subgroups — C3×C5×#1559 #14274593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1559 #14274593