Subgroups — C₃×C₅×#1481 #14274515

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₅×#1481	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₃×C₅×#125	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
11	C₃×C₅×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₃×C₄×C₅×Q₈	—	2	1	no	—
13	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
14	C₃×C₅×#117	—	2	1	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	—	2	1	no	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₃×#1481	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#533	C₂×C₃	1	1	yes	characteristic
23	C₅×#431	C₂×C₃	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
25	C₅×Q₈⋊C₈	C₂×C₃	1	4	yes	complemented
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	4	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	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×#533	C₂×C₅	1	1	yes	characteristic
47	C₃×#431	C₂×C₅	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
49	C₃×Q₈⋊C₈	C₂×C₅	1	4	yes	complemented
50	C₂²×C₅×C₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
52	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
53	C₅×#125	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₅×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
58	C₅×#117	—	2	1	no	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₅×C₄⋊C₈	—	2	1	no	—
61	C₅×C₄⋊C₈	—	2	1	no	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
65	#1481	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	—	4	2	no	—
82	C₃×C₅×C₈	—	4	2	no	—
83	C₂²×C₃×C₈	—	2	1	no	—
84	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
85	C₃×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
86	C₃×#125	—	2	1	no	—
87	C₃×C₂²⋊Q₈	—	2	1	no	—
88	C₃×C₂²⋊Q₈	C₄×C₅	1	2	yes	—
89	C₃×C₄×Q₈	—	2	1	no	—
90	C₃×C₄×Q₈	C₂²×C₅	1	2	yes	—
91	C₃×#117	—	2	1	no	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₈	—	2	1	no	—
94	C₃×C₄⋊C₈	—	2	1	no	—
95	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
96	C₃×C₂²⋊C₈	—	2	1	no	—
97	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	4	2	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	#533	C₂×C₃×C₅	1	1	yes	characteristic
119	#431	C₂×C₃×C₅	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
121	Q₈⋊C₈	C₂×C₃×C₅	1	4	yes	complemented
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	#103	1	2	yes	—
125	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	—	2	1	no	—
128	C₃×C₄×C₅	—	2	1	no	—
129	C₃×C₄×C₅	—	4	2	no	—
130	C₃×C₄×C₅	—	4	2	no	—
131	C₃×C₄×C₅	#103	1	2	yes	—
132	C₃×C₄×C₅	#131	1	2	yes	—
133	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	—	4	2	no	—
143	C₃×C₄⋊C₄	—	2	1	no	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₄²	—	2	2	no	—
151	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
152	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
154	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
155	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₅×Q₈	—	2	4	no	—
157	C₅×Q₈	—	4	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
166	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
167	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₅×C₈	—	4	2	no	—
171	C₅×C₈	—	4	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
174	C₄⋊Q₈	C₃×C₄×C₅	1	1	yes	characteristic
175	#125	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	C₂²⋊Q₈	C₃×C₄×C₅	1	2	yes	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	C₂²×C₃×C₅	1	2	yes	—
180	#117	—	2	1	no	—
181	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
187	C₂×C₃×C₅	—	2	2	no	—
188	C₂×C₃×C₅	#408	1	1	yes	characteristic
189	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
190	C₂×C₃×C₅	#421	1	1	yes	characteristic
191	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₃×Q₈	—	2	4	no	—
193	C₃×Q₈	—	4	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
202	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
203	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×C₈	—	4	2	no	—
207	C₃×C₈	—	4	2	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	C₃×#103	1	2	yes	—
211	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	4	2	no	—
216	C₄×C₅	—	4	2	no	—
217	C₄×C₅	C₃×#103	1	2	yes	—
218	C₄×C₅	C₃×#131	1	2	yes	—
219	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
220	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
238	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
239	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
240	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
241	C₃×C₅	#1481	1	1	yes	characteristic, direct product factor, Hall
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	C₅×#103	1	2	yes	—
245	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	4	2	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	C₅×#103	1	2	yes	—
252	C₃×C₄	C₅×#131	1	2	yes	—
253	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
254	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	C₃×#408	1	1	yes	characteristic
257	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
258	C₂×C₅	C₃×#421	1	1	yes	characteristic
259	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
270	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
271	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
272	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	C₅×#408	1	1	yes	characteristic
278	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
279	C₂×C₃	C₅×#421	1	1	yes	characteristic
280	C₅	C₃×#1481	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₃×C₅×#103	1	2	yes	—
284	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	C₃×C₅×#103	1	2	yes	—
291	C₄	C₃×C₅×#131	1	2	yes	—
292	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
293	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
294	C₃	C₅×#1481	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	2	no	—
296	C₂	C₃×C₅×#408	1	1	yes	characteristic
297	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
298	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	C₂³×C₃×C₅	C₂²⋊C₄
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
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. C₅×#1481	C₃	direct (N × H)
21. C₃×#1481	C₅	direct (N × H)
25. C₅×Q₈⋊C₈	C₂×C₃	semidirect
49. C₃×Q₈⋊C₈	C₂×C₅	semidirect
65. #1481	C₃×C₅	direct (N × H)
121. Q₈⋊C₈	C₂×C₃×C₅	semidirect
241. C₃×C₅	#1481	direct (N × H)
280. C₅	C₃×#1481	direct (N × H)
294. C₃	C₅×#1481	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×#431	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₅×Q₈⋊C₈	2
C₅×#1481	3
C₃×#1481	5

Subgroups — C3×C5×#1481 #14274515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1481 #14274515