Subgroups — C₂×C₃×C₅×#487 #14276163

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#487	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#103	C₂	1	4	yes	maximal
5	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#487	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	—	2	1	no	—
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
11	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
15	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
16	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#103	C₂²	1	16	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
21	C₂×C₃×#487	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₅×#487	C₂×C₃	1	8	yes	direct product factor
25	C₂×C₅×#103	C₂×C₃	1	4	yes	—
26	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₅×Q₈	—	2	4	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	2	8	no	—
30	C₂×C₃×C₅×Q₈	—	4	4	no	—
31	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	4	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	4	no	—
40	C₃×C₅×C₄⋊C₄	—	2	4	no	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₄²×C₅	D₄	1	2	yes	—
45	C₃×C₄²×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
48	C₃×#487	C₂×C₅	1	8	yes	direct product factor
49	C₂×C₃×#103	C₂×C₅	1	4	yes	—
50	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
51	C₂²×C₅×Q₈	—	2	1	no	—
52	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
53	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
54	C₅×C₄⋊Q₈	—	2	2	no	—
55	C₅×C₄⋊Q₈	—	2	2	no	—
56	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
58	C₂×C₅×C₄⋊C₄	—	2	1	no	—
59	C₂×C₅×C₄⋊C₄	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	—	2	1	no	—
61	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₅×#103	C₂²×C₃	1	16	yes	—
64	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
65	C₂×#487	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	8	no	—
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₃×C₅×Q₈	—	4	8	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
75	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₃×C₅×C₈	—	2	4	no	—
82	C₂²×C₃×Q₈	—	2	1	no	—
83	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
84	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
85	C₃×C₄⋊Q₈	—	2	2	no	—
86	C₃×C₄⋊Q₈	—	2	2	no	—
87	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
89	C₂×C₃×C₄⋊C₄	—	2	1	no	—
90	C₂×C₃×C₄⋊C₄	—	2	1	no	—
91	C₂×C₃×C₄⋊C₄	—	2	1	no	—
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
94	C₃×#103	C₂²×C₅	1	16	yes	—
95	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	2	4	no	—
98	C₂×C₅×Q₈	—	2	8	no	—
99	C₂×C₅×Q₈	—	4	4	no	—
100	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₅×C₈	—	2	4	no	—
107	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
113	C₄²×C₅	C₃×D₄	1	2	yes	—
114	C₄²×C₅	C₂³×C₃	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
116	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
117	#487	C₂×C₃×C₅	1	8	yes	direct product factor
118	C₂×#103	C₂×C₃×C₅	1	4	yes	—
119	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
120	C₂²×C₃×C₅	#124	1	2	yes	—
121	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
122	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
123	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₃×C₄×C₅	—	4	4	no	—
125	C₃×C₄×C₅	—	4	4	no	—
126	C₃×C₄×C₅	#124	1	2	yes	—
127	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
128	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
129	C₃×C₄×C₅	C₂×#41	1	2	yes	—
130	C₂×C₃×Q₈	—	2	2	no	—
131	C₂×C₃×Q₈	—	2	4	no	—
132	C₂×C₃×Q₈	—	2	8	no	—
133	C₂×C₃×Q₈	—	4	4	no	—
134	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₃×C₈	—	2	4	no	—
141	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
142	C₃×C₄⋊C₄	—	2	4	no	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
147	C₃×C₄²	C₅×D₄	1	2	yes	—
148	C₃×C₄²	C₂³×C₅	1	2	yes	—
149	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₅×Q₈	—	2	8	no	—
151	C₅×Q₈	—	4	4	no	—
152	C₅×Q₈	—	4	8	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
158	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₃×#41	1	4	yes	—
163	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
164	C₅×C₈	—	2	4	no	—
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
167	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
171	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
176	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
177	#103	C₂²×C₃×C₅	1	16	yes	—
178	C₄×C₈	C₂²×C₃×C₅	1	4	yes	—
179	C₂×C₃×C₅	#487	1	4	yes	direct product factor
180	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
181	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
182	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
183	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
184	C₃×Q₈	—	2	8	no	—
185	C₃×Q₈	—	4	4	no	—
186	C₃×Q₈	—	4	8	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	4	4	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
192	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
195	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
196	C₂×C₃×C₄	C₅×#41	1	4	yes	—
197	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
198	C₃×C₈	—	2	4	no	—
199	C₂²×C₅	C₃×#124	1	2	yes	—
200	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
201	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
202	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	C₃×#124	1	2	yes	—
206	C₄×C₅	C₂×C₃×SD₁₆	1	4	yes	—
207	C₄×C₅	C₂×C₃×Q₁₆	1	4	yes	—
208	C₄×C₅	C₂×C₃×#41	1	2	yes	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
226	C₄²	C₃×C₅×D₄	1	2	yes	—
227	C₄²	C₂³×C₃×C₅	1	2	yes	—
228	C₃×C₅	C₂×#487	1	1	yes	characteristic, direct product factor, Hall
229	C₂²×C₃	C₅×#124	1	2	yes	—
230	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
231	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
232	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
233	C₃×C₄	—	4	4	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	C₅×#124	1	2	yes	—
236	C₃×C₄	C₂×C₅×SD₁₆	1	4	yes	—
237	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	—
238	C₃×C₄	C₂×C₅×#41	1	2	yes	—
239	C₂×C₅	C₃×#487	1	4	yes	direct product factor
240	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
241	C₂×C₅	C₂²×C₃×SD₁₆	1	1	yes	characteristic
242	C₂×C₅	C₂²×C₃×Q₁₆	1	1	yes	characteristic
243	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
244	Q₈	—	2	8	no	—
245	Q₈	—	4	4	no	—
246	Q₈	—	4	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
252	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	—
253	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
255	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
256	C₂×C₄	C₃×C₅×#41	1	4	yes	—
257	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₂×C₃	C₅×#487	1	4	yes	direct product factor
260	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
261	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
262	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
263	C₅	C₂×C₃×#487	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	C₃×C₅×#124	1	2	yes	—
265	C₂²	C₂×C₃×C₅×SD₁₆	1	2	yes	—
266	C₂²	C₂×C₃×C₅×Q₁₆	1	2	yes	—
267	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	C₃×C₅×#124	1	2	yes	—
271	C₄	C₂×C₃×C₅×SD₁₆	1	4	yes	—
272	C₄	C₂×C₃×C₅×Q₁₆	1	4	yes	—
273	C₄	C₂×C₃×C₅×#41	1	2	yes	—
274	C₃	C₂×C₅×#487	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	C₃×C₅×#487	1	4	yes	direct product factor
276	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
277	C₂	C₂²×C₃×C₅×SD₁₆	1	1	yes	characteristic
278	C₂	C₂²×C₃×C₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#487	C₂	direct (N × H)
6. C₂×C₅×#487	C₃	direct (N × H)
21. C₂×C₃×#487	C₅	direct (N × H)
24. C₅×#487	C₂×C₃	direct (N × H)
48. C₃×#487	C₂×C₅	direct (N × H)
65. C₂×#487	C₃×C₅	direct (N × H)
117. #487	C₂×C₃×C₅	direct (N × H)
179. C₂×C₃×C₅	#487	direct (N × H)
228. C₃×C₅	C₂×#487	direct (N × H)
239. C₂×C₅	C₃×#487	direct (N × H)
259. C₂×C₃	C₅×#487	direct (N × H)
263. C₅	C₂×C₃×#487	direct (N × H)
274. C₃	C₂×C₅×#487	direct (N × H)
275. C₂	C₃×C₅×#487	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#487	2
C₂×C₃×C₅×#103	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#487	3
C₂×C₃×#487	5

Subgroups — C2×C3×C5×#487 #14276163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#487 #14276163