Subgroups — C₂×C₃×D₅×D₁₆ #14478195

Order 1920.

880 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×D₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3958	C₂	1	2	yes	maximal, complemented
4	C₃×D₅×D₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₅×D₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×D₄×D₅	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
10	C₂×C₃×#686	—	2	2	no	—
11	C₃×D₅×D₈	—	2	4	no	—
12	C₃×D₅×D₈	C₂²	1	8	yes	complemented
13	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
14	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₃×#3958	C₂²	1	8	yes	complemented
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic, complemented
19	C₃×D₈₀	C₂²	1	4	yes	complemented
20	C₃×D₅×C₁₆	C₂²	1	4	yes	complemented
21	C₂²×C₃×D₁₆	—	5	1	no	maximal, Hall
22	C₂×D₅×D₈	C₂×C₃	1	2	yes	complemented
23	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3958	C₂×C₃	1	2	yes	complemented
25	D₅×D₁₆	C₂×C₃	1	8	yes	direct product factor
26	C₂×D₈₀	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×D₅	—	4	2	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₂²⋊D₅	—	4	2	no	—
31	C₃×D₄×D₅	—	2	8	no	—
32	C₃×D₄×D₅	—	4	4	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
35	C₃×C₅×D₈	—	2	2	no	—
36	C₃×C₅×D₈	C₂³	1	4	yes	complemented
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×#686	—	2	8	no	—
39	C₂×C₃×#168	D₄	1	1	yes	characteristic
40	C₃×D₄₀	—	2	2	no	—
41	C₃×D₄₀	C₂³	1	4	yes	complemented
42	C₃×C₈×D₅	D₄	1	2	yes	—
43	C₃×C₈×D₅	C₂³	1	2	yes	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	complemented
45	C₃×#672	C₂³	1	2	yes	complemented
46	C₂²×C₃×D₈	—	5	2	no	—
47	C₂×C₃×D₁₆	—	5	2	no	—
48	C₂×C₃×D₁₆	—	5	8	no	—
49	C₂×C₃×D₁₆	—	5	1	no	—
50	C₂×C₃×D₁₆	D₅	1	1	yes	characteristic, direct product factor
51	C₂²×C₃×C₁₆	—	5	1	no	—
52	C₂×D₄×D₅	—	2	2	no	—
53	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
54	C₂×#686	—	2	2	no	—
55	D₅×D₈	—	2	4	no	—
56	D₅×D₈	C₂²×C₃	1	8	yes	complemented
57	C₂×D₄₀	C₂²×C₃	1	2	yes	complemented
58	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
59	C₅×D₁₆	C₂²×C₃	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
61	#3958	C₂²×C₃	1	8	yes	complemented
62	C₂×#672	C₂²×C₃	1	1	yes	characteristic, complemented
63	D₈₀	C₂²×C₃	1	4	yes	complemented
64	D₅×C₁₆	C₂²×C₃	1	4	yes	complemented
65	C₂²×D₁₆	—	5	1	no	Sylow
66	C₂³×C₃×C₅	—	4	2	no	—
67	C₂²×C₃×D₅	—	4	2	no	—
68	C₂²×C₃×D₅	—	4	8	no	—
69	C₂²×C₃×D₅	—	8	8	no	—
70	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
71	C₃×C₅×D₄	—	2	4	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₂²⋊D₅	—	4	8	no	—
75	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
76	C₃×D₂₀	—	2	4	no	—
77	C₃×D₂₀	—	4	2	no	—
78	C₃×C₄×D₅	D₈	1	2	yes	—
79	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
80	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
82	C₃×#168	C₂×D₄	1	2	yes	—
83	C₂²×C₃×D₄	—	10	2	no	—
84	C₂×C₃×D₈	—	5	8	no	—
85	C₂×C₃×D₈	—	5	2	no	—
86	C₂×C₃×D₈	—	10	4	no	—
87	C₂×C₃×D₈	—	10	2	no	—
88	C₂×C₃×D₈	C₂×D₅	1	2	yes	complemented
89	C₂²×C₃×C₈	—	5	1	no	—
90	C₃×D₁₆	—	5	8	no	—
91	C₃×D₁₆	—	5	4	no	—
92	C₃×D₁₆	C₂×D₅	1	4	yes	direct product factor
93	C₂×C₃×C₁₆	—	5	1	no	—
94	C₂×C₃×C₁₆	—	5	4	no	—
95	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
96	C₂³×D₅	—	4	2	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₂²⋊D₅	—	4	2	no	—
99	D₄×D₅	—	2	8	no	—
100	D₄×D₅	—	4	4	no	—
101	C₂×D₂₀	—	2	2	no	—
102	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
103	C₅×D₈	—	2	2	no	—
104	C₅×D₈	C₂³×C₃	1	4	yes	complemented
105	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
106	#686	—	2	8	no	—
107	C₂×#168	C₃×D₄	1	1	yes	characteristic
108	D₄₀	—	2	2	no	—
109	D₄₀	C₂³×C₃	1	4	yes	complemented
110	C₈×D₅	C₃×D₄	1	2	yes	—
111	C₈×D₅	C₂³×C₃	1	2	yes	—
112	C₅×C₁₆	C₂³×C₃	1	2	yes	complemented
113	#672	C₂³×C₃	1	2	yes	complemented
114	C₂²×D₈	—	5	2	no	—
115	C₂×D₁₆	—	5	2	no	—
116	C₂×D₁₆	—	5	8	no	—
117	C₂×D₁₆	—	5	1	no	—
118	C₂×D₁₆	C₃×D₅	1	1	yes	characteristic, direct product factor
119	C₂²×C₁₆	—	5	1	no	—
120	C₂²×C₃×C₅	—	4	4	no	—
121	C₂²×C₃×C₅	—	8	4	no	—
122	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₃×D₅	—	4	4	no	—
124	C₂×C₃×D₅	—	8	4	no	—
125	C₂×C₃×D₅	—	8	16	no	—
126	C₂×C₃×D₅	D₁₆	1	4	yes	direct product factor
127	C₂×C₃×D₅	C₂×D₈	1	2	yes	—
128	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₃×Dic₅	C₂×D₈	1	2	yes	—
131	C₂⁴×C₃	—	20	2	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	—	10	8	no	—
134	C₂×C₃×D₄	—	10	2	no	—
135	C₂×C₃×D₄	—	20	4	no	—
136	C₂×C₃×D₄	—	20	2	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₃×D₈	—	2	2	no	—
139	C₃×D₈	—	5	4	no	—
140	C₃×D₈	—	10	8	no	—
141	C₃×D₈	—	10	2	no	—
142	C₃×D₈	C₂²×D₅	1	4	yes	complemented
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	—	5	2	no	—
145	C₂×C₃×C₈	—	5	2	no	—
146	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
147	C₃×C₁₆	—	5	2	no	—
148	C₃×C₁₆	C₂²×D₅	1	2	yes	complemented
149	C₂³×C₅	—	4	2	no	—
150	C₂²×D₅	—	4	2	no	—
151	C₂²×D₅	—	4	8	no	—
152	C₂²×D₅	—	8	8	no	—
153	C₂²×D₅	C₃×D₈	1	1	yes	characteristic
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	4	2	no	—
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂²⋊D₅	—	4	8	no	—
158	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
159	D₂₀	—	2	4	no	—
160	D₂₀	—	4	2	no	—
161	C₄×D₅	C₃×D₈	1	2	yes	—
162	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
163	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
164	C₅×C₈	C₂⁴×C₃	1	1	yes	characteristic
165	#168	C₂×C₃×D₄	1	2	yes	—
166	C₂²×D₄	—	10	2	no	—
167	C₂×D₈	—	5	8	no	—
168	C₂×D₈	—	5	2	no	—
169	C₂×D₈	—	10	4	no	—
170	C₂×D₈	—	10	2	no	—
171	C₂×D₈	C₂×C₃×D₅	1	2	yes	complemented
172	C₂²×C₈	—	5	1	no	—
173	D₁₆	—	5	8	no	—
174	D₁₆	—	5	4	no	—
175	D₁₆	C₂×C₃×D₅	1	4	yes	direct product factor
176	C₂×C₁₆	—	5	1	no	—
177	C₂×C₁₆	—	5	4	no	—
178	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic, complemented
179	C₂×C₃×C₅	—	8	4	no	—
180	C₂×C₃×C₅	C₂×D₁₆	1	2	yes	complemented
181	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
182	C₃×D₅	—	8	4	no	—
183	C₃×D₅	C₂×D₁₆	1	4	yes	direct product factor
184	C₂³×C₃	—	4	2	no	—
185	C₂³×C₃	—	5	1	no	—
186	C₂³×C₃	—	20	8	no	—
187	C₂³×C₃	—	20	2	no	—
188	C₂³×C₃	—	40	8	no	—
189	C₃×D₄	—	2	4	no	—
190	C₃×D₄	—	4	2	no	—
191	C₃×D₄	—	10	4	no	—
192	C₃×D₄	—	20	8	no	—
193	C₃×D₄	—	20	2	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	2	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
198	C₃×C₈	—	5	2	no	—
199	C₃×C₈	D₄×D₅	1	1	yes	characteristic
200	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
201	C₂²×C₅	—	4	4	no	—
202	C₂²×C₅	—	8	4	no	—
203	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
204	C₂×D₅	—	4	4	no	—
205	C₂×D₅	—	8	4	no	—
206	C₂×D₅	—	8	16	no	—
207	C₂×D₅	C₃×D₁₆	1	4	yes	direct product factor
208	C₂×D₅	C₂×C₃×D₈	1	2	yes	—
209	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
210	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
211	Dic₅	C₂×C₃×D₈	1	2	yes	—
212	C₂⁴	—	20	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	10	8	no	—
215	C₂×D₄	—	10	2	no	—
216	C₂×D₄	—	20	4	no	—
217	C₂×D₄	—	20	2	no	—
218	C₂²×C₄	—	5	1	no	—
219	D₈	—	2	2	no	—
220	D₈	—	5	4	no	—
221	D₈	—	10	8	no	—
222	D₈	—	10	2	no	—
223	D₈	C₂²×C₃×D₅	1	4	yes	complemented
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
228	C₁₆	—	5	2	no	—
229	C₁₆	C₂²×C₃×D₅	1	2	yes	complemented
230	C₃×C₅	C₂²×D₁₆	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	4	4	no	—
232	C₂²×C₃	—	5	2	no	—
233	C₂²×C₃	—	5	4	no	—
234	C₂²×C₃	—	8	4	no	—
235	C₂²×C₃	—	20	4	no	—
236	C₂²×C₃	—	40	16	no	—
237	C₂²×C₃	—	40	4	no	—
238	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
239	C₃×C₄	—	5	2	no	—
240	C₃×C₄	D₅×D₈	1	1	yes	characteristic
241	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
242	C₂×C₅	—	8	4	no	—
243	C₂×C₅	C₂×C₃×D₁₆	1	2	yes	complemented
244	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
245	D₅	—	8	4	no	—
246	D₅	C₂×C₃×D₁₆	1	4	yes	direct product factor
247	C₂³	—	4	2	no	—
248	C₂³	—	5	1	no	—
249	C₂³	—	20	8	no	—
250	C₂³	—	20	2	no	—
251	C₂³	—	40	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	2	no	—
254	D₄	—	10	4	no	—
255	D₄	—	20	8	no	—
256	D₄	—	20	2	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	5	2	no	—
260	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
261	C₈	—	5	2	no	—
262	C₈	C₃×D₄×D₅	1	1	yes	characteristic
263	C₈	C₂³×C₃×D₅	1	1	yes	characteristic
264	C₂×C₃	—	5	4	no	—
265	C₂×C₃	—	8	4	no	—
266	C₂×C₃	—	40	4	no	—
267	C₂×C₃	D₅×D₁₆	1	2	yes	direct product factor
268	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
269	C₅	C₂²×C₃×D₁₆	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	4	4	no	—
271	C₂²	—	5	4	no	—
272	C₂²	—	5	2	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	20	4	no	—
275	C₂²	—	40	16	no	—
276	C₂²	—	40	4	no	—
277	C₂²	C₃×D₅×D₈	1	1	yes	characteristic
278	C₄	—	5	2	no	—
279	C₄	C₃×D₅×D₈	1	1	yes	characteristic
280	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
281	C₃	C₂×D₅×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	5	4	no	—
283	C₂	—	8	4	no	—
284	C₂	—	40	4	no	—
285	C₂	C₃×D₅×D₁₆	1	2	yes	direct product factor
286	C₂	C₂×C₃×D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×D₈
commutator, cosocle	C₅×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₃×D₁₆	D₅
hypocenter	C₅	C₂²×C₃×D₁₆
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₅×D₈	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₂×C₃×#3958	C₂	semidirect
4. C₃×D₅×D₁₆	C₂	direct (N × H)
5. C₂×C₃×D₈₀	C₂	semidirect
6. C₂×C₃×D₅×C₁₆	C₂	semidirect
7. C₂×D₅×D₁₆	C₃	direct (N × H)
9. C₂×C₃×C₅×D₈	C₂²	semidirect
12. C₃×D₅×D₈	C₂²	semidirect
13. C₂×C₃×D₄₀	C₂²	semidirect
15. C₃×C₅×D₁₆	C₂²	semidirect
16. C₂×C₃×C₅×C₁₆	C₂²	semidirect
17. C₃×#3958	C₂²	semidirect
18. C₂×C₃×#672	C₂²	semidirect
19. C₃×D₈₀	C₂²	semidirect
20. C₃×D₅×C₁₆	C₂²	semidirect
22. C₂×D₅×D₈	C₂×C₃	semidirect
23. C₂×C₅×D₁₆	C₂×C₃	semidirect
24. C₂×#3958	C₂×C₃	semidirect
25. D₅×D₁₆	C₂×C₃	direct (N × H)
26. C₂×D₈₀	C₂×C₃	semidirect
27. C₂×D₅×C₁₆	C₂×C₃	semidirect
36. C₃×C₅×D₈	C₂³	semidirect
41. C₃×D₄₀	C₂³	semidirect
44. C₃×C₅×C₁₆	C₂³	semidirect
45. C₃×#672	C₂³	semidirect
50. C₂×C₃×D₁₆	D₅	direct (N × H)
53. C₂×C₅×D₈	C₂²×C₃	semidirect
56. D₅×D₈	C₂²×C₃	semidirect
57. C₂×D₄₀	C₂²×C₃	semidirect
59. C₅×D₁₆	C₂²×C₃	semidirect
60. C₂×C₅×C₁₆	C₂²×C₃	semidirect
61. #3958	C₂²×C₃	semidirect
62. C₂×#672	C₂²×C₃	semidirect
63. D₈₀	C₂²×C₃	semidirect
64. D₅×C₁₆	C₂²×C₃	semidirect
88. C₂×C₃×D₈	C₂×D₅	semidirect
92. C₃×D₁₆	C₂×D₅	direct (N × H)
95. C₂×C₃×C₁₆	C₂×D₅	semidirect
104. C₅×D₈	C₂³×C₃	semidirect
109. D₄₀	C₂³×C₃	semidirect
112. C₅×C₁₆	C₂³×C₃	semidirect
113. #672	C₂³×C₃	semidirect
118. C₂×D₁₆	C₃×D₅	direct (N × H)
126. C₂×C₃×D₅	D₁₆	direct (N × H)
142. C₃×D₈	C₂²×D₅	semidirect
148. C₃×C₁₆	C₂²×D₅	semidirect
171. C₂×D₈	C₂×C₃×D₅	semidirect
175. D₁₆	C₂×C₃×D₅	direct (N × H)
178. C₂×C₁₆	C₂×C₃×D₅	semidirect
180. C₂×C₃×C₅	C₂×D₁₆	semidirect
183. C₃×D₅	C₂×D₁₆	direct (N × H)
207. C₂×D₅	C₃×D₁₆	direct (N × H)
223. D₈	C₂²×C₃×D₅	semidirect
229. C₁₆	C₂²×C₃×D₅	semidirect
230. C₃×C₅	C₂²×D₁₆	semidirect
243. C₂×C₅	C₂×C₃×D₁₆	semidirect
246. D₅	C₂×C₃×D₁₆	direct (N × H)
267. C₂×C₃	D₅×D₁₆	direct (N × H)
269. C₅	C₂²×C₃×D₁₆	semidirect
281. C₃	C₂×D₅×D₁₆	direct (N × H)
285. C₂	C₃×D₅×D₁₆	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×D₅×D₈	2
C₂×C₃×C₅×D₁₆	2
C₂×C₃×#3958	2
C₃×D₅×D₁₆	2
C₂×C₃×D₈₀	2
C₂×C₃×D₅×C₁₆	2
C₂×D₅×D₁₆	3
C₂²×C₃×D₁₆	5

Subgroups — C2×C3×D5×D16 #14478195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₅×D₁₆ #14478195