Subgroups — C₂×C₅×D₃×D₁₆ #14478194

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×C₅×D₃×D₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#826	C₂	1	2	yes	maximal, complemented
4	C₅×D₃×D₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×D₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₁₆	—	3	1	no	maximal, Hall
8	C₂×C₅×D₃×D₄	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
10	C₂×C₅×#213	—	2	2	no	—
11	C₅×D₃×D₈	—	2	4	no	—
12	C₅×D₃×D₈	C₂²	1	8	yes	complemented
13	C₂×C₅×D₂₄	C₂²	1	2	yes	complemented
14	C₂×C₅×D₃×C₈	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	C₅×#826	C₂²	1	8	yes	complemented
18	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₅×D₄₈	C₂²	1	4	yes	complemented
20	C₅×D₃×C₁₆	C₂²	1	4	yes	complemented
21	C₂×D₃×D₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₅×D₈	—	3	2	no	—
23	C₂×C₅×D₁₆	—	3	8	no	—
24	C₂×C₅×D₁₆	—	3	1	no	—
25	C₂×C₅×D₁₆	—	3	2	no	—
26	C₂×C₅×D₁₆	D₃	1	1	yes	characteristic, direct product factor
27	C₂²×C₅×C₁₆	—	3	1	no	—
28	C₂³×C₅×D₃	—	4	2	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₅×#67	—	4	2	no	—
31	C₅×D₃×D₄	—	2	8	no	—
32	C₅×D₃×D₄	—	4	4	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₄×C₅×D₃	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	C₅×#213	—	2	8	no	—
39	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
40	C₅×D₂₄	—	2	2	no	—
41	C₅×D₂₄	C₂³	1	4	yes	complemented
42	C₅×D₃×C₈	D₄	1	2	yes	—
43	C₅×D₃×C₈	C₂³	1	2	yes	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	complemented
45	C₅×C₃⋊C₁₆	C₂³	1	2	yes	complemented
46	C₂×D₃×D₈	C₂×C₅	1	2	yes	complemented
47	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#826	C₂×C₅	1	2	yes	complemented
49	D₃×D₁₆	C₂×C₅	1	8	yes	direct product factor
50	C₂×D₄₈	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×D₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂²×C₅×D₄	—	6	2	no	—
53	C₂×C₅×D₈	—	3	8	no	—
54	C₂×C₅×D₈	—	3	2	no	—
55	C₂×C₅×D₈	—	6	4	no	—
56	C₂×C₅×D₈	—	6	2	no	—
57	C₂×C₅×D₈	C₂×D₃	1	2	yes	complemented
58	C₂²×C₅×C₈	—	3	1	no	—
59	C₅×D₁₆	—	3	8	no	—
60	C₅×D₁₆	—	3	4	no	—
61	C₅×D₁₆	C₂×D₃	1	4	yes	direct product factor
62	C₂×C₅×C₁₆	—	3	1	no	—
63	C₂×C₅×C₁₆	—	3	4	no	—
64	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
65	C₂²×D₁₆	—	3	1	no	Sylow
66	C₂³×C₃×C₅	—	4	2	no	—
67	C₂²×C₅×D₃	—	4	2	no	—
68	C₂²×C₅×D₃	—	4	8	no	—
69	C₂²×C₅×D₃	—	8	8	no	—
70	C₂²×C₅×D₃	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	C₅×#67	—	4	8	no	—
75	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
76	C₅×D₁₂	—	2	4	no	—
77	C₅×D₁₂	—	4	2	no	—
78	C₄×C₅×D₃	D₈	1	2	yes	—
79	C₄×C₅×D₃	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	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
83	C₂×D₃×D₄	—	2	2	no	—
84	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
85	C₂×#213	—	2	2	no	—
86	D₃×D₈	—	2	4	no	—
87	D₃×D₈	C₂²×C₅	1	8	yes	complemented
88	C₂×D₂₄	C₂²×C₅	1	2	yes	complemented
89	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₃×D₁₆	C₂²×C₅	1	4	yes	complemented
91	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
92	#826	C₂²×C₅	1	8	yes	complemented
93	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
94	D₄₈	C₂²×C₅	1	4	yes	complemented
95	D₃×C₁₆	C₂²×C₅	1	4	yes	complemented
96	C₂⁴×C₅	—	12	2	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	6	8	no	—
99	C₂×C₅×D₄	—	6	2	no	—
100	C₂×C₅×D₄	—	12	4	no	—
101	C₂×C₅×D₄	—	12	2	no	—
102	C₂²×C₄×C₅	—	3	1	no	—
103	C₅×D₈	—	2	2	no	—
104	C₅×D₈	—	3	4	no	—
105	C₅×D₈	—	6	8	no	—
106	C₅×D₈	—	6	2	no	—
107	C₅×D₈	C₂²×D₃	1	4	yes	complemented
108	C₂×C₅×C₈	—	3	1	no	—
109	C₂×C₅×C₈	—	3	2	no	—
110	C₂×C₅×C₈	—	3	2	no	—
111	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₅×C₁₆	—	3	2	no	—
113	C₅×C₁₆	C₂²×D₃	1	2	yes	complemented
114	C₂²×D₈	—	3	2	no	—
115	C₂×D₁₆	—	3	2	no	—
116	C₂×D₁₆	—	3	8	no	—
117	C₂×D₁₆	—	3	1	no	—
118	C₂×D₁₆	C₅×D₃	1	1	yes	characteristic, direct product factor
119	C₂²×C₁₆	—	3	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×C₅×D₃	—	4	4	no	—
124	C₂×C₅×D₃	—	8	4	no	—
125	C₂×C₅×D₃	—	8	16	no	—
126	C₂×C₅×D₃	D₁₆	1	4	yes	direct product factor
127	C₂×C₅×D₃	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	C₅×Dic₃	C₂×D₈	1	2	yes	—
131	C₂³×D₃	—	4	2	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×#67	—	4	2	no	—
134	D₃×D₄	—	2	8	no	—
135	D₃×D₄	—	4	4	no	—
136	C₂×D₁₂	—	2	2	no	—
137	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
138	C₃×D₈	—	2	2	no	—
139	C₃×D₈	C₂³×C₅	1	4	yes	complemented
140	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
141	#213	—	2	8	no	—
142	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
143	D₂₄	—	2	2	no	—
144	D₂₄	C₂³×C₅	1	4	yes	complemented
145	D₃×C₈	C₅×D₄	1	2	yes	—
146	D₃×C₈	C₂³×C₅	1	2	yes	—
147	C₃×C₁₆	C₂³×C₅	1	2	yes	complemented
148	C₃⋊C₁₆	C₂³×C₅	1	2	yes	complemented
149	C₂³×C₅	—	3	1	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	12	8	no	—
152	C₂³×C₅	—	12	2	no	—
153	C₂³×C₅	—	24	8	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	6	4	no	—
157	C₅×D₄	—	12	8	no	—
158	C₅×D₄	—	12	2	no	—
159	C₂×C₄×C₅	—	3	1	no	—
160	C₂×C₄×C₅	—	3	2	no	—
161	C₂×C₄×C₅	—	3	2	no	—
162	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
163	C₅×C₈	—	3	2	no	—
164	C₅×C₈	D₃×D₄	1	1	yes	characteristic
165	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
166	C₂²×D₄	—	6	2	no	—
167	C₂×D₈	—	3	8	no	—
168	C₂×D₈	—	3	2	no	—
169	C₂×D₈	—	6	4	no	—
170	C₂×D₈	—	6	2	no	—
171	C₂×D₈	C₂×C₅×D₃	1	2	yes	complemented
172	C₂²×C₈	—	3	1	no	—
173	D₁₆	—	3	8	no	—
174	D₁₆	—	3	4	no	—
175	D₁₆	C₂×C₅×D₃	1	4	yes	direct product factor
176	C₂×C₁₆	—	3	1	no	—
177	C₂×C₁₆	—	3	4	no	—
178	C₂×C₁₆	C₂×C₅×D₃	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	C₅×D₃	—	8	4	no	—
183	C₅×D₃	C₂×D₁₆	1	4	yes	direct product factor
184	C₂³×C₃	—	4	2	no	—
185	C₂²×D₃	—	4	2	no	—
186	C₂²×D₃	—	4	8	no	—
187	C₂²×D₃	—	8	8	no	—
188	C₂²×D₃	C₅×D₈	1	1	yes	characteristic
189	C₃×D₄	—	2	4	no	—
190	C₃×D₄	—	4	2	no	—
191	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	#67	—	4	8	no	—
193	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
194	D₁₂	—	2	4	no	—
195	D₁₂	—	4	2	no	—
196	C₄×D₃	C₅×D₈	1	2	yes	—
197	C₄×D₃	C₂×C₅×D₄	1	2	yes	—
198	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
199	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
200	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
201	C₂²×C₅	—	3	4	no	—
202	C₂²×C₅	—	3	2	no	—
203	C₂²×C₅	—	4	4	no	—
204	C₂²×C₅	—	8	4	no	—
205	C₂²×C₅	—	12	4	no	—
206	C₂²×C₅	—	24	16	no	—
207	C₂²×C₅	—	24	4	no	—
208	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
209	C₄×C₅	—	3	2	no	—
210	C₄×C₅	D₃×D₈	1	1	yes	characteristic
211	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
212	C₂⁴	—	12	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	6	8	no	—
215	C₂×D₄	—	6	2	no	—
216	C₂×D₄	—	12	4	no	—
217	C₂×D₄	—	12	2	no	—
218	C₂²×C₄	—	3	1	no	—
219	D₈	—	2	2	no	—
220	D₈	—	3	4	no	—
221	D₈	—	6	8	no	—
222	D₈	—	6	2	no	—
223	D₈	C₂²×C₅×D₃	1	4	yes	complemented
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	3	2	no	—
227	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
228	C₁₆	—	3	2	no	—
229	C₁₆	C₂²×C₅×D₃	1	2	yes	complemented
230	C₃×C₅	C₂²×D₁₆	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	4	4	no	—
232	C₂²×C₃	—	8	4	no	—
233	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
234	C₂×D₃	—	4	4	no	—
235	C₂×D₃	—	8	4	no	—
236	C₂×D₃	—	8	16	no	—
237	C₂×D₃	C₅×D₁₆	1	4	yes	direct product factor
238	C₂×D₃	C₂×C₅×D₈	1	2	yes	—
239	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
240	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
241	Dic₃	C₂×C₅×D₈	1	2	yes	—
242	C₂×C₅	—	3	4	no	—
243	C₂×C₅	—	8	4	no	—
244	C₂×C₅	—	24	4	no	—
245	C₂×C₅	D₃×D₁₆	1	2	yes	direct product factor
246	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
247	C₂³	—	3	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	12	8	no	—
250	C₂³	—	12	2	no	—
251	C₂³	—	24	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	2	no	—
254	D₄	—	6	4	no	—
255	D₄	—	12	8	no	—
256	D₄	—	12	2	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
261	C₈	—	3	2	no	—
262	C₈	C₅×D₃×D₄	1	1	yes	characteristic
263	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
264	C₂×C₃	—	8	4	no	—
265	C₂×C₃	C₂×C₅×D₁₆	1	2	yes	complemented
266	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
267	D₃	—	8	4	no	—
268	D₃	C₂×C₅×D₁₆	1	4	yes	direct product factor
269	C₅	C₂×D₃×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	3	4	no	—
271	C₂²	—	3	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	12	4	no	—
275	C₂²	—	24	16	no	—
276	C₂²	—	24	4	no	—
277	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
278	C₄	—	3	2	no	—
279	C₄	C₅×D₃×D₈	1	1	yes	characteristic
280	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
281	C₃	C₂²×C₅×D₁₆	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	3	4	no	—
283	C₂	—	8	4	no	—
284	C₂	—	24	4	no	—
285	C₂	C₅×D₃×D₁₆	1	2	yes	direct product factor
286	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₅×D₁₆	D₃
hypocenter	C₃	C₂²×C₅×D₁₆
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₃×D₈	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₂×C₅×#826	C₂	semidirect
4. C₅×D₃×D₁₆	C₂	direct (N × H)
5. C₂×C₅×D₄₈	C₂	semidirect
6. C₂×C₅×D₃×C₁₆	C₂	semidirect
9. C₂×C₃×C₅×D₈	C₂²	semidirect
12. C₅×D₃×D₈	C₂²	semidirect
13. C₂×C₅×D₂₄	C₂²	semidirect
15. C₃×C₅×D₁₆	C₂²	semidirect
16. C₂×C₃×C₅×C₁₆	C₂²	semidirect
17. C₅×#826	C₂²	semidirect
18. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
19. C₅×D₄₈	C₂²	semidirect
20. C₅×D₃×C₁₆	C₂²	semidirect
21. C₂×D₃×D₁₆	C₅	direct (N × H)
26. C₂×C₅×D₁₆	D₃	direct (N × H)
36. C₃×C₅×D₈	C₂³	semidirect
41. C₅×D₂₄	C₂³	semidirect
44. C₃×C₅×C₁₆	C₂³	semidirect
45. C₅×C₃⋊C₁₆	C₂³	semidirect
46. C₂×D₃×D₈	C₂×C₅	semidirect
47. C₂×C₃×D₁₆	C₂×C₅	semidirect
48. C₂×#826	C₂×C₅	semidirect
49. D₃×D₁₆	C₂×C₅	direct (N × H)
50. C₂×D₄₈	C₂×C₅	semidirect
51. C₂×D₃×C₁₆	C₂×C₅	semidirect
57. C₂×C₅×D₈	C₂×D₃	semidirect
61. C₅×D₁₆	C₂×D₃	direct (N × H)
64. C₂×C₅×C₁₆	C₂×D₃	semidirect
84. C₂×C₃×D₈	C₂²×C₅	semidirect
87. D₃×D₈	C₂²×C₅	semidirect
88. C₂×D₂₄	C₂²×C₅	semidirect
90. C₃×D₁₆	C₂²×C₅	semidirect
91. C₂×C₃×C₁₆	C₂²×C₅	semidirect
92. #826	C₂²×C₅	semidirect
93. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
94. D₄₈	C₂²×C₅	semidirect
95. D₃×C₁₆	C₂²×C₅	semidirect
107. C₅×D₈	C₂²×D₃	semidirect
113. C₅×C₁₆	C₂²×D₃	semidirect
118. C₂×D₁₆	C₅×D₃	direct (N × H)
126. C₂×C₅×D₃	D₁₆	direct (N × H)
139. C₃×D₈	C₂³×C₅	semidirect
144. D₂₄	C₂³×C₅	semidirect
147. C₃×C₁₆	C₂³×C₅	semidirect
148. C₃⋊C₁₆	C₂³×C₅	semidirect
171. C₂×D₈	C₂×C₅×D₃	semidirect
175. D₁₆	C₂×C₅×D₃	direct (N × H)
178. C₂×C₁₆	C₂×C₅×D₃	semidirect
180. C₂×C₃×C₅	C₂×D₁₆	semidirect
183. C₅×D₃	C₂×D₁₆	direct (N × H)
223. D₈	C₂²×C₅×D₃	semidirect
229. C₁₆	C₂²×C₅×D₃	semidirect
230. C₃×C₅	C₂²×D₁₆	semidirect
237. C₂×D₃	C₅×D₁₆	direct (N × H)
245. C₂×C₅	D₃×D₁₆	direct (N × H)
265. C₂×C₃	C₂×C₅×D₁₆	semidirect
268. D₃	C₂×C₅×D₁₆	direct (N × H)
269. C₅	C₂×D₃×D₁₆	direct (N × H)
281. C₃	C₂²×C₅×D₁₆	semidirect
285. C₂	C₅×D₃×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×C₅×D₃×D₈	2
C₂×C₃×C₅×D₁₆	2
C₂×C₅×#826	2
C₅×D₃×D₁₆	2
C₂×C₅×D₄₈	2
C₂×C₅×D₃×C₁₆	2
C₂²×C₅×D₁₆	3
C₂×D₃×D₁₆	5

Subgroups — C2×C5×D3×D16 #14478194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₃×D₁₆ #14478194