Subgroups — C₂×D₃×D₈₀ #14478209

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×D₃×D₄₀	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90938	C₂	1	2	yes	maximal, complemented
6	D₃×D₈₀	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₈₀	—	3	1	no	maximal, Hall
8	C₂×D₃×D₂₀	—	2	2	no	—
9	C₂×D₁₂₀	C₂²	1	2	yes	complemented
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
12	C₂×#6653	—	2	2	no	—
13	D₃×D₄₀	—	2	4	no	—
14	D₃×D₄₀	C₂²	1	8	yes	complemented
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
16	D₂₄₀	C₂²	1	4	yes	complemented
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₅×D₃×C₁₆	C₂²	1	4	yes	complemented
19	C₃×D₈₀	C₂²	1	4	yes	complemented
20	#90938	C₂²	1	8	yes	complemented
21	C₂×D₃×D₁₆	—	5	1	no	maximal, Hall
22	C₂²×D₄₀	—	3	2	no	—
23	C₂²×C₅×C₁₆	—	3	1	no	—
24	C₂×D₈₀	—	3	2	no	—
25	C₂×D₈₀	—	3	1	no	—
26	C₂×D₈₀	—	3	8	no	—
27	C₂×D₈₀	D₃	1	1	yes	characteristic, direct product factor
28	C₂²×D₃×D₅	—	4	2	no	—
29	C₂×D₆₀	—	2	2	no	—
30	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
31	C₂×C₃×D₂₀	—	2	2	no	—
32	C₂×#1202	—	4	2	no	—
33	D₃×D₂₀	—	2	8	no	—
34	D₃×D₂₀	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	D₁₂₀	—	2	2	no	—
37	D₁₂₀	C₂³	1	4	yes	complemented
38	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
39	C₅×D₃×C₈	D₄	1	2	yes	—
40	C₅×D₃×C₈	C₂³	1	2	yes	—
41	C₃×D₄₀	—	2	2	no	—
42	C₃×D₄₀	C₂³	1	4	yes	complemented
43	#6653	—	2	8	no	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	complemented
45	C₅×C₃⋊C₁₆	C₂³	1	2	yes	complemented
46	C₂×D₃×D₈	—	5	2	no	—
47	C₂×C₃×D₁₆	—	5	1	no	—
48	C₂×#826	—	5	2	no	—
49	D₃×D₁₆	—	5	8	no	—
50	C₂×D₄₈	—	5	1	no	—
51	C₂×D₃×C₁₆	D₅	1	1	yes	characteristic, complemented
52	C₂²×D₂₀	—	6	2	no	—
53	C₂²×C₅×C₈	—	3	1	no	—
54	C₂×D₄₀	—	3	2	no	—
55	C₂×D₄₀	—	3	8	no	—
56	C₂×D₄₀	—	6	4	no	—
57	C₂×D₄₀	—	6	2	no	—
58	C₂×D₄₀	C₂×D₃	1	2	yes	complemented
59	C₂×C₅×C₁₆	—	3	1	no	—
60	C₂×C₅×C₁₆	—	3	4	no	—
61	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
62	D₈₀	—	3	8	no	—
63	D₈₀	—	3	4	no	—
64	D₈₀	C₂×D₃	1	4	yes	direct product factor
65	C₂²×D₁₆	—	15	1	no	Sylow
66	C₂²×D₁₅	—	4	2	no	—
67	C₂²×C₅×D₃	D₈	1	1	yes	characteristic
68	C₂²×C₃×D₅	—	4	2	no	—
69	C₂×D₃×D₅	—	4	8	no	—
70	C₂×D₃×D₅	—	8	8	no	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	D₆₀	—	2	4	no	—
73	D₆₀	—	4	2	no	—
74	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
75	C₄×C₅×D₃	D₈	1	2	yes	—
76	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
77	C₃×D₂₀	—	2	4	no	—
78	C₃×D₂₀	—	4	2	no	—
79	#1202	—	4	8	no	—
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	—	10	2	no	—
84	C₂×C₃×D₈	—	5	2	no	—
85	C₂×#213	—	10	2	no	—
86	D₃×D₈	—	5	8	no	—
87	D₃×D₈	—	10	4	no	—
88	C₂×D₂₄	—	5	2	no	—
89	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
90	C₃×D₁₆	—	5	4	no	—
91	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
92	#826	—	5	8	no	—
93	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
94	D₄₈	—	5	4	no	—
95	D₃×C₁₆	C₂×D₅	1	4	yes	complemented
96	C₂³×D₅	—	12	2	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂×D₂₀	—	2	2	no	—
99	C₂×D₂₀	—	6	8	no	—
100	C₂×D₂₀	—	6	2	no	—
101	C₂×D₂₀	—	12	4	no	—
102	C₂×D₂₀	—	12	2	no	—
103	C₂×C₅×C₈	—	3	1	no	—
104	C₂×C₅×C₈	—	3	2	no	—
105	C₂×C₅×C₈	—	3	2	no	—
106	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
107	D₄₀	—	2	2	no	—
108	D₄₀	—	3	4	no	—
109	D₄₀	—	6	8	no	—
110	D₄₀	—	6	2	no	—
111	D₄₀	C₂²×D₃	1	4	yes	complemented
112	C₅×C₁₆	—	3	2	no	—
113	C₅×C₁₆	C₂²×D₃	1	2	yes	complemented
114	C₂²×D₈	—	15	2	no	—
115	C₂×D₁₆	—	5	1	no	—
116	C₂×D₁₆	—	15	8	no	—
117	C₂×D₁₆	—	15	2	no	—
118	C₂×D₁₆	—	15	1	no	—
119	C₂²×C₁₆	—	3	1	no	—
120	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×D₁₅	—	4	4	no	—
122	C₂×D₁₅	—	8	4	no	—
123	C₂×C₅×D₃	D₁₆	1	4	yes	complemented
124	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
125	C₂×C₃×D₅	—	4	4	no	—
126	C₂×C₃×D₅	—	8	4	no	—
127	D₃×D₅	—	8	16	no	—
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	—	20	2	no	—
132	C₂×C₃×D₄	—	10	2	no	—
133	C₂×#67	—	20	2	no	—
134	D₃×D₄	—	10	8	no	—
135	D₃×D₄	—	20	4	no	—
136	C₂×D₁₂	—	10	2	no	—
137	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
138	C₃×D₈	—	5	4	no	—
139	C₃×D₈	—	10	2	no	—
140	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
141	#213	—	10	8	no	—
142	C₂×C₃⋊C₈	D₂₀	1	1	yes	characteristic
143	D₂₄	—	5	4	no	—
144	D₂₄	—	10	2	no	—
145	D₃×C₈	D₂₀	1	2	yes	—
146	D₃×C₈	C₂²×D₅	1	2	yes	—
147	C₃×C₁₆	C₂²×D₅	1	2	yes	complemented
148	C₃⋊C₁₆	C₂²×D₅	1	2	yes	complemented
149	C₂³×C₅	—	3	1	no	—
150	C₂²×D₅	—	4	2	no	—
151	C₂²×D₅	—	12	8	no	—
152	C₂²×D₅	—	12	2	no	—
153	C₂²×D₅	—	24	8	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	2	no	—
156	C₂×C₄×C₅	—	3	2	no	—
157	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
158	D₂₀	—	2	4	no	—
159	D₂₀	—	4	2	no	—
160	D₂₀	—	6	4	no	—
161	D₂₀	—	12	8	no	—
162	D₂₀	—	12	2	no	—
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	—	30	2	no	—
167	C₂×D₈	—	5	2	no	—
168	C₂×D₈	—	15	8	no	—
169	C₂×D₈	—	15	2	no	—
170	C₂×D₈	—	30	4	no	—
171	C₂×D₈	—	30	2	no	—
172	C₂²×C₈	—	3	1	no	—
173	D₁₆	—	5	4	no	—
174	D₁₆	—	15	8	no	—
175	D₁₆	—	15	4	no	—
176	C₂×C₁₆	—	3	1	no	—
177	C₂×C₁₆	—	3	4	no	—
178	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic, complemented
179	C₂×C₃×C₅	C₂×D₁₆	1	2	yes	complemented
180	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
181	D₁₅	—	8	4	no	—
182	C₃×D₅	—	8	4	no	—
183	C₅×D₃	C₂×D₁₆	1	4	yes	complemented
184	C₂³×C₃	—	20	2	no	—
185	C₂²×D₃	—	20	2	no	—
186	C₂²×D₃	—	20	8	no	—
187	C₂²×D₃	—	40	8	no	—
188	C₂²×D₃	D₄₀	1	1	yes	characteristic
189	C₃×D₄	—	10	4	no	—
190	C₃×D₄	—	20	2	no	—
191	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
192	#67	—	20	8	no	—
193	C₂×Dic₃	D₄₀	1	1	yes	characteristic
194	D₁₂	—	10	4	no	—
195	D₁₂	—	20	2	no	—
196	C₄×D₃	D₄₀	1	2	yes	—
197	C₄×D₃	C₂×D₂₀	1	2	yes	—
198	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
199	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
200	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
201	C₂²×C₅	—	3	4	no	—
202	C₂²×C₅	—	3	2	no	—
203	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
204	C₂×D₅	—	4	4	no	—
205	C₂×D₅	—	8	4	no	—
206	C₂×D₅	—	12	4	no	—
207	C₂×D₅	—	24	16	no	—
208	C₂×D₅	—	24	4	no	—
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	—	60	2	no	—
213	C₂×D₄	—	10	2	no	—
214	C₂×D₄	—	30	8	no	—
215	C₂×D₄	—	30	2	no	—
216	C₂×D₄	—	60	4	no	—
217	C₂×D₄	—	60	2	no	—
218	C₂²×C₄	—	3	1	no	—
219	D₈	—	5	4	no	—
220	D₈	—	10	2	no	—
221	D₈	—	15	4	no	—
222	D₈	—	30	8	no	—
223	D₈	—	30	2	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	3	2	no	—
227	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
228	C₁₆	—	3	2	no	—
229	C₁₆	C₂×D₃×D₅	1	2	yes	complemented
230	C₃×C₅	C₂²×D₁₆	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	20	4	no	—
232	C₂²×C₃	—	40	4	no	—
233	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
234	C₂×D₃	—	20	4	no	—
235	C₂×D₃	—	40	4	no	—
236	C₂×D₃	—	40	16	no	—
237	C₂×D₃	D₈₀	1	4	yes	direct product factor
238	C₂×D₃	C₂×D₄₀	1	2	yes	—
239	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
240	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
241	Dic₃	C₂×D₄₀	1	2	yes	—
242	C₂×C₅	—	3	4	no	—
243	C₂×C₅	D₃×D₁₆	1	2	yes	complemented
244	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
245	D₅	—	8	4	no	—
246	D₅	—	24	4	no	—
247	C₂³	—	3	1	no	—
248	C₂³	—	20	2	no	—
249	C₂³	—	60	8	no	—
250	C₂³	—	60	2	no	—
251	C₂³	—	120	8	no	—
252	D₄	—	10	4	no	—
253	D₄	—	20	2	no	—
254	D₄	—	30	4	no	—
255	D₄	—	60	8	no	—
256	D₄	—	60	2	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
261	C₈	—	3	2	no	—
262	C₈	D₃×D₂₀	1	1	yes	characteristic
263	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₂×C₃	—	40	4	no	—
265	C₂×C₃	C₂×D₈₀	1	2	yes	complemented
266	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
267	D₃	—	40	4	no	—
268	D₃	C₂×D₈₀	1	4	yes	direct product factor
269	C₅	C₂×D₃×D₁₆	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	3	4	no	—
271	C₂²	—	3	2	no	—
272	C₂²	—	20	4	no	—
273	C₂²	—	40	4	no	—
274	C₂²	—	60	4	no	—
275	C₂²	—	120	16	no	—
276	C₂²	—	120	4	no	—
277	C₂²	D₃×D₄₀	1	1	yes	characteristic
278	C₄	—	3	2	no	—
279	C₄	D₃×D₄₀	1	1	yes	characteristic
280	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
281	C₃	C₂²×D₈₀	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	3	4	no	—
283	C₂	—	40	4	no	—
284	C₂	—	120	4	no	—
285	C₂	D₃×D₈₀	1	2	yes	direct product factor
286	C₂	C₂×D₃×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄₀
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×C₁₆	C₂²
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	C₂²×D₁₆
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₄₀	C₂	semidirect
2. C₂×D₂₄₀	C₂	semidirect
3. C₂×C₅×D₃×C₁₆	C₂	semidirect
4. C₂×C₃×D₈₀	C₂	semidirect
5. C₂×#90938	C₂	semidirect
6. D₃×D₈₀	C₂	direct (N × H)
9. C₂×D₁₂₀	C₂²	semidirect
11. C₂×C₃×D₄₀	C₂²	semidirect
14. D₃×D₄₀	C₂²	semidirect
15. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. D₂₄₀	C₂²	semidirect
17. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
18. C₅×D₃×C₁₆	C₂²	semidirect
19. C₃×D₈₀	C₂²	semidirect
20. #90938	C₂²	semidirect
27. C₂×D₈₀	D₃	direct (N × H)
37. D₁₂₀	C₂³	semidirect
42. C₃×D₄₀	C₂³	semidirect
44. C₃×C₅×C₁₆	C₂³	semidirect
45. C₅×C₃⋊C₁₆	C₂³	semidirect
51. C₂×D₃×C₁₆	D₅	semidirect
58. C₂×D₄₀	C₂×D₃	semidirect
61. C₂×C₅×C₁₆	C₂×D₃	semidirect
64. D₈₀	C₂×D₃	direct (N × H)
91. C₂×C₃×C₁₆	C₂×D₅	semidirect
93. C₂×C₃⋊C₁₆	C₂×D₅	semidirect
95. D₃×C₁₆	C₂×D₅	semidirect
111. D₄₀	C₂²×D₃	semidirect
113. C₅×C₁₆	C₂²×D₃	semidirect
123. C₂×C₅×D₃	D₁₆	semidirect
147. C₃×C₁₆	C₂²×D₅	semidirect
148. C₃⋊C₁₆	C₂²×D₅	semidirect
178. C₂×C₁₆	D₃×D₅	semidirect
179. C₂×C₃×C₅	C₂×D₁₆	semidirect
183. C₅×D₃	C₂×D₁₆	semidirect
229. C₁₆	C₂×D₃×D₅	semidirect
230. C₃×C₅	C₂²×D₁₆	semidirect
237. C₂×D₃	D₈₀	direct (N × H)
243. C₂×C₅	D₃×D₁₆	semidirect
265. C₂×C₃	C₂×D₈₀	semidirect
268. D₃	C₂×D₈₀	direct (N × H)
269. C₅	C₂×D₃×D₁₆	semidirect
281. C₃	C₂²×D₈₀	semidirect
285. C₂	D₃×D₈₀	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×D₃×D₄₀	2
C₂×D₂₄₀	2
C₂×C₅×D₃×C₁₆	2
C₂×C₃×D₈₀	2
C₂×#90938	2
D₃×D₈₀	2
C₂²×D₈₀	3
C₂×D₃×D₁₆	5

Subgroups — C2×D3×D80 #14478209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×D₈₀ #14478209