Subgroups — C₂×D₃×Dic₄₀ #14478256

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Dic₂₀	C₂	1	2	yes	maximal
2	C₂×Dic₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Dic₄₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90947	C₂	1	2	yes	maximal, complemented
6	D₃×Dic₄₀	C₂	1	8	yes	maximal, direct product factor
7	C₂²×Dic₄₀	—	3	1	no	maximal, Hall
8	C₂×D₃×Dic₁₀	—	2	2	no	—
9	C₂×Dic₆₀	C₂²	1	2	yes	—
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₂₀	C₂²	1	2	yes	—
12	C₂×#6662	—	2	2	no	—
13	D₃×Dic₂₀	—	2	4	no	—
14	D₃×Dic₂₀	C₂²	1	8	yes	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	Dic₁₂₀	C₂²	1	4	yes	complemented
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₅×D₃×C₁₆	C₂²	1	4	yes	—
19	C₃×Dic₄₀	C₂²	1	4	yes	complemented
20	#90947	C₂²	1	8	yes	complemented
21	C₂×D₃×Q₃₂	—	5	1	no	maximal, Hall
22	C₂²×Dic₂₀	—	3	2	no	—
23	C₂²×C₅×C₁₆	—	3	1	no	—
24	C₂×Dic₄₀	—	3	2	no	—
25	C₂×Dic₄₀	—	3	1	no	—
26	C₂×Dic₄₀	—	3	8	no	—
27	C₂×Dic₄₀	D₃	1	1	yes	characteristic, direct product factor
28	C₂×Dic₃₀	—	2	2	no	—
29	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
30	C₂×C₃×Dic₁₀	—	2	2	no	—
31	C₂×#1204	—	4	2	no	—
32	C₂×D₃×Dic₅	—	4	2	no	—
33	D₃×Dic₁₀	—	2	8	no	—
34	D₃×Dic₁₀	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	Dic₆₀	—	2	2	no	—
37	Dic₆₀	C₂³	1	4	yes	—
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×Dic₂₀	—	2	2	no	—
42	C₃×Dic₂₀	C₂³	1	4	yes	—
43	#6662	—	2	8	no	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	—
45	C₅×C₃⋊C₁₆	C₂³	1	2	yes	—
46	C₂×D₃×Q₁₆	—	5	2	no	—
47	C₂×C₃×Q₃₂	—	5	1	no	—
48	C₂×#829	—	5	2	no	—
49	D₃×Q₃₂	—	5	8	no	—
50	C₂×Dic₂₄	—	5	1	no	—
51	C₂×D₃×C₁₆	D₅	1	1	yes	characteristic
52	C₂²×Dic₁₀	—	6	2	no	—
53	C₂²×C₅×C₈	—	3	1	no	—
54	C₂×Dic₂₀	—	3	2	no	—
55	C₂×Dic₂₀	—	3	8	no	—
56	C₂×Dic₂₀	—	6	4	no	—
57	C₂×Dic₂₀	—	6	2	no	—
58	C₂×Dic₂₀	C₂×D₃	1	2	yes	—
59	C₂×C₅×C₁₆	—	3	1	no	—
60	C₂×C₅×C₁₆	—	3	4	no	—
61	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
62	Dic₄₀	—	3	8	no	—
63	Dic₄₀	—	3	4	no	—
64	Dic₄₀	C₂×D₃	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	15	1	no	Sylow
66	C₂²×C₅×D₃	D₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₁₅	—	4	2	no	—
69	Dic₃₀	—	2	4	no	—
70	Dic₃₀	—	4	2	no	—
71	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
72	C₄×C₅×D₃	D₈	1	2	yes	—
73	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
74	C₂×C₃×Dic₅	—	4	2	no	—
75	C₃×Dic₁₀	—	2	4	no	—
76	C₃×Dic₁₀	—	4	2	no	—
77	#1204	—	4	8	no	—
78	D₃×Dic₅	—	4	8	no	—
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
81	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
82	C₂×D₃×Q₈	—	10	2	no	—
83	C₂×C₃×Q₁₆	—	5	2	no	—
84	C₂×#216	—	10	2	no	—
85	D₃×Q₁₆	—	5	8	no	—
86	D₃×Q₁₆	—	10	4	no	—
87	C₂×Dic₁₂	—	5	2	no	—
88	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₃×Q₃₂	—	5	4	no	—
90	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
91	#829	—	5	8	no	—
92	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
93	Dic₂₄	—	5	4	no	—
94	D₃×C₁₆	C₂×D₅	1	4	yes	—
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂²×Dic₅	—	12	2	no	—
97	C₂×Dic₁₀	—	2	2	no	—
98	C₂×Dic₁₀	—	6	8	no	—
99	C₂×Dic₁₀	—	6	2	no	—
100	C₂×Dic₁₀	—	12	4	no	—
101	C₂×Dic₁₀	—	12	2	no	—
102	C₂×C₅×C₈	—	3	1	no	—
103	C₂×C₅×C₈	—	3	2	no	—
104	C₂×C₅×C₈	—	3	2	no	—
105	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
106	Dic₂₀	—	2	2	no	—
107	Dic₂₀	—	3	4	no	—
108	Dic₂₀	—	6	8	no	—
109	Dic₂₀	—	6	2	no	—
110	Dic₂₀	C₂²×D₃	1	4	yes	—
111	C₅×C₁₆	—	3	2	no	—
112	C₅×C₁₆	C₂²×D₃	1	2	yes	—
113	C₂²×Q₁₆	—	15	2	no	—
114	C₂×Q₃₂	—	5	1	no	—
115	C₂×Q₃₂	—	15	8	no	—
116	C₂×Q₃₂	—	15	2	no	—
117	C₂×Q₃₂	—	15	1	no	—
118	C₂²×C₁₆	—	3	1	no	—
119	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₅×D₃	Q₃₂	1	4	yes	complemented
121	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
122	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	Dic₁₅	—	4	4	no	—
125	C₃×Dic₅	—	4	4	no	—
126	C₅×Dic₃	C₂×D₈	1	2	yes	—
127	C₂×C₃×Q₈	—	10	2	no	—
128	D₃×Q₈	—	10	8	no	—
129	D₃×Q₈	—	20	4	no	—
130	C₂×C₄×D₃	—	20	2	no	—
131	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
132	C₂×Dic₆	—	10	2	no	—
133	C₂×Dic₆	—	20	2	no	—
134	C₃×Q₁₆	—	5	4	no	—
135	C₃×Q₁₆	—	10	2	no	—
136	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
137	#216	—	10	8	no	—
138	C₂×C₃⋊C₈	D₂₀	1	1	yes	characteristic
139	Dic₁₂	—	5	4	no	—
140	Dic₁₂	—	10	2	no	—
141	D₃×C₈	D₂₀	1	2	yes	—
142	D₃×C₈	C₂²×D₅	1	2	yes	—
143	C₃×C₁₆	C₂²×D₅	1	2	yes	—
144	C₃⋊C₁₆	C₂²×D₅	1	2	yes	—
145	C₂³×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	2	no	—
148	C₂×C₄×C₅	—	3	2	no	—
149	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
150	C₂×Dic₅	—	4	2	no	—
151	C₂×Dic₅	—	12	8	no	—
152	C₂×Dic₅	—	12	2	no	—
153	Dic₁₀	—	2	4	no	—
154	Dic₁₀	—	4	2	no	—
155	Dic₁₀	—	6	4	no	—
156	Dic₁₀	—	12	8	no	—
157	Dic₁₀	—	12	2	no	—
158	C₅×C₈	—	3	2	no	—
159	C₅×C₈	D₃×D₄	1	1	yes	characteristic
160	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
161	C₂²×Q₈	—	30	2	no	—
162	C₂×Q₁₆	—	5	2	no	—
163	C₂×Q₁₆	—	15	8	no	—
164	C₂×Q₁₆	—	15	2	no	—
165	C₂×Q₁₆	—	30	4	no	—
166	C₂×Q₁₆	—	30	2	no	—
167	C₂²×C₈	—	3	1	no	—
168	Q₃₂	—	5	4	no	—
169	Q₃₂	—	15	8	no	—
170	Q₃₂	—	15	4	no	—
171	C₂×C₁₆	—	3	1	no	—
172	C₂×C₁₆	—	3	4	no	—
173	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×Q₃₂	1	2	yes	complemented
175	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
176	C₅×D₃	C₂×Q₃₂	1	4	yes	complemented
177	C₂²×D₃	D₄₀	1	1	yes	characteristic
178	C₃×Q₈	—	10	4	no	—
179	C₃×Q₈	—	20	2	no	—
180	C₂×C₃×C₄	—	20	2	no	—
181	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
182	C₂×Dic₃	—	20	2	no	—
183	C₂×Dic₃	D₄₀	1	1	yes	characteristic
184	C₄×D₃	—	20	8	no	—
185	C₄×D₃	D₄₀	1	2	yes	—
186	C₄×D₃	C₂×D₂₀	1	2	yes	—
187	Dic₆	—	10	4	no	—
188	Dic₆	—	20	2	no	—
189	Dic₆	—	20	8	no	—
190	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
191	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
192	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
193	C₂²×C₅	—	3	4	no	—
194	C₂²×C₅	—	3	2	no	—
195	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
196	C₄×C₅	—	3	2	no	—
197	C₄×C₅	D₃×D₈	1	1	yes	characteristic
198	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
199	Dic₅	—	4	4	no	—
200	Dic₅	—	12	4	no	—
201	C₂×Q₈	—	10	2	no	—
202	C₂×Q₈	—	30	8	no	—
203	C₂×Q₈	—	30	2	no	—
204	C₂×Q₈	—	60	2	no	—
205	C₂×Q₈	—	60	4	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	60	2	no	—
208	Q₁₆	—	5	4	no	—
209	Q₁₆	—	10	2	no	—
210	Q₁₆	—	15	4	no	—
211	Q₁₆	—	30	8	no	—
212	Q₁₆	—	30	2	no	—
213	C₂×C₈	—	3	1	no	—
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	2	no	—
216	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
217	C₁₆	—	3	2	no	—
218	C₁₆	C₂×D₃×D₅	1	2	yes	—
219	C₃×C₅	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
221	C₂×D₃	Dic₄₀	1	4	yes	direct product factor
222	C₂×D₃	C₂×D₄₀	1	2	yes	—
223	C₃×C₄	—	20	4	no	—
224	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
225	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
226	Dic₃	—	20	4	no	—
227	Dic₃	C₂×D₄₀	1	2	yes	—
228	C₂×C₅	—	3	4	no	—
229	C₂×C₅	D₃×Q₃₂	1	2	yes	complemented
230	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
231	C₂³	—	3	1	no	—
232	Q₈	—	10	4	no	—
233	Q₈	—	20	2	no	—
234	Q₈	—	30	4	no	—
235	Q₈	—	60	8	no	—
236	Q₈	—	60	2	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	20	2	no	—
241	C₂×C₄	—	60	8	no	—
242	C₂×C₄	—	60	2	no	—
243	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
244	C₈	—	3	2	no	—
245	C₈	D₃×D₂₀	1	1	yes	characteristic
246	C₈	C₂²×D₃×D₅	1	1	yes	characteristic
247	C₂×C₃	C₂×Dic₄₀	1	2	yes	complemented
248	C₂×C₃	C₂²×D₄₀	1	1	yes	characteristic
249	D₃	C₂×Dic₄₀	1	4	yes	direct product factor
250	C₅	C₂×D₃×Q₃₂	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	3	4	no	—
252	C₂²	—	3	2	no	—
253	C₂²	D₃×D₄₀	1	1	yes	characteristic
254	C₄	—	3	2	no	—
255	C₄	—	20	4	no	—
256	C₄	—	60	4	no	—
257	C₄	D₃×D₄₀	1	1	yes	characteristic
258	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
259	C₃	C₂²×Dic₄₀	1	1	yes	characteristic, complemented, Sylow
260	C₂	—	3	4	no	—
261	C₂	D₃×Dic₄₀	1	2	yes	direct product factor
262	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
prime radical	C₂²×C₅×D₃	D₈
hypercenter	C₂×C₁₆	D₃×D₅
hypocenter	C₃×C₅	C₂²×Q₃₂
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
2. C₂×Dic₁₂₀	C₂	semidirect
4. C₂×C₃×Dic₄₀	C₂	semidirect
5. C₂×#90947	C₂	semidirect
6. D₃×Dic₄₀	C₂	direct (N × H)
16. Dic₁₂₀	C₂²	semidirect
19. C₃×Dic₄₀	C₂²	semidirect
20. #90947	C₂²	semidirect
27. C₂×Dic₄₀	D₃	direct (N × H)
64. Dic₄₀	C₂×D₃	direct (N × H)
120. C₂×C₅×D₃	Q₃₂	semidirect
174. C₂×C₃×C₅	C₂×Q₃₂	semidirect
176. C₅×D₃	C₂×Q₃₂	semidirect
219. C₃×C₅	C₂²×Q₃₂	semidirect
221. C₂×D₃	Dic₄₀	direct (N × H)
229. C₂×C₅	D₃×Q₃₂	semidirect
247. C₂×C₃	C₂×Dic₄₀	semidirect
249. D₃	C₂×Dic₄₀	direct (N × H)
250. C₅	C₂×D₃×Q₃₂	semidirect
259. C₃	C₂²×Dic₄₀	semidirect
261. C₂	D₃×Dic₄₀	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×Dic₂₀	2
C₂×Dic₁₂₀	2
C₂×C₅×D₃×C₁₆	2
C₂×C₃×Dic₄₀	2
C₂×#90947	2
D₃×Dic₄₀	2
C₂²×Dic₄₀	3
C₂×D₃×Q₃₂	5

Subgroups — C2×D3×Dic40 #14478256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×Dic₄₀ #14478256