Subgroups — C₂²×Q₈×D₂₀ #13006217

Order 1280.

11798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₅	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈×D₂₀	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#4043	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×D₂₀	C₂	1	3	yes	maximal
6	C₂²×#4015	C₂	1	3	yes	maximal, complemented
7	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×D₅	—	2	32	no	—
9	C₂²×Q₈×D₅	C₂²	1	48	yes	complemented
10	C₂³×D₂₀	C₂²	1	1	yes	characteristic
11	C₂³×C₄×D₅	C₂²	1	6	yes	—
12	C₂³×Dic₁₀	C₂²	1	6	yes	—
13	C₂×C₄×C₅×Q₈	C₂²	1	24	yes	complemented
14	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₄²×C₅	C₂²	1	3	yes	—
16	C₂²×#685	C₂²	1	6	yes	—
17	C₂²×C₄⋊Dic₅	C₂²	1	6	yes	—
18	C₂²×C₄⋊Dic₅	C₂²	1	3	yes	—
19	Q₈×D₂₀	C₂²	1	256	yes	direct product factor
20	C₂×#4043	C₂²	1	144	yes	complemented
21	C₂×C₄×D₂₀	C₂²	1	72	yes	—
22	C₂×#4015	C₂²	1	72	yes	complemented
23	C₂²×D₄×Q₈	—	5	1	no	maximal, Sylow
24	C₂⁴×D₅	C₂³	1	2	yes	—
25	C₂²×C₅×Q₈	D₄	1	16	yes	complemented
26	C₂²×C₅×Q₈	C₂³	1	12	yes	—
27	C₂³×C₄×C₅	C₂³	1	3	yes	—
28	C₂³×C₄×C₅	C₂³	1	3	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂³×Dic₅	C₂³	1	6	yes	—
31	C₂×Q₈×D₅	—	2	384	no	—
32	C₂×Q₈×D₅	C₂³	1	128	yes	complemented
33	C₂²×D₂₀	Q₈	1	16	yes	direct product factor
34	C₂²×D₂₀	C₂³	1	12	yes	—
35	C₂²×C₄×D₅	—	2	48	no	—
36	C₂²×C₄×D₅	C₂³	1	72	yes	—
37	C₂²×Dic₁₀	—	2	48	no	—
38	C₂²×Dic₁₀	C₂³	1	72	yes	—
39	C₄×C₅×Q₈	C₂³	1	64	yes	complemented
40	C₂×C₅×C₄⋊C₄	C₂³	1	36	yes	—
41	C₂×C₄²×C₅	C₂³	1	36	yes	—
42	C₂×#685	C₂³	1	72	yes	—
43	C₂×C₄⋊Dic₅	C₂³	1	36	yes	—
44	C₂×C₄⋊Dic₅	C₂³	1	72	yes	—
45	#4043	C₂³	1	384	yes	complemented
46	C₄×D₂₀	C₂³	1	192	yes	—
47	#4015	C₂³	1	192	yes	complemented
48	C₂⁴×Q₈	—	5	2	no	—
49	C₂×D₄×Q₈	—	5	48	no	—
50	C₂²×C₄⋊Q₈	—	5	3	no	—
51	C₂²×C₂²⋊Q₈	—	5	6	no	—
52	C₂²×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
53	C₂²×C₄×D₄	—	5	3	no	—
54	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
55	C₂³×D₅	—	2	8	no	—
56	C₂³×D₅	—	2	8	no	—
57	C₂³×D₅	C₂×Q₈	1	16	yes	complemented
58	C₂³×D₅	C₂⁴	1	12	yes	—
59	C₂×C₅×Q₈	C₂×D₄	1	96	yes	complemented
60	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
61	C₂²×C₄×C₅	—	2	12	no	—
62	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
63	C₂²×C₄×C₅	C₂×Q₈	1	8	yes	complemented
64	C₂²×C₄×C₅	C₂⁴	1	18	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	18	yes	—
66	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
67	C₂²×Dic₅	—	2	24	no	—
68	C₂²×Dic₅	C₂⁴	1	36	yes	—
69	Q₈×D₅	—	2	512	no	—
70	C₂×D₂₀	C₂×Q₈	1	96	yes	direct product factor
71	C₂×D₂₀	C₂⁴	1	16	yes	—
72	C₂×C₄×D₅	—	2	288	no	—
73	C₂×C₄×D₅	C₂⁴	1	96	yes	—
74	C₂×Dic₁₀	—	2	288	no	—
75	C₂×Dic₁₀	C₂⁴	1	96	yes	—
76	C₅×C₄⋊C₄	C₂⁴	1	48	yes	—
77	C₄²×C₅	C₂⁴	1	48	yes	—
78	#685	C₂⁴	1	96	yes	—
79	C₄⋊Dic₅	C₂⁴	1	48	yes	—
80	C₄⋊Dic₅	C₂⁴	1	96	yes	—
81	C₂³×Q₈	—	5	48	no	—
82	C₂³×Q₈	—	5	6	no	—
83	C₂³×Q₈	—	10	32	no	—
84	C₂³×Q₈	C₂×D₅	1	1	yes	characteristic
85	C₂³×D₄	—	5	1	no	—
86	C₂⁴×C₄	—	5	6	no	—
87	D₄×Q₈	—	5	256	no	—
88	C₂×C₄⋊Q₈	—	5	72	no	—
89	C₂×C₂²⋊Q₈	—	5	144	no	—
90	C₂×C₄×Q₈	C₂×D₅	1	24	yes	complemented
91	C₂×C₄×D₄	—	5	72	no	—
92	C₂²×C₄⋊C₄	—	5	3	no	—
93	C₂²×C₄⋊C₄	—	5	6	no	—
94	C₂²×C₄⋊C₄	C₂×D₅	1	3	yes	—
95	C₂²×C₂²⋊C₄	—	5	6	no	—
96	C₂²×C₄²	C₂×D₅	1	3	yes	—
97	C₂³×C₅	C₂²×D₄	1	4	yes	—
98	C₂³×C₅	C₂²×Q₈	1	4	yes	—
99	C₂³×C₅	2₋¹⁺⁴	1	4	yes	—
100	C₂³×C₅	C₂⁵	1	3	yes	—
101	C₂²×D₅	—	2	64	no	—
102	C₂²×D₅	—	2	24	no	—
103	C₂²×D₅	—	2	24	no	—
104	C₂²×D₅	C₂²×Q₈	1	48	yes	complemented
105	C₂²×D₅	C₂⁵	1	8	yes	—
106	C₅×Q₈	C₂²×D₄	1	64	yes	complemented
107	C₂×C₄×C₅	—	2	36	no	—
108	C₂×C₄×C₅	C₂²×D₄	1	72	yes	—
109	C₂×C₄×C₅	C₂²×Q₈	1	24	yes	complemented
110	C₂×C₄×C₅	C₂⁵	1	12	yes	—
111	C₂×C₄×C₅	C₂⁵	1	12	yes	—
112	C₂×C₄×C₅	C₂⁵	1	4	yes	—
113	C₂×Dic₅	—	2	72	no	—
114	C₂×Dic₅	C₂⁵	1	24	yes	—
115	D₂₀	C₂²×Q₈	1	64	yes	direct product factor
116	C₄×D₅	—	2	192	no	—
117	Dic₁₀	—	2	192	no	—
118	C₂⁵	—	5	2	no	—
119	C₂²×Q₈	—	5	128	no	—
120	C₂²×Q₈	—	5	72	no	—
121	C₂²×Q₈	—	10	384	no	—
122	C₂²×Q₈	—	10	48	no	—
123	C₂²×Q₈	D₂₀	1	16	yes	direct product factor
124	C₂²×Q₈	C₂²×D₅	1	12	yes	—
125	C₂²×D₄	—	5	16	no	—
126	C₂²×D₄	—	5	12	no	—
127	C₂³×C₄	—	5	6	no	—
128	C₂³×C₄	—	5	72	no	—
129	C₂³×C₄	—	10	48	no	—
130	C₂³×C₄	C₂²×D₅	1	3	yes	—
131	C₂³×C₄	C₂²×D₅	1	3	yes	—
132	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₄⋊Q₈	—	5	192	no	—
134	C₂²⋊Q₈	—	5	384	no	—
135	C₄×Q₈	C₂²×D₅	1	64	yes	complemented
136	C₄×D₄	—	5	192	no	—
137	C₂×C₄⋊C₄	—	5	36	no	—
138	C₂×C₄⋊C₄	—	5	72	no	—
139	C₂×C₄⋊C₄	C₂²×D₅	1	36	yes	—
140	C₂×C₂²⋊C₄	—	5	72	no	—
141	C₂×C₄²	C₂²×D₅	1	36	yes	—
142	C₂²×C₅	D₄×Q₈	1	16	yes	complemented
143	C₂²×C₅	C₂³×D₄	1	6	yes	—
144	C₂²×C₅	C₂³×Q₈	1	6	yes	—
145	C₂²×C₅	C₂×2₋¹⁺⁴	1	6	yes	—
146	C₂²×C₅	C₂⁶	1	1	yes	characteristic
147	C₂×D₅	—	2	8	no	—
148	C₂×D₅	—	2	8	no	—
149	C₂×D₅	—	2	96	no	—
150	C₂×D₅	C₂³×Q₈	1	16	yes	complemented
151	C₄×C₅	—	2	12	no	—
152	C₄×C₅	C₂³×D₄	1	24	yes	—
153	C₄×C₅	C₂³×Q₈	1	8	yes	complemented
154	Dic₅	—	2	24	no	—
155	C₂⁴	—	5	16	no	—
156	C₂⁴	—	5	12	no	—
157	C₂⁴	—	10	8	no	—
158	C₂⁴	—	10	8	no	—
159	C₂⁴	C₂³×D₅	1	1	yes	characteristic
160	C₂×Q₈	—	5	96	no	—
161	C₂×Q₈	—	10	512	no	—
162	C₂×Q₈	—	10	288	no	—
163	C₂×Q₈	C₂×D₂₀	1	96	yes	direct product factor
164	C₂×Q₈	C₂³×D₅	1	16	yes	—
165	C₂×D₄	—	5	96	no	—
166	C₂×D₄	—	5	16	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	5	36	no	—
169	C₂²×C₄	—	5	96	no	—
170	C₂²×C₄	—	10	24	no	—
171	C₂²×C₄	—	10	288	no	—
172	C₂²×C₄	C₂×D₂₀	1	24	yes	—
173	C₂²×C₄	Q₈×D₅	1	8	yes	complemented
174	C₂²×C₄	C₂³×D₅	1	18	yes	—
175	C₂²×C₄	C₂³×D₅	1	18	yes	—
176	C₂²×C₄	C₂³×D₅	1	6	yes	—
177	C₄⋊C₄	—	5	48	no	—
178	C₄⋊C₄	—	5	96	no	—
179	C₄⋊C₄	C₂³×D₅	1	48	yes	—
180	C₂²⋊C₄	—	5	96	no	—
181	C₄²	C₂³×D₅	1	48	yes	—
182	C₂×C₅	C₂×D₄×Q₈	1	12	yes	complemented
183	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
184	C₂×C₅	C₂⁴×Q₈	1	1	yes	characteristic
185	C₂×C₅	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
186	D₅	—	2	16	no	—
187	C₂³	—	5	48	no	—
188	C₂³	—	5	8	no	—
189	C₂³	—	10	64	no	—
190	C₂³	—	10	24	no	—
191	C₂³	—	10	24	no	—
192	C₂³	C₂²×D₂₀	1	4	yes	—
193	C₂³	C₂×Q₈×D₅	1	4	yes	—
194	C₂³	#4150	1	4	yes	—
195	C₂³	C₂⁴×D₅	1	3	yes	—
196	Q₈	—	10	192	no	—
197	Q₈	C₂²×D₂₀	1	64	yes	direct product factor
198	D₄	—	5	64	no	—
199	C₂×C₄	—	2	36	no	—
200	C₂×C₄	—	5	24	no	—
201	C₂×C₄	—	10	72	no	—
202	C₂×C₄	—	10	192	no	—
203	C₂×C₄	C₂²×D₂₀	1	72	yes	—
204	C₂×C₄	C₂×Q₈×D₅	1	24	yes	complemented
205	C₂×C₄	C₂⁴×D₅	1	12	yes	—
206	C₂×C₄	C₂⁴×D₅	1	12	yes	—
207	C₂×C₄	C₂⁴×D₅	1	4	yes	—
208	C₅	C₂²×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	5	16	no	—
210	C₂²	—	10	8	no	—
211	C₂²	—	10	8	no	—
212	C₂²	—	10	96	no	—
213	C₂²	Q₈×D₂₀	1	16	yes	direct product factor
214	C₂²	C₂³×D₂₀	1	6	yes	—
215	C₂²	C₂²×Q₈×D₅	1	6	yes	—
216	C₂²	C₂×#4150	1	6	yes	—
217	C₂²	C₂⁵×D₅	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	10	24	no	—
220	C₄	C₂³×D₂₀	1	24	yes	—
221	C₄	C₂²×Q₈×D₅	1	8	yes	complemented
222	C₂	—	10	16	no	—
223	C₂	C₂×Q₈×D₂₀	1	12	yes	direct product factor
224	C₂	C₂⁴×D₂₀	1	1	yes	characteristic
225	C₂	C₂³×Q₈×D₅	1	1	yes	characteristic
226	C₂	C₂²×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₄×C₅×Q₈	C₂
socle	C₂⁴×C₅	C₂⁴
prime radical	C₂³×D₂₀	C₂²
hypercenter	C₂²×C₄×Q₈	D₅
hypocenter	C₅	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅	C₂	semidirect
2. C₂²×C₄×C₅×Q₈	C₂	semidirect
3. C₂×Q₈×D₂₀	C₂	direct (N × H)
4. C₂²×#4043	C₂	semidirect
6. C₂²×#4015	C₂	semidirect
9. C₂²×Q₈×D₅	C₂²	semidirect
13. C₂×C₄×C₅×Q₈	C₂²	semidirect
19. Q₈×D₂₀	C₂²	direct (N × H)
20. C₂×#4043	C₂²	semidirect
22. C₂×#4015	C₂²	semidirect
25. C₂²×C₅×Q₈	D₄	semidirect
32. C₂×Q₈×D₅	C₂³	semidirect
33. C₂²×D₂₀	Q₈	direct (N × H)
39. C₄×C₅×Q₈	C₂³	semidirect
45. #4043	C₂³	semidirect
47. #4015	C₂³	semidirect
52. C₂²×C₄×Q₈	D₅	semidirect
57. C₂³×D₅	C₂×Q₈	semidirect
59. C₂×C₅×Q₈	C₂×D₄	semidirect
63. C₂²×C₄×C₅	C₂×Q₈	semidirect
70. C₂×D₂₀	C₂×Q₈	direct (N × H)
90. C₂×C₄×Q₈	C₂×D₅	semidirect
104. C₂²×D₅	C₂²×Q₈	semidirect
106. C₅×Q₈	C₂²×D₄	semidirect
109. C₂×C₄×C₅	C₂²×Q₈	semidirect
115. D₂₀	C₂²×Q₈	direct (N × H)
123. C₂²×Q₈	D₂₀	direct (N × H)
135. C₄×Q₈	C₂²×D₅	semidirect
142. C₂²×C₅	D₄×Q₈	semidirect
150. C₂×D₅	C₂³×Q₈	semidirect
153. C₄×C₅	C₂³×Q₈	semidirect
163. C₂×Q₈	C₂×D₂₀	direct (N × H)
173. C₂²×C₄	Q₈×D₅	semidirect
182. C₂×C₅	C₂×D₄×Q₈	semidirect
197. Q₈	C₂²×D₂₀	direct (N × H)
204. C₂×C₄	C₂×Q₈×D₅	semidirect
208. C₅	C₂²×D₄×Q₈	semidirect
213. C₂²	Q₈×D₂₀	direct (N × H)
221. C₄	C₂²×Q₈×D₅	semidirect
223. C₂	C₂×Q₈×D₂₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₅	2
C₂²×C₄×C₅×Q₈	2
C₂×Q₈×D₂₀	2
C₂²×#4043	2
C₂²×C₄×D₂₀	2
C₂²×#4015	2
C₂²×D₄×Q₈	5

Subgroups — C22×Q8×D20 #13006217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×D₂₀ #13006217