Subgroups — C₂×C₄×C₅×D₂₀ #13106627

Order 1600.

960 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×D₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×D₂₀	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₅×D₅	C₂²	1	2	yes	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
12	C₂×C₅×D₂₀	C₄	1	8	yes	direct product factor
13	C₂×C₅×D₂₀	C₂²	1	4	yes	—
14	C₂×C₄×C₅×D₅	—	2	8	no	—
15	C₂×C₄×C₅×D₅	C₂²	1	8	yes	complemented
16	C₄²×C₅²	C₂²	1	4	yes	complemented
17	C₅×#685	C₂²	1	8	yes	complemented
18	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
19	C₂×C₄×C₅×D₄	—	5	1	no	maximal
20	C₂×C₄×D₂₀	C₅	1	1	yes	maximal, characteristic, direct product factor
21	C₂³×C₅²	C₂³	1	1	yes	characteristic
22	C₂²×C₅×D₅	—	2	4	no	—
23	C₂²×C₅×D₅	—	2	4	no	—
24	C₂²×C₅×D₅	C₂×C₄	1	8	yes	complemented
25	C₂²×C₅×D₅	C₂³	1	4	yes	—
26	C₂×C₄×C₅²	—	2	2	no	—
27	C₂×C₄×C₅²	C₂×C₄	1	4	yes	complemented
28	C₂×C₄×C₅²	D₄	1	4	yes	complemented
29	C₂×C₄×C₅²	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂×C₄×C₅²	C₂³	1	2	yes	—
32	C₂×C₅×Dic₅	—	2	4	no	—
33	C₂×C₅×Dic₅	C₂³	1	4	yes	—
34	C₅×D₂₀	C₂×C₄	1	16	yes	direct product factor
35	C₄×C₅×D₅	—	2	16	no	—
36	C₂²×C₅×D₄	—	5	1	no	—
37	C₂³×C₄×C₅	—	5	2	no	—
38	C₂²×D₂₀	C₂×C₅	1	1	yes	characteristic
39	C₂²×C₄×D₅	C₂×C₅	1	2	yes	complemented
40	C₄×C₅×D₄	—	5	8	no	—
41	C₂×C₅×C₄⋊C₄	—	5	1	no	—
42	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
43	C₂×C₄²×C₅	—	2	2	no	—
44	C₂×C₄²×C₅	D₅	1	1	yes	characteristic, complemented
45	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×#685	C₂×C₅	1	2	yes	complemented
47	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₄×D₂₀	C₂×C₅	1	8	yes	direct product factor
49	C₂²×C₅²	C₂²×C₄	1	2	yes	—
50	C₂²×C₅²	C₂×D₄	1	2	yes	—
51	C₂²×C₅²	#41	1	2	yes	—
52	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
53	C₂×C₅×D₅	—	2	16	no	—
54	C₂×C₅×D₅	—	2	4	no	—
55	C₂×C₅×D₅	—	2	4	no	—
56	C₂×C₅×D₅	C₂²×C₄	1	8	yes	complemented
57	C₄×C₅²	—	2	2	no	—
58	C₄×C₅²	C₂²×C₄	1	4	yes	complemented
59	C₄×C₅²	C₂×D₄	1	4	yes	complemented
60	C₅×Dic₅	—	2	4	no	—
61	C₂⁴×C₅	—	5	2	no	—
62	C₂³×D₅	C₂²×C₅	1	2	yes	—
63	C₂×C₅×D₄	—	5	8	no	—
64	C₂×C₅×D₄	—	5	4	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	5	2	no	—
69	C₂²×C₄×C₅	—	5	8	no	—
70	C₂²×C₄×C₅	—	10	8	no	—
71	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
77	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
78	C₂×D₂₀	C₄×C₅	1	8	yes	direct product factor
79	C₂×D₂₀	C₂²×C₅	1	4	yes	—
80	C₂×C₄×D₅	—	2	8	no	—
81	C₂×C₄×D₅	C₂²×C₅	1	8	yes	complemented
82	C₅×C₄⋊C₄	—	5	4	no	—
83	C₅×C₂²⋊C₄	—	5	8	no	—
84	C₄²×C₅	—	2	8	no	—
85	C₄²×C₅	C₂×D₅	1	4	yes	complemented
86	C₄²×C₅	C₂²×C₅	1	4	yes	complemented
87	#685	C₂²×C₅	1	8	yes	complemented
88	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
89	C₂×C₄×D₄	—	5	1	no	Sylow
90	C₂×C₅²	C₄×D₄	1	4	yes	complemented
91	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₅²	C₂×#41	1	1	yes	characteristic
94	C₅×D₅	—	2	8	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	—	5	8	no	—
97	C₂³×C₅	—	5	4	no	—
98	C₂³×C₅	—	10	4	no	—
99	C₂³×C₅	—	10	4	no	—
100	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
101	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
102	C₂²×D₅	—	2	4	no	—
103	C₂²×D₅	—	2	4	no	—
104	C₂²×D₅	C₂×C₄×C₅	1	8	yes	complemented
105	C₂²×D₅	C₂³×C₅	1	4	yes	—
106	C₅×D₄	—	5	16	no	—
107	C₂×C₄×C₅	—	2	8	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	5	4	no	—
116	C₂×C₄×C₅	—	10	4	no	—
117	C₂×C₄×C₅	—	10	16	no	—
118	C₂×C₄×C₅	C₄×D₅	1	4	yes	complemented
119	C₂×C₄×C₅	D₂₀	1	4	yes	direct product factor
120	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	complemented
121	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
122	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
123	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
124	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
126	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
127	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
128	C₂×Dic₅	—	2	4	no	—
129	C₂×Dic₅	C₂³×C₅	1	4	yes	—
130	D₂₀	C₂×C₄×C₅	1	16	yes	direct product factor
131	C₄×D₅	—	2	16	no	—
132	C₂²×D₄	—	5	1	no	—
133	C₂³×C₄	—	5	2	no	—
134	C₄×D₄	—	5	8	no	—
135	C₂×C₄⋊C₄	—	5	1	no	—
136	C₂×C₂²⋊C₄	—	5	2	no	—
137	C₂×C₄²	C₅×D₅	1	1	yes	characteristic, complemented
138	C₅²	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	4	no	—
141	C₂²×C₅	—	2	4	no	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	—	5	8	no	—
144	C₂²×C₅	—	10	4	no	—
145	C₂²×C₅	—	10	4	no	—
146	C₂²×C₅	—	10	16	no	—
147	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
148	C₂²×C₅	C₂×D₂₀	1	2	yes	—
149	C₂²×C₅	#709	1	2	yes	—
150	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
151	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
152	C₂²×C₅	C₅×#41	1	2	yes	—
153	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
154	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
155	C₂×D₅	—	2	16	no	—
156	C₂×D₅	—	2	4	no	—
157	C₂×D₅	—	2	4	no	—
158	C₂×D₅	C₂²×C₄×C₅	1	8	yes	complemented
159	C₄×C₅	—	2	8	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	8	no	—
163	C₄×C₅	—	2	8	no	—
164	C₄×C₅	—	10	4	no	—
165	C₄×C₅	C₂×C₄×D₅	1	4	yes	complemented
166	C₄×C₅	C₂×D₂₀	1	4	yes	direct product factor
167	C₄×C₅	C₂²×C₄×C₅	1	4	yes	complemented
168	C₄×C₅	C₂×C₅×D₄	1	4	yes	complemented
169	Dic₅	—	2	4	no	—
170	C₂⁴	—	5	2	no	—
171	C₂×D₄	—	5	8	no	—
172	C₂×D₄	—	5	4	no	—
173	C₂²×C₄	—	5	2	no	—
174	C₂²×C₄	—	5	8	no	—
175	C₂²×C₄	—	10	8	no	—
176	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
179	C₄⋊C₄	—	5	4	no	—
180	C₂²⋊C₄	—	5	8	no	—
181	C₄²	C₂×C₅×D₅	1	4	yes	complemented
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	—	2	2	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	—	2	8	no	—
186	C₂×C₅	—	10	8	no	—
187	C₂×C₅	C₄×D₂₀	1	4	yes	direct product factor
188	C₂×C₅	C₄×C₅×D₄	1	4	yes	complemented
189	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
190	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
191	C₂×C₅	C₂×#709	1	1	yes	characteristic
192	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
193	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
194	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
195	D₅	—	2	8	no	—
196	C₂³	—	5	8	no	—
197	C₂³	—	5	4	no	—
198	C₂³	—	10	4	no	—
199	C₂³	—	10	4	no	—
200	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
201	D₄	—	5	16	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	5	4	no	—
204	C₂×C₄	—	10	4	no	—
205	C₂×C₄	—	10	16	no	—
206	C₂×C₄	C₄×C₅×D₅	1	4	yes	complemented
207	C₂×C₄	C₅×D₂₀	1	4	yes	direct product factor
208	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
209	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
210	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
211	C₅	—	2	2	no	—
212	C₅	C₂×C₄×D₂₀	1	1	yes	characteristic, direct product factor
213	C₅	C₂×C₄×C₅×D₄	1	1	yes	characteristic, complemented
214	C₂²	—	5	8	no	—
215	C₂²	—	10	4	no	—
216	C₂²	—	10	4	no	—
217	C₂²	—	10	16	no	—
218	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
219	C₂²	C₂×C₅×D₂₀	1	2	yes	—
220	C₂²	C₅×#709	1	2	yes	—
221	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	10	4	no	—
224	C₄	C₂×C₄×C₅×D₅	1	4	yes	complemented
225	C₄	C₂×C₅×D₂₀	1	4	yes	direct product factor
226	C₂	—	10	8	no	—
227	C₂	C₄×C₅×D₂₀	1	4	yes	direct product factor
228	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
229	C₂	C₂²×C₅×D₂₀	1	1	yes	characteristic
230	C₂	C₂×C₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂²×C₅×D₂₀	C₂
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₂×C₄×C₅×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅×D₅	C₂	semidirect
3. C₂×C₄²×C₅²	C₂	semidirect
4. C₂×C₅×#685	C₂	semidirect
5. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
6. C₄×C₅×D₂₀	C₂	direct (N × H)
12. C₂×C₅×D₂₀	C₄	direct (N × H)
15. C₂×C₄×C₅×D₅	C₂²	semidirect
16. C₄²×C₅²	C₂²	semidirect
17. C₅×#685	C₂²	semidirect
18. C₅×C₄⋊Dic₅	C₂²	semidirect
20. C₂×C₄×D₂₀	C₅	direct (N × H)
24. C₂²×C₅×D₅	C₂×C₄	semidirect
27. C₂×C₄×C₅²	C₂×C₄	semidirect
28. C₂×C₄×C₅²	D₄	semidirect
34. C₅×D₂₀	C₂×C₄	direct (N × H)
39. C₂²×C₄×D₅	C₂×C₅	semidirect
44. C₂×C₄²×C₅	D₅	semidirect
45. C₂×C₄²×C₅	C₂×C₅	semidirect
46. C₂×#685	C₂×C₅	semidirect
47. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
48. C₄×D₂₀	C₂×C₅	direct (N × H)
56. C₂×C₅×D₅	C₂²×C₄	semidirect
58. C₄×C₅²	C₂²×C₄	semidirect
59. C₄×C₅²	C₂×D₄	semidirect
78. C₂×D₂₀	C₄×C₅	direct (N × H)
81. C₂×C₄×D₅	C₂²×C₅	semidirect
85. C₄²×C₅	C₂×D₅	semidirect
86. C₄²×C₅	C₂²×C₅	semidirect
87. #685	C₂²×C₅	semidirect
88. C₄⋊Dic₅	C₂²×C₅	semidirect
90. C₂×C₅²	C₄×D₄	semidirect
104. C₂²×D₅	C₂×C₄×C₅	semidirect
118. C₂×C₄×C₅	C₄×D₅	semidirect
119. C₂×C₄×C₅	D₂₀	direct (N × H)
120. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
121. C₂×C₄×C₅	C₅×D₄	semidirect
130. D₂₀	C₂×C₄×C₅	direct (N × H)
137. C₂×C₄²	C₅×D₅	semidirect
138. C₅²	C₂×C₄×D₄	semidirect
158. C₂×D₅	C₂²×C₄×C₅	semidirect
165. C₄×C₅	C₂×C₄×D₅	semidirect
166. C₄×C₅	C₂×D₂₀	direct (N × H)
167. C₄×C₅	C₂²×C₄×C₅	semidirect
168. C₄×C₅	C₂×C₅×D₄	semidirect
181. C₄²	C₂×C₅×D₅	semidirect
187. C₂×C₅	C₄×D₂₀	direct (N × H)
188. C₂×C₅	C₄×C₅×D₄	semidirect
206. C₂×C₄	C₄×C₅×D₅	semidirect
207. C₂×C₄	C₅×D₂₀	direct (N × H)
212. C₅	C₂×C₄×D₂₀	direct (N × H)
213. C₅	C₂×C₄×C₅×D₄	semidirect
224. C₄	C₂×C₄×C₅×D₅	semidirect
225. C₄	C₂×C₅×D₂₀	direct (N × H)
227. C₂	C₄×C₅×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
C₂²×C₅×D₂₀	2
C₂²×C₄×C₅×D₅	2
C₂×C₄²×C₅²	2
C₂×C₅×#685	2
C₂×C₅×C₄⋊Dic₅	2
C₄×C₅×D₂₀	2
C₂×C₄×C₅×D₄	5
C₂×C₄×D₂₀	5

Subgroups — C2×C4×C5×D20 #13106627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₂₀ #13106627