Subgroups — C₂×C₅×#4150 #13107867

Order 1600.

1752 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4150	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅×Q₈×D₅	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#711	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#709	C₂	1	3	yes	maximal, complemented
6	C₂²×C₅×Dic₁₀	C₂	1	3	yes	maximal, complemented
7	C₅²×#41	C₂²	1	8	yes	complemented
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅²×D₄	C₂²	1	3	yes	—
10	C₂²×C₄×C₅²	C₂²	1	3	yes	—
11	C₂×C₅×C₂²⋊D₅	C₂²	1	6	yes	—
12	C₂²×C₅×Dic₅	C₂²	1	6	yes	—
13	C₅×Q₈×D₅	C₂²	1	16	yes	complemented
14	C₅×#711	C₂²	1	48	yes	complemented
15	C₅×#709	C₂²	1	24	yes	complemented
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	6	yes	—
18	C₂×C₅×Dic₁₀	C₂²	1	3	yes	—
19	C₂×C₅×Dic₁₀	C₂²	1	6	yes	—
20	C₂×C₅×Dic₁₀	C₂²	1	24	yes	complemented
21	C₂×C₅×2₋¹⁺⁴	—	5	1	no	maximal
22	C₂×#4150	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	C₂³	1	3	yes	—
24	C₂²×C₅×D₅	C₂³	1	2	yes	—
25	C₅²×Q₈	C₂³	1	4	yes	—
26	C₅²×D₄	C₂³	1	12	yes	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂³	1	3	yes	—
29	C₂×C₄×C₅²	C₂³	1	12	yes	—
30	C₅×C₂²⋊D₅	C₂³	1	24	yes	—
31	C₂×C₅×Dic₅	C₂³	1	6	yes	—
32	C₂×C₅×Dic₅	C₂³	1	24	yes	—
33	C₅×D₂₀	C₂³	1	4	yes	—
34	C₄×C₅×D₅	C₂³	1	24	yes	—
35	C₅×Dic₁₀	C₂³	1	12	yes	—
36	C₅×Dic₁₀	C₂³	1	24	yes	—
37	C₅×2₋¹⁺⁴	—	5	16	no	—
38	C₂×C₅×#41	—	2	2	no	—
39	C₂×C₅×#41	—	5	6	no	—
40	C₂×C₅×#41	—	5	3	no	—
41	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
42	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂²×C₅×Q₈	—	5	2	no	—
44	C₂²×C₅×Q₈	—	5	3	no	—
45	#4150	C₂×C₅	1	16	yes	direct product factor
46	C₂×Q₈×D₅	C₂×C₅	1	2	yes	complemented
47	C₂×#711	C₂×C₅	1	6	yes	complemented
48	C₂×#709	C₂×C₅	1	3	yes	complemented
49	C₂²×Dic₁₀	C₂×C₅	1	3	yes	complemented
50	C₂²×C₅²	—	2	6	no	—
51	C₂²×C₅²	C₂⁴	1	6	yes	—
52	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
53	C₂×C₅×D₅	—	2	4	no	—
54	C₂×C₅×D₅	C₂⁴	1	4	yes	—
55	C₄×C₅²	C₂⁴	1	2	yes	—
56	C₄×C₅²	C₂⁴	1	6	yes	—
57	C₅×Dic₅	C₂⁴	1	12	yes	—
58	C₅×#41	—	2	16	no	—
59	C₅×#41	—	5	48	no	—
60	C₅×#41	—	5	24	no	—
61	C₅×#41	C₂×D₅	1	8	yes	complemented
62	C₅×#41	C₂²×C₅	1	8	yes	complemented
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂×C₅×Q₈	—	5	16	no	—
65	C₂×C₅×Q₈	—	5	6	no	—
66	C₂×C₅×Q₈	—	5	3	no	—
67	C₂×C₅×Q₈	—	5	24	no	—
68	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
69	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₅×D₄	—	2	6	no	—
71	C₂×C₅×D₄	—	5	6	no	—
72	C₂×C₅×D₄	—	5	1	no	—
73	C₂×C₅×D₄	C₂×D₅	1	3	yes	—
74	C₂×C₅×D₄	C₂²×C₅	1	3	yes	—
75	C₂²×C₄×C₅	—	2	6	no	—
76	C₂²×C₄×C₅	—	5	6	no	—
77	C₂²×C₄×C₅	—	5	6	no	—
78	C₂²×C₄×C₅	C₂×D₅	1	3	yes	—
79	C₂²×C₄×C₅	C₂²×C₅	1	3	yes	—
80	C₂×C₂²⋊D₅	C₂²×C₅	1	6	yes	—
81	C₂²×Dic₅	C₂²×C₅	1	6	yes	—
82	Q₈×D₅	C₂²×C₅	1	16	yes	complemented
83	#711	C₂²×C₅	1	48	yes	complemented
84	#709	C₂²×C₅	1	24	yes	complemented
85	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂²×C₅	1	6	yes	—
87	C₂×Dic₁₀	C₂²×C₅	1	3	yes	—
88	C₂×Dic₁₀	C₂²×C₅	1	6	yes	—
89	C₂×Dic₁₀	C₂²×C₅	1	24	yes	complemented
90	C₂×2₋¹⁺⁴	—	5	1	no	Sylow
91	C₂×C₅²	—	2	6	no	—
92	C₂×C₅²	2₋¹⁺⁴	1	2	yes	complemented
93	C₂×C₅²	C₂⁵	1	1	yes	characteristic
94	C₅×D₅	—	2	4	no	—
95	C₂³×C₅	—	2	6	no	—
96	C₂³×C₅	—	5	2	no	—
97	C₂³×C₅	C₂²×D₅	1	3	yes	—
98	C₂³×C₅	C₂³×C₅	1	3	yes	—
99	C₂²×D₅	C₂³×C₅	1	2	yes	—
100	C₅×Q₈	—	2	8	no	—
101	C₅×Q₈	—	5	24	no	—
102	C₅×Q₈	—	5	12	no	—
103	C₅×Q₈	C₂²×D₅	1	4	yes	—
104	C₅×Q₈	C₂³×C₅	1	4	yes	—
105	C₅×D₄	—	2	24	no	—
106	C₅×D₄	—	5	24	no	—
107	C₅×D₄	—	5	4	no	—
108	C₅×D₄	C₂²×D₅	1	12	yes	—
109	C₅×D₄	C₂³×C₅	1	12	yes	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	6	no	—
112	C₂×C₄×C₅	—	2	24	no	—
113	C₂×C₄×C₅	—	5	24	no	—
114	C₂×C₄×C₅	—	5	6	no	—
115	C₂×C₄×C₅	—	5	24	no	—
116	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
117	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
118	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
121	C₂×C₄×C₅	C₂³×C₅	1	12	yes	—
122	C₂²⋊D₅	C₂³×C₅	1	24	yes	—
123	C₂×Dic₅	C₂³×C₅	1	6	yes	—
124	C₂×Dic₅	C₂³×C₅	1	24	yes	—
125	D₂₀	C₂³×C₅	1	4	yes	—
126	C₄×D₅	C₂³×C₅	1	24	yes	—
127	Dic₁₀	C₂³×C₅	1	12	yes	—
128	Dic₁₀	C₂³×C₅	1	24	yes	—
129	2₋¹⁺⁴	—	5	16	no	—
130	C₂×#41	—	5	6	no	—
131	C₂×#41	—	5	3	no	—
132	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
133	C₂²×Q₈	—	5	2	no	—
134	C₂²×Q₈	—	5	3	no	—
135	C₅²	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₅	—	2	6	no	—
137	C₂²×C₅	—	2	6	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	12	no	—
140	C₂²×C₅	—	4	12	no	—
141	C₂²×C₅	—	5	4	no	—
142	C₂²×C₅	—	10	4	no	—
143	C₂²×C₅	C₂³×D₅	1	6	yes	—
144	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
145	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
146	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
147	C₂×D₅	—	2	4	no	—
148	C₂×D₅	C₂⁴×C₅	1	4	yes	—
149	C₄×C₅	—	2	4	no	—
150	C₄×C₅	—	2	12	no	—
151	C₄×C₅	—	5	12	no	—
152	C₄×C₅	C₂³×D₅	1	6	yes	—
153	C₄×C₅	C₂³×D₅	1	2	yes	—
154	C₄×C₅	C₂⁴×C₅	1	2	yes	—
155	C₄×C₅	C₂⁴×C₅	1	6	yes	—
156	Dic₅	C₂⁴×C₅	1	12	yes	—
157	#41	—	5	48	no	—
158	#41	—	5	24	no	—
159	#41	C₂×C₅×D₅	1	8	yes	complemented
160	C₂×Q₈	—	5	16	no	—
161	C₂×Q₈	—	5	6	no	—
162	C₂×Q₈	—	5	3	no	—
163	C₂×Q₈	—	5	24	no	—
164	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
165	C₂×D₄	—	5	6	no	—
166	C₂×D₄	—	5	1	no	—
167	C₂×D₄	C₂×C₅×D₅	1	3	yes	—
168	C₂²×C₄	—	5	6	no	—
169	C₂²×C₄	—	5	6	no	—
170	C₂²×C₄	C₂×C₅×D₅	1	3	yes	—
171	C₂×C₅	—	2	6	no	—
172	C₂×C₅	—	2	6	no	—
173	C₂×C₅	—	2	4	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	4	12	no	—
176	C₂×C₅	—	10	4	no	—
177	C₂×C₅	#4150	1	2	yes	direct product factor
178	C₂×C₅	C₅×2₋¹⁺⁴	1	2	yes	complemented
179	C₂×C₅	C₂⁴×D₅	1	1	yes	characteristic
180	C₂×C₅	C₂⁵×C₅	1	1	yes	characteristic
181	D₅	—	2	4	no	—
182	C₂³	—	5	2	no	—
183	C₂³	C₂²×C₅×D₅	1	3	yes	—
184	Q₈	—	5	24	no	—
185	Q₈	—	5	12	no	—
186	Q₈	C₂²×C₅×D₅	1	4	yes	—
187	D₄	—	5	24	no	—
188	D₄	—	5	4	no	—
189	D₄	C₂²×C₅×D₅	1	12	yes	—
190	C₂×C₄	—	5	6	no	—
191	C₂×C₄	—	5	24	no	—
192	C₂×C₄	—	5	24	no	—
193	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
194	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₅×D₅	1	12	yes	—
196	C₅	—	2	2	no	—
197	C₅	C₂×#4150	1	1	yes	characteristic, direct product factor
198	C₅	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic, complemented
199	C₂²	—	2	6	no	—
200	C₂²	—	5	4	no	—
201	C₂²	—	10	4	no	—
202	C₂²	C₂³×C₅×D₅	1	6	yes	—
203	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
204	C₄	—	5	12	no	—
205	C₄	C₂³×C₅×D₅	1	6	yes	—
206	C₄	C₂³×C₅×D₅	1	2	yes	—
207	C₂	—	2	6	no	—
208	C₂	—	10	4	no	—
209	C₂	C₅×#4150	1	2	yes	direct product factor
210	C₂	C₂⁴×C₅×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂×C₅	C₂⁵×C₅
Frattini	C₂	C₂⁴×C₅×D₅
Fitting	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₅×#4150	C₂	direct (N × H)
3. C₂×C₅×Q₈×D₅	C₂	semidirect
4. C₂×C₅×#711	C₂	semidirect
5. C₂×C₅×#709	C₂	semidirect
6. C₂²×C₅×Dic₁₀	C₂	semidirect
7. C₅²×#41	C₂²	semidirect
13. C₅×Q₈×D₅	C₂²	semidirect
14. C₅×#711	C₂²	semidirect
15. C₅×#709	C₂²	semidirect
20. C₂×C₅×Dic₁₀	C₂²	semidirect
22. C₂×#4150	C₅	direct (N × H)
41. C₂×C₅×#41	D₅	semidirect
42. C₂×C₅×#41	C₂×C₅	semidirect
45. #4150	C₂×C₅	direct (N × H)
46. C₂×Q₈×D₅	C₂×C₅	semidirect
47. C₂×#711	C₂×C₅	semidirect
48. C₂×#709	C₂×C₅	semidirect
49. C₂²×Dic₁₀	C₂×C₅	semidirect
61. C₅×#41	C₂×D₅	semidirect
62. C₅×#41	C₂²×C₅	semidirect
82. Q₈×D₅	C₂²×C₅	semidirect
83. #711	C₂²×C₅	semidirect
84. #709	C₂²×C₅	semidirect
89. C₂×Dic₁₀	C₂²×C₅	semidirect
92. C₂×C₅²	2₋¹⁺⁴	semidirect
132. C₂×#41	C₅×D₅	semidirect
135. C₅²	C₂×2₋¹⁺⁴	semidirect
159. #41	C₂×C₅×D₅	semidirect
177. C₂×C₅	#4150	direct (N × H)
178. C₂×C₅	C₅×2₋¹⁺⁴	semidirect
197. C₅	C₂×#4150	direct (N × H)
198. C₅	C₂×C₅×2₋¹⁺⁴	semidirect
209. C₂	C₅×#4150	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×C₅²×#41	2
C₅×#4150	2
C₂×C₅×Q₈×D₅	2
C₂×C₅×#711	2
C₂×C₅×#709	2
C₂²×C₅×Dic₁₀	2
C₂×C₅×2₋¹⁺⁴	5
C₂×#4150	5

Subgroups — C2×C5×#4150 #13107867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4150 #13107867