Subgroups — C₂²×C₅×#711 #13107860

Order 1600.

2016 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂⁵×C₅
Frattini	C₂	C₂⁴×C₅×D₅
Fitting	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂²×C₅×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×D₄	C₂	semidirect
2. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
3. C₂³×C₅×Dic₅	C₂	semidirect
4. C₂×C₅×#711	C₂	direct (N × H)
5. C₂²×C₄×C₅×D₅	C₂	semidirect
6. C₂²×C₅×Dic₁₀	C₂	semidirect
9. C₂×C₅²×D₄	C₂²	semidirect
11. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
14. C₂²×C₅×Dic₅	C₂²	semidirect
15. C₅×#711	C₂²	direct (N × H)
16. C₂×C₄×C₅×D₅	C₂²	semidirect
17. C₂×C₅×Dic₁₀	C₂²	semidirect
19. C₂²×#711	C₅	direct (N × H)
25. C₅²×D₄	C₂³	semidirect
27. C₅×C₂²⋊D₅	C₂³	semidirect
30. C₂×C₅×Dic₅	C₂³	semidirect
31. C₄×C₅×D₅	C₂³	semidirect
32. C₅×Dic₁₀	C₂³	semidirect
37. C₂²×C₅×D₄	D₅	semidirect
38. C₂²×C₅×D₄	C₂×C₅	semidirect
41. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
42. C₂³×Dic₅	C₂×C₅	semidirect
43. C₂×#711	C₂×C₅	direct (N × H)
44. C₂²×C₄×D₅	C₂×C₅	semidirect
45. C₂²×Dic₁₀	C₂×C₅	semidirect
47. C₂²×C₅²	#41	semidirect
64. C₂×C₅×D₄	C₂×D₅	semidirect
65. C₂×C₅×D₄	C₂²×C₅	semidirect
73. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
76. C₂²×Dic₅	C₂²×C₅	semidirect
77. #711	C₂²×C₅	direct (N × H)
78. C₂×C₄×D₅	C₂²×C₅	semidirect
79. C₂×Dic₁₀	C₂²×C₅	semidirect
82. C₂×C₅²	C₂×#41	semidirect
101. C₅×D₄	C₂²×D₅	semidirect
102. C₅×D₄	C₂³×C₅	semidirect
110. C₂²⋊D₅	C₂³×C₅	semidirect
113. C₂×Dic₅	C₂³×C₅	semidirect
114. C₄×D₅	C₂³×C₅	semidirect
115. Dic₁₀	C₂³×C₅	semidirect
119. C₂²×D₄	C₅×D₅	semidirect
122. C₅²	C₂²×#41	semidirect
131. C₂²×C₅	#711	direct (N × H)
132. C₂²×C₅	C₅×#41	semidirect
151. C₂×D₄	C₂×C₅×D₅	semidirect
163. C₂×C₅	C₂×#711	direct (N × H)
164. C₂×C₅	C₂×C₅×#41	semidirect
175. D₄	C₂²×C₅×D₅	semidirect
182. C₅	C₂²×#711	direct (N × H)
183. C₅	C₂²×C₅×#41	semidirect
187. C₂²	C₅×#711	direct (N × H)
195. C₂	C₂×C₅×#711	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×C₅²×D₄	2
C₂²×C₅×C₂²⋊D₅	2
C₂³×C₅×Dic₅	2
C₂×C₅×#711	2
C₂²×C₄×C₅×D₅	2
C₂²×C₅×Dic₁₀	2
C₂²×C₅×#41	5
C₂²×#711	5

Subgroups — C22×C5×#711 #13107860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#711 #13107860