Subgroups — C₅×D₅×#131 #13106903

Order 1600.

594 conjugacy classes of subgroups.

The subgroup lattice has 227 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₅²×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4094	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4053	C₂	1	3	yes	maximal, complemented
5	C₅×D₅×M₁₆	C₂	1	3	yes	maximal, complemented
6	C₅×#4047	C₂	1	3	yes	maximal, complemented
7	C₂×C₅×C₈×D₅	C₂	1	3	yes	maximal, complemented
8	C₅²×#41	C₂²	1	1	yes	characteristic
9	C₅×#713	C₄	1	1	yes	characteristic
10	C₅×Q₈×D₅	C₄	1	1	yes	characteristic
11	C₅×#711	C₄	1	3	yes	—
12	C₅×D₄×D₅	C₄	1	3	yes	—
13	C₅×#709	C₂²	1	3	yes	—
14	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
15	C₅²×M₁₆	C₂²	1	3	yes	complemented
16	C₂×C₅²×C₈	C₂²	1	3	yes	complemented
17	C₅×#681	C₂²	1	3	yes	complemented
18	C₂×C₅×#168	C₂²	1	3	yes	complemented
19	C₅×#676	C₂²	1	6	yes	complemented
20	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
21	C₅×C₈×D₅	C₂²	1	3	yes	—
22	C₅×C₈×D₅	C₂²	1	3	yes	complemented
23	C₅×C₈×D₅	C₂²	1	3	yes	complemented
24	C₂×C₅×#131	—	5	1	no	maximal
25	D₅×#131	C₅	1	1	yes	maximal, characteristic, direct product factor
26	C₂²×C₅×D₅	C₂×C₄	1	3	yes	—
27	C₅²×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₅²×D₄	C₂×C₄	1	3	yes	—
29	C₂×C₄×C₅²	C₂³	1	3	yes	—
30	C₅×C₂²⋊D₅	C₂×C₄	1	6	yes	—
31	C₂×C₅×Dic₅	C₂×C₄	1	3	yes	—
32	C₅×D₂₀	C₂×C₄	1	3	yes	—
33	C₄×C₅×D₅	C₂×C₄	1	3	yes	—
34	C₄×C₅×D₅	C₂×C₄	1	3	yes	—
35	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
36	C₄×C₅×D₅	C₂³	1	3	yes	—
37	C₅×Dic₁₀	C₂×C₄	1	3	yes	—
38	C₅²×C₈	C₂³	1	3	yes	—
39	C₅²×C₈	C₂³	1	1	yes	characteristic
40	C₅×#168	C₂³	1	1	yes	characteristic
41	C₅×#168	C₂³	1	3	yes	—
42	C₂×C₅×#41	—	5	1	no	—
43	D₅×#41	C₂×C₅	1	1	yes	characteristic
44	C₅×#131	—	2	2	no	—
45	C₅×#131	—	5	1	no	—
46	C₅×#131	—	5	3	no	—
47	C₅×#131	—	5	3	no	—
48	C₅×#131	D₅	1	1	yes	characteristic, direct product factor
49	C₅×#131	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₅×M₁₆	—	5	3	no	—
51	C₂²×C₅×C₈	—	5	3	no	—
52	#4094	C₂×C₅	1	1	yes	characteristic, complemented
53	#4053	C₂×C₅	1	3	yes	complemented
54	D₅×M₁₆	C₂×C₅	1	3	yes	complemented
55	#4047	C₂×C₅	1	3	yes	complemented
56	C₂×C₈×D₅	C₂×C₅	1	3	yes	complemented
57	C₂²×C₅²	C₂²×C₄	1	3	yes	—
58	C₂×C₅×D₅	—	2	6	no	—
59	C₂×C₅×D₅	C₂²×C₄	1	3	yes	—
60	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
61	C₄×C₅²	C₂²×C₄	1	3	yes	—
62	C₄×C₅²	C₂⁴	1	1	yes	characteristic
63	C₅×Dic₅	C₂²×C₄	1	3	yes	—
64	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
65	C₅×#41	—	2	2	no	—
66	C₅×#41	—	5	1	no	—
67	C₅×#41	—	5	3	no	—
68	C₅×#41	—	5	3	no	—
69	C₅×#41	C₂×D₅	1	1	yes	characteristic
70	C₅×#41	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	5	1	no	—
72	C₂×C₅×D₄	—	5	3	no	—
73	C₂²×C₄×C₅	—	5	3	no	—
74	#713	C₄×C₅	1	1	yes	characteristic
75	Q₈×D₅	C₄×C₅	1	1	yes	characteristic
76	#711	C₄×C₅	1	3	yes	—
77	D₄×D₅	C₄×C₅	1	3	yes	—
78	#709	C₂²×C₅	1	3	yes	—
79	C₂×C₄×D₅	C₂²×C₅	1	3	yes	—
80	C₅×M₁₆	—	2	6	no	—
81	C₅×M₁₆	—	5	3	no	—
82	C₅×M₁₆	—	5	6	no	—
83	C₅×M₁₆	C₂×D₅	1	3	yes	complemented
84	C₅×M₁₆	C₂²×C₅	1	3	yes	complemented
85	C₂×C₅×C₈	—	2	6	no	—
86	C₂×C₅×C₈	—	5	3	no	—
87	C₂×C₅×C₈	—	5	3	no	—
88	C₂×C₅×C₈	—	5	3	no	—
89	C₂×C₅×C₈	—	5	1	no	—
90	C₂×C₅×C₈	—	5	3	no	—
91	C₂×C₅×C₈	C₂×D₅	1	3	yes	complemented
92	C₂×C₅×C₈	C₂²×C₅	1	3	yes	complemented
93	#681	C₂²×C₅	1	3	yes	complemented
94	C₂×#168	C₂²×C₅	1	3	yes	complemented
95	#676	C₂²×C₅	1	6	yes	complemented
96	C₈×D₅	C₂²×C₅	1	1	yes	characteristic
97	C₈×D₅	C₂²×C₅	1	3	yes	—
98	C₈×D₅	C₂²×C₅	1	3	yes	complemented
99	C₈×D₅	C₂²×C₅	1	3	yes	complemented
100	C₂×#131	—	5	1	no	Sylow
101	C₂×C₅²	—	2	3	no	—
102	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
103	C₅×D₅	—	2	3	no	—
104	C₅×D₅	#131	1	2	yes	direct product factor
105	C₂³×C₅	—	5	3	no	—
106	C₂²×D₅	C₂×C₄×C₅	1	3	yes	—
107	C₅×Q₈	—	2	2	no	—
108	C₅×Q₈	—	5	3	no	—
109	C₅×Q₈	C₄×D₅	1	1	yes	characteristic
110	C₅×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
111	C₅×D₄	—	2	6	no	—
112	C₅×D₄	—	5	6	no	—
113	C₅×D₄	—	5	3	no	—
114	C₅×D₄	C₄×D₅	1	3	yes	—
115	C₅×D₄	C₂×C₄×C₅	1	3	yes	—
116	C₂×C₄×C₅	—	2	6	no	—
117	C₂×C₄×C₅	—	5	3	no	—
118	C₂×C₄×C₅	—	5	3	no	—
119	C₂×C₄×C₅	—	5	3	no	—
120	C₂×C₄×C₅	—	5	1	no	—
121	C₂×C₄×C₅	—	5	3	no	—
122	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
123	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
124	C₂²⋊D₅	C₂×C₄×C₅	1	6	yes	—
125	C₂×Dic₅	C₂×C₄×C₅	1	3	yes	—
126	D₂₀	C₂×C₄×C₅	1	3	yes	—
127	C₄×D₅	C₂×C₄×C₅	1	3	yes	—
128	C₄×D₅	C₂×C₄×C₅	1	3	yes	—
129	C₄×D₅	C₂³×C₅	1	1	yes	characteristic
130	C₄×D₅	C₂³×C₅	1	3	yes	—
131	Dic₁₀	C₂×C₄×C₅	1	3	yes	—
132	C₅×C₈	—	2	2	no	—
133	C₅×C₈	—	2	6	no	—
134	C₅×C₈	—	5	1	no	—
135	C₅×C₈	—	5	3	no	—
136	C₅×C₈	C₂²×D₅	1	3	yes	—
137	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
138	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
139	C₅×C₈	C₂³×C₅	1	3	yes	—
140	#168	C₂³×C₅	1	1	yes	characteristic
141	#168	C₂³×C₅	1	3	yes	—
142	C₂×#41	—	5	1	no	—
143	#131	—	5	1	no	—
144	#131	—	5	3	no	—
145	#131	—	5	3	no	—
146	#131	C₅×D₅	1	1	yes	characteristic, direct product factor
147	C₂×M₁₆	—	5	3	no	—
148	C₂²×C₈	—	5	3	no	—
149	C₅²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₅	—	2	6	no	—
151	C₂²×C₅	—	5	1	no	—
152	C₂²×C₅	—	5	3	no	—
153	C₂²×C₅	—	10	6	no	—
154	C₂²×C₅	C₂×C₄×D₅	1	3	yes	—
155	C₂²×C₅	C₂²×C₄×C₅	1	3	yes	—
156	C₂×D₅	—	2	6	no	—
157	C₂×D₅	C₂²×C₄×C₅	1	3	yes	—
158	C₂×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₄×C₅	—	2	6	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	5	1	no	—
162	C₄×C₅	—	5	3	no	—
163	C₄×C₅	C₂×C₄×D₅	1	3	yes	—
164	C₄×C₅	C₂²×C₄×C₅	1	3	yes	—
165	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
166	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
167	Dic₅	C₂²×C₄×C₅	1	3	yes	—
168	Dic₅	C₂²×C₄×C₅	1	1	yes	characteristic
169	#41	—	5	1	no	—
170	#41	—	5	3	no	—
171	#41	—	5	3	no	—
172	#41	C₂×C₅×D₅	1	1	yes	characteristic
173	C₂×Q₈	—	5	1	no	—
174	C₂×D₄	—	5	3	no	—
175	C₂²×C₄	—	5	3	no	—
176	M₁₆	—	5	3	no	—
177	M₁₆	—	5	6	no	—
178	M₁₆	C₂×C₅×D₅	1	3	yes	complemented
179	C₂×C₈	—	5	3	no	—
180	C₂×C₈	—	5	1	no	—
181	C₂×C₈	—	5	3	no	—
182	C₂×C₈	—	5	3	no	—
183	C₂×C₈	—	5	3	no	—
184	C₂×C₈	C₂×C₅×D₅	1	3	yes	complemented
185	C₂×C₅	—	2	3	no	—
186	C₂×C₅	—	2	3	no	—
187	C₂×C₅	—	2	2	no	—
188	C₂×C₅	—	4	6	no	—
189	C₂×C₅	—	5	2	no	—
190	C₂×C₅	—	10	3	no	—
191	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
192	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
193	D₅	—	2	3	no	—
194	D₅	C₅×#131	1	2	yes	direct product factor
195	C₂³	—	5	3	no	—
196	Q₈	—	5	3	no	—
197	Q₈	C₄×C₅×D₅	1	1	yes	characteristic
198	D₄	—	5	6	no	—
199	D₄	—	5	3	no	—
200	D₄	C₄×C₅×D₅	1	3	yes	—
201	C₂×C₄	—	5	3	no	—
202	C₂×C₄	—	5	3	no	—
203	C₂×C₄	—	5	3	no	—
204	C₂×C₄	—	5	1	no	—
205	C₂×C₄	—	5	3	no	—
206	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
207	C₈	—	5	1	no	—
208	C₈	—	5	3	no	—
209	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
210	C₈	C₂²×C₅×D₅	1	3	yes	—
211	C₅	—	2	2	no	—
212	C₅	D₅×#131	1	1	yes	characteristic, direct product factor
213	C₅	C₂×C₅×#131	1	1	yes	characteristic, complemented
214	C₂²	—	5	1	no	—
215	C₂²	—	5	3	no	—
216	C₂²	—	10	6	no	—
217	C₂²	C₂×C₄×C₅×D₅	1	3	yes	—
218	C₄	—	5	1	no	—
219	C₄	—	5	3	no	—
220	C₄	C₂×C₄×C₅×D₅	1	3	yes	—
221	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
222	C₂	—	2	3	no	—
223	C₂	—	5	2	no	—
224	C₂	—	10	3	no	—
225	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₅²×#131	C₂
socle	C₂×C₅²	C₂³×C₄
prime radical	C₅×D₅×#41	C₂
cosocle	C₄×C₅	C₂⁴×C₅
hypercenter	C₅×#131	D₅
hypocenter	C₅	C₂×C₅×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₅×C₈ ◁ C₅×#131

Complements

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

N	H ≅ G/N	type
2. C₅²×#131	C₂	semidirect
3. C₅×#4094	C₂	semidirect
4. C₅×#4053	C₂	semidirect
5. C₅×D₅×M₁₆	C₂	semidirect
6. C₅×#4047	C₂	semidirect
7. C₂×C₅×C₈×D₅	C₂	semidirect
15. C₅²×M₁₆	C₂²	semidirect
16. C₂×C₅²×C₈	C₂²	semidirect
17. C₅×#681	C₂²	semidirect
18. C₂×C₅×#168	C₂²	semidirect
19. C₅×#676	C₂²	semidirect
22. C₅×C₈×D₅	C₂²	semidirect
23. C₅×C₈×D₅	C₂²	semidirect
25. D₅×#131	C₅	direct (N × H)
48. C₅×#131	D₅	direct (N × H)
49. C₅×#131	C₂×C₅	semidirect
52. #4094	C₂×C₅	semidirect
53. #4053	C₂×C₅	semidirect
54. D₅×M₁₆	C₂×C₅	semidirect
55. #4047	C₂×C₅	semidirect
56. C₂×C₈×D₅	C₂×C₅	semidirect
83. C₅×M₁₆	C₂×D₅	semidirect
84. C₅×M₁₆	C₂²×C₅	semidirect
91. C₂×C₅×C₈	C₂×D₅	semidirect
92. C₂×C₅×C₈	C₂²×C₅	semidirect
93. #681	C₂²×C₅	semidirect
94. C₂×#168	C₂²×C₅	semidirect
95. #676	C₂²×C₅	semidirect
98. C₈×D₅	C₂²×C₅	semidirect
99. C₈×D₅	C₂²×C₅	semidirect
104. C₅×D₅	#131	direct (N × H)
146. #131	C₅×D₅	direct (N × H)
149. C₅²	C₂×#131	semidirect
178. M₁₆	C₂×C₅×D₅	semidirect
184. C₂×C₈	C₂×C₅×D₅	semidirect
194. D₅	C₅×#131	direct (N × H)
212. C₅	D₅×#131	direct (N × H)
213. C₅	C₂×C₅×#131	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₅×#41	2
C₅²×#131	2
C₅×#4094	2
C₅×#4053	2
C₅×D₅×M₁₆	2
C₅×#4047	2
C₂×C₅×C₈×D₅	2
C₂×C₅×#131	5
D₅×#131	5

Subgroups — C5×D5×#131 #13106903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×#131 #13106903