Subgroups — C₅×#65718 #13104478

Order 1600.

271 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4130	C₂	1	1	yes	maximal, characteristic
2	C₅×D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₅×#3992	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
7	C₅×C₂²⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
8	C₅×#704	C₄	1	2	yes	complemented
9	C₅×C₄⋊AGL(1,𝔽₅)	C₄	1	2	yes	—
10	C₄×C₅×AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	C₅×#700	C₄	1	2	yes	complemented
12	C₅×D₅⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅²×C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×#168	C₄	1	1	yes	characteristic, complemented
15	C₅×C₈×D₅	C₄	1	1	yes	characteristic, complemented
16	C₅×C₈×D₅	C₄	1	1	yes	characteristic, complemented
17	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
18	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
19	#65718	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₅×#405	—	5	1	no	maximal
21	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
29	C₅×#170	C₂×C₄	1	4	yes	—
30	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₅×#168	C₂×C₄	1	1	yes	characteristic
33	C₅×#168	C₂×C₄	1	1	yes	characteristic
34	#4130	C₂×C₅	1	1	yes	characteristic
35	D₅⋊M₁₆	C₂×C₅	1	1	yes	characteristic
36	C₂×C₅×M₁₆	—	5	1	no	—
37	C₂²×C₅×C₈	—	5	1	no	—
38	C₅×#117	—	5	1	no	—
39	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic
40	#3992	C₂×C₅	1	2	yes	complemented
41	C₈×AGL(1,𝔽₅)	C₂×C₅	1	2	yes	complemented
42	C₅×#97	—	5	2	no	—
43	C₄×C₅×C₈	—	5	2	no	—
44	C₂²×C₅²	C₄²	1	1	yes	characteristic
45	C₂×C₅×D₅	—	2	2	no	—
46	C₂×C₅×D₅	C₄²	1	1	yes	characteristic
47	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
48	C₅×AGL(1,𝔽₅)	—	2	4	no	—
49	C₄×C₅²	C₄²	1	1	yes	characteristic
50	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
51	C₅×Dic₅	C₄²	1	1	yes	characteristic
52	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	5	1	no	—
54	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
55	C₂²⋊AGL(1,𝔽₅)	C₄×C₅	1	2	yes	—
56	#704	C₄×C₅	1	2	yes	complemented
57	C₄⋊AGL(1,𝔽₅)	C₄×C₅	1	2	yes	—
58	C₄×AGL(1,𝔽₅)	C₂²×C₅	1	2	yes	—
59	#700	C₄×C₅	1	2	yes	complemented
60	D₅⋊C₈	C₂²×C₅	1	2	yes	—
61	C₅×M₁₆	—	5	2	no	—
62	C₅×M₁₆	—	5	2	no	—
63	C₂×C₅×C₈	—	4	1	no	—
64	C₂×C₅×C₈	—	5	2	no	—
65	C₂×C₅×C₈	—	5	1	no	—
66	C₂×C₅×C₈	—	5	1	no	—
67	C₂×C₅×C₈	—	5	1	no	—
68	C₂×C₅×C₈	—	5	1	no	—
69	C₂×C₅×C₈	—	5	1	no	—
70	C₂×C₅×C₈	C₄×C₅	1	1	yes	characteristic, complemented
71	C₂×C₅×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
72	C₅×C₄⋊C₄	—	5	2	no	—
73	C₅×C₂²⋊C₄	—	5	2	no	—
74	C₄²×C₅	—	5	2	no	—
75	C₂×#168	C₄×C₅	1	1	yes	characteristic, complemented
76	C₈×D₅	C₄×C₅	1	1	yes	characteristic, complemented
77	C₈×D₅	C₄×C₅	1	1	yes	characteristic, complemented
78	C₈×D₅	C₂²×C₅	1	1	yes	characteristic
79	C₈×D₅	C₂²×C₅	1	1	yes	characteristic
80	#405	—	5	1	no	Sylow
81	C₂×C₅²	—	2	1	no	—
82	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
83	C₅×D₅	—	2	1	no	—
84	C₅×D₅	#131	1	2	yes	—
85	C₂³×C₅	—	5	1	no	—
86	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	1	4	yes	—
88	C₂×C₄×C₅	—	4	1	no	—
89	C₂×C₄×C₅	—	5	1	no	—
90	C₂×C₄×C₅	—	5	1	no	—
91	C₂×C₄×C₅	—	5	1	no	—
92	C₂×C₄×C₅	—	5	1	no	—
93	C₂×C₄×C₅	—	5	1	no	—
94	C₂×C₄×C₅	—	5	4	no	—
95	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
98	C₄×D₅	C₂×C₄×C₅	1	1	yes	characteristic
99	C₄×D₅	C₂×C₄×C₅	1	1	yes	characteristic
100	C₄×D₅	C₂×C₄×C₅	1	1	yes	characteristic
101	C₄×D₅	C₂³×C₅	1	1	yes	characteristic
102	#170	C₂×C₄×C₅	1	4	yes	—
103	C₅×C₈	—	4	1	no	—
104	C₅×C₈	—	4	1	no	—
105	C₅×C₈	—	5	4	no	—
106	C₅×C₈	—	5	1	no	—
107	C₅×C₈	—	5	1	no	—
108	C₅×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₅×C₈	C₂×C₄×C₅	1	1	yes	characteristic
110	C₅×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₅×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
112	#168	C₂×C₄×C₅	1	1	yes	characteristic
113	#168	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂×M₁₆	—	5	1	no	—
115	C₂²×C₈	—	5	1	no	—
116	#117	—	5	1	no	—
117	#97	—	5	2	no	—
118	C₄×C₈	—	5	2	no	—
119	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₅	—	4	1	no	—
121	C₂²×C₅	—	5	1	no	—
122	C₂²×C₅	—	5	1	no	—
123	C₂²×C₅	—	10	2	no	—
124	C₂²×C₅	C₄²×C₅	1	1	yes	characteristic
125	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂×D₅	—	2	2	no	—
127	C₂×D₅	C₄²×C₅	1	1	yes	characteristic
128	C₂×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
129	AGL(1,𝔽₅)	—	2	4	no	—
130	C₄×C₅	—	4	1	no	—
131	C₄×C₅	—	4	1	no	—
132	C₄×C₅	—	5	1	no	—
133	C₄×C₅	—	5	1	no	—
134	C₄×C₅	—	10	4	no	—
135	C₄×C₅	C₄²×C₅	1	1	yes	characteristic
136	C₄×C₅	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
138	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
139	Dic₅	C₄²×C₅	1	1	yes	characteristic
140	Dic₅	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂²×C₄	—	5	1	no	—
142	M₁₆	—	5	2	no	—
143	M₁₆	—	5	2	no	—
144	C₂×C₈	—	5	2	no	—
145	C₂×C₈	—	5	1	no	—
146	C₂×C₈	—	5	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	1	no	—
149	C₂×C₈	—	5	1	no	—
150	C₂×C₈	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
151	C₄⋊C₄	—	5	2	no	—
152	C₂²⋊C₄	—	5	2	no	—
153	C₄²	—	5	2	no	—
154	C₂×C₅	—	2	1	no	—
155	C₂×C₅	—	2	1	no	—
156	C₂×C₅	—	4	2	no	—
157	C₂×C₅	—	4	1	no	—
158	C₂×C₅	—	5	2	no	—
159	C₂×C₅	—	10	1	no	—
160	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
161	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
162	D₅	—	2	1	no	—
163	D₅	C₅×#131	1	2	yes	—
164	C₂³	—	5	1	no	—
165	C₂×C₄	—	5	1	no	—
166	C₂×C₄	—	5	1	no	—
167	C₂×C₄	—	5	1	no	—
168	C₂×C₄	—	5	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	5	4	no	—
171	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₈	—	5	4	no	—
173	C₈	—	5	1	no	—
174	C₈	—	5	1	no	—
175	C₈	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₈	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₅	—	4	1	no	—
178	C₅	C₅×#405	1	1	yes	characteristic, complemented
179	C₅	#65718	1	1	yes	characteristic, direct product factor
180	C₂²	—	5	1	no	—
181	C₂²	—	5	1	no	—
182	C₂²	—	10	2	no	—
183	C₂²	C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₄	—	5	1	no	—
185	C₄	—	5	1	no	—
186	C₄	—	10	4	no	—
187	C₄	C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₄	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	5	2	no	—
191	C₂	—	10	1	no	—
192	C₂	C₂×C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₄	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₂×C₅²×C₈	C₄
socle	C₂×C₅²	C₂×C₄²
prime radical	C₂²×C₅×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³×C₅
hypercenter	C₂×C₅×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₅×#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#3992	C₂	semidirect
5. C₅×C₈×AGL(1,𝔽₅)	C₂	semidirect
8. C₅×#704	C₄	semidirect
11. C₅×#700	C₄	semidirect
13. C₂×C₅²×C₈	C₄	semidirect
14. C₂×C₅×#168	C₄	semidirect
15. C₅×C₈×D₅	C₄	semidirect
16. C₅×C₈×D₅	C₄	semidirect
19. #65718	C₅	direct (N × H)
40. #3992	C₂×C₅	semidirect
41. C₈×AGL(1,𝔽₅)	C₂×C₅	semidirect
56. #704	C₄×C₅	semidirect
59. #700	C₄×C₅	semidirect
70. C₂×C₅×C₈	C₄×C₅	semidirect
71. C₂×C₅×C₈	AGL(1,𝔽₅)	semidirect
75. C₂×#168	C₄×C₅	semidirect
76. C₈×D₅	C₄×C₅	semidirect
77. C₈×D₅	C₄×C₅	semidirect
119. C₅²	#405	semidirect
150. C₂×C₈	C₅×AGL(1,𝔽₅)	semidirect
178. C₅	C₅×#405	semidirect
179. C₅	#65718	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#4130	2
C₅×D₅⋊M₁₆	2
C₂×C₅×C₈×D₅	2
C₅×#3992	2
C₅×C₈×AGL(1,𝔽₅)	2
#65718	5
C₅×#405	5

Subgroups — C5×#65718 #13104478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65718 #13104478