Subgroups — C₂×C₅×C₈×AGL(1,𝔽₅) #13104476

Order 1600.

335 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
4. C₅×C₈×AGL(1,𝔽₅)	C₂	direct (N × H)
7. C₂×C₅×#170	C₄	semidirect
10. C₅×D₅⋊C₈	C₄	semidirect
12. C₂×C₅²×C₈	C₄	semidirect
13. C₂×C₅×#168	C₄	semidirect
14. C₅×C₈×D₅	C₄	semidirect
16. C₂×C₈×AGL(1,𝔽₅)	C₅	direct (N × H)
19. C₂×C₅×AGL(1,𝔽₅)	C₈	direct (N × H)
27. C₅×#170	C₂×C₄	semidirect
28. C₅²×C₈	C₂×C₄	semidirect
29. C₅×#168	C₂×C₄	semidirect
36. C₈×AGL(1,𝔽₅)	C₂×C₅	direct (N × H)
42. C₅×AGL(1,𝔽₅)	C₂×C₈	direct (N × H)
51. C₂×#170	C₄×C₅	semidirect
54. D₅⋊C₈	C₄×C₅	semidirect
63. C₂×C₅×C₈	C₄×C₅	semidirect
64. C₂×C₅×C₈	AGL(1,𝔽₅)	direct (N × H)
67. C₂×#168	C₄×C₅	semidirect
68. C₈×D₅	C₄×C₅	semidirect
71. C₂×C₅²	C₄×C₈	semidirect
73. C₅×D₅	C₄×C₈	semidirect
77. C₂×AGL(1,𝔽₅)	C₅×C₈	direct (N × H)
94. #170	C₂×C₄×C₅	semidirect
98. C₅×C₈	C₂×C₄×C₅	semidirect
99. C₅×C₈	C₂×AGL(1,𝔽₅)	direct (N × H)
100. #168	C₂×C₄×C₅	semidirect
105. C₅²	C₂×C₄×C₈	semidirect
115. AGL(1,𝔽₅)	C₂×C₅×C₈	direct (N × H)
135. C₂×C₈	C₅×AGL(1,𝔽₅)	direct (N × H)
142. C₂×C₅	C₄×C₅×C₈	semidirect
143. C₂×C₅	C₈×AGL(1,𝔽₅)	direct (N × H)
146. D₅	C₄×C₅×C₈	semidirect
159. C₈	C₂×C₅×AGL(1,𝔽₅)	direct (N × H)
161. C₅	C₂×C₄×C₅×C₈	semidirect
162. C₅	C₂×C₈×AGL(1,𝔽₅)	direct (N × H)
174. C₂	C₅×C₈×AGL(1,𝔽₅)	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×C₄×C₅×AGL(1,𝔽₅)	2
C₂×C₅×D₅⋊C₈	2
C₂×C₅×C₈×D₅	2
C₅×C₈×AGL(1,𝔽₅)	2
C₂×C₈×AGL(1,𝔽₅)	5
C₂×C₄×C₅×C₈	5

Subgroups — C2×C5×C8×AGL(1,𝔽5) #13104476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₈×AGL(1,𝔽₅) #13104476