Group #13100087 — subgroups

Subgroups — C₅×#64897 #13100087

Order 1600.

183 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
4	C₅×#704	—	2	2	no	—
5	C₅×#700	—	2	2	no	—
6	C₅×D₅⋊C₈	C₄	1	4	yes	—
7	C₂×C₅²×C₈	C₄	1	1	yes	characteristic
8	C₂×C₅×#168	C₄	1	1	yes	characteristic
9	C₅×#676	—	2	2	no	—
10	#64897	C₅	1	1	yes	maximal, characteristic, direct product factor
11	C₅×#338	—	5	1	no	maximal
12	C₂²×C₅×D₅	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
15	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
16	C₄×C₅×D₅	D₄	1	2	yes	—
17	C₅×#170	—	2	4	no	—
18	C₅²×C₈	—	2	1	no	—
19	C₅×#168	—	2	1	no	—
20	D₅⋊M₁₆	C₂×C₅	1	2	yes	—
21	C₂×C₅×M₁₆	—	5	2	no	—
22	C₂×C₅×M₁₆	—	5	1	no	—
23	C₂×#676	C₂×C₅	1	1	yes	characteristic
24	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₅×D₅	—	4	1	no	—
26	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
27	C₄×C₅²	C₄²	1	1	yes	characteristic
28	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
29	C₅×Dic₅	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₅	—	5	1	no	—
31	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
32	#704	—	2	2	no	—
33	#700	—	2	2	no	—
34	D₅⋊C₈	C₄×C₅	1	4	yes	—
35	C₅×M₁₆	—	10	2	no	—
36	C₅×M₁₆	—	10	2	no	—
37	C₅×M₁₆	—	10	2	no	—
38	C₂×C₅×C₈	—	4	1	no	—
39	C₂×C₅×C₈	—	5	4	no	—
40	C₂×C₅×C₈	—	5	1	no	—
41	C₂×C₅×C₈	C₄×C₅	1	1	yes	characteristic
42	C₂×C₅×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₂×#168	C₄×C₅	1	1	yes	characteristic
44	#676	—	2	2	no	—
45	#338	—	5	1	no	Sylow
46	C₂×C₅²	—	2	1	no	—
47	C₂×C₅²	#95	1	1	yes	characteristic
48	C₅×D₅	—	2	2	no	—
49	C₂³×C₅	—	5	1	no	—
50	C₂²×D₅	C₅×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅	—	4	1	no	—
52	C₂×C₄×C₅	—	5	1	no	—
53	C₂×C₄×C₅	—	5	2	no	—
54	C₂×C₄×C₅	—	5	2	no	—
55	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×Dic₅	C₅×D₄	1	1	yes	characteristic
58	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
59	C₄×D₅	C₅×D₄	1	2	yes	—
60	#170	—	2	4	no	—
61	C₅×C₈	—	2	1	no	—
62	C₅×C₈	—	2	1	no	—
63	C₅×C₈	—	4	2	no	—
64	C₅×C₈	—	10	4	no	—
65	C₅×C₈	—	10	1	no	—
66	#168	—	2	1	no	—
67	C₂×M₁₆	—	5	2	no	—
68	C₂×M₁₆	—	5	1	no	—
69	C₅²	#338	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₅	—	4	1	no	—
71	C₂²×C₅	—	5	2	no	—
72	C₂²×C₅	—	20	1	no	—
73	C₂²×C₅	C₅×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₂×D₅	—	4	1	no	—
76	C₂×D₅	C₅×C₄⋊C₄	1	2	yes	—
77	C₄×C₅	—	4	1	no	—
78	C₄×C₅	—	4	1	no	—
79	C₄×C₅	—	5	2	no	—
80	C₄×C₅	C₄²×C₅	1	1	yes	characteristic
81	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₅	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
84	Dic₅	C₅×C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄	—	5	1	no	—
86	M₁₆	—	10	2	no	—
87	M₁₆	—	10	2	no	—
88	M₁₆	—	10	2	no	—
89	C₂×C₈	—	5	4	no	—
90	C₂×C₈	—	5	1	no	—
91	C₂×C₈	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂×C₅	—	2	1	no	—
93	C₂×C₅	—	2	1	no	—
94	C₂×C₅	—	4	2	no	—
95	C₂×C₅	—	4	1	no	—
96	C₂×C₅	—	10	2	no	—
97	C₂×C₅	C₅×#95	1	1	yes	characteristic
98	C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
99	D₅	—	2	2	no	—
100	C₂³	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	2	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	10	4	no	—
107	C₈	—	10	1	no	—
108	C₅	—	4	1	no	—
109	C₅	C₅×#338	1	1	yes	characteristic, complemented
110	C₅	#64897	1	1	yes	characteristic, direct product factor
111	C₂²	—	5	2	no	—
112	C₂²	—	20	1	no	—
113	C₂²	C₅×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₄	—	5	2	no	—
115	C₄	C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₄	C₅×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	10	2	no	—
119	C₂	C₅×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×D₅⋊M₁₆

C₂

yes

maximal

C₂×C₅×#676

C₂

yes

maximal, characteristic

C₂×C₄×C₅×D₅

C₂²

yes

characteristic

C₅×#704

—

C₅×#700

—

C₅×D₅⋊C₈

C₄

yes

—

C₂×C₅²×C₈

C₄

yes

characteristic

C₂×C₅×#168

C₄

yes

characteristic

C₅×#676

—

#64897

C₅

yes

maximal, characteristic, direct product factor

C₅×#338

—

maximal

C₂²×C₅×D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

D₄

yes

characteristic

C₄×C₅×D₅

C₂×C₄

yes

—

C₄×C₅×D₅

D₄

yes

—

C₅×#170

—

C₅²×C₈

—

C₅×#168

—

D₅⋊M₁₆

C₂×C₅

yes

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

—

C₂×#676

C₂×C₅

yes

characteristic

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

C₄⋊C₄

yes

—

C₄×C₅²

C₄²

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₅×Dic₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

—

C₂×C₄×D₅

C₂²×C₅

yes

characteristic

#704

—

#700

—

D₅⋊C₈

C₄×C₅

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×C₅

yes

C₂×C₅×C₈

yes

C₄×C₅

yes

—

—

C₂×C₅²

—

C₂×C₅²

#95

yes

characteristic

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

C₅×Q₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

C₅×D₄

yes

characteristic

C₄×D₅

C₂×C₄×C₅

yes

—

C₄×D₅

C₅×D₄

yes

—

#170

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₅²

#338

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₅×C₄⋊C₄

yes

characteristic

C₂²×C₅

C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₂×D₅

—

C₂×D₅

C₅×C₄⋊C₄

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄²×C₅

yes

characteristic

C₄×C₅

C₅×C₂²⋊C₄

yes

characteristic

C₄×C₅

C₄×AGL(1,𝔽₅)

yes

characteristic

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

Dic₅

C₅×C₂²⋊C₄

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₅×AGL(1,𝔽₅)

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₅×#95

yes

characteristic

C₂×C₅

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

D₅

—

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₅×AGL(1,𝔽₅)

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₅

—

109

C₅

C₅×#338

yes

characteristic, complemented

110

C₅

#64897

yes

characteristic, direct product factor

111

C₂²

—

112

C₂²

—

113

C₂²

C₅×C₄⋊AGL(1,𝔽₅)

yes

characteristic

114

C₄

—

115

C₄

C₄×C₅×AGL(1,𝔽₅)

yes

characteristic

116

C₄

C₅×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

C₅×(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄×C₅	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²×C₅
Frattini	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)
Fitting	C₂×C₅²×C₈	C₄
socle	C₂×C₅²	#95
prime radical	C₂²×C₅×D₅	Q₈
cosocle	C₂×C₄×D₅	C₂²×C₅
hypercenter	C₂×C₅×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₅×#338
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈

subgroup

G/H

center

C₄×C₅

C₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₄²×C₅

Frattini

C₂×C₄

C₂×C₅×AGL(1,𝔽₅)

Fitting

C₂×C₅²×C₈

C₄

socle

C₂×C₅²

#95

prime radical

C₂²×C₅×D₅

Q₈

cosocle

C₂×C₄×D₅

C₂²×C₅

C₂×C₅×C₈

C₅

G ▷ C₄×C₅ ▷ C₅

C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈

N	H ≅ G/N	type
10. #64897	C₅	direct (N × H)
69. C₅²	#338	semidirect
109. C₅	C₅×#338	semidirect
110. C₅	#64897	direct (N × H)

H ≅ G/N

type

10. #64897

C₅

direct (N × H)

69. C₅²

#338

semidirect

109. C₅

C₅×#338

semidirect

110. C₅

#64897

direct (N × H)

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

subgroup	index
C₅×D₅⋊M₁₆	2
C₂×C₅×#676	2
#64897	5
C₅×#338	5

subgroup

index

C₅×D₅⋊M₁₆

C₂×C₅×#676

#64897

C₅×#338

Subgroups — C₅×#64897 #13100087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#64897 #13100087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#64897 #13100087