Group #13102119 — subgroups

Subgroups — C₅×#65147 #13102119

Order 1600.

208 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 86 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×#4036	C₂	1	2	yes	maximal
2	C₅²×#106	C₂	1	1	yes	maximal, characteristic
3	C₅×#3949	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₅×#3939	C₂	1	2	yes	maximal
6	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
7	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
8	C₅×C₄⋊Dic₅	—	2	2	no	—
9	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₅×#683	—	2	2	no	—
11	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#168	C₂²	1	1	yes	characteristic
13	C₅×#499	—	5	1	no	maximal
14	#65147	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×C₅×Dic₅	—	2	2	no	—
18	C₂×C₅×Dic₅	D₄	1	2	yes	—
19	C₅²×C₈	Q₈	1	2	yes	—
20	C₅×#168	Q₈	1	2	yes	—
21	C₅×#125	—	5	2	no	—
22	#4036	C₂×C₅	1	2	yes	—
23	C₅×#107	—	5	2	no	—
24	C₅×#106	—	2	2	no	—
25	C₅×#106	—	5	1	no	—
26	C₅×#106	D₅	1	1	yes	characteristic
27	C₅×#106	C₂×C₅	1	1	yes	characteristic
28	C₄×C₅×C₈	—	5	1	no	—
29	#3949	C₂×C₅	1	1	yes	characteristic
30	C₈×Dic₅	C₂×C₅	1	1	yes	characteristic
31	#3939	C₂×C₅	1	2	yes	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅²	—	4	2	no	—
34	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
36	C₅×Dic₅	—	2	2	no	—
37	C₅×Dic₅	—	4	2	no	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	—	5	1	no	—
40	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
42	C₅×C₄⋊C₄	—	2	4	no	—
43	C₅×C₄⋊C₄	—	5	2	no	—
44	C₅×C₄⋊C₄	—	10	2	no	—
45	C₅×C₄⋊C₄	—	10	2	no	—
46	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
47	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
48	C₄²×C₅	—	5	1	no	—
49	C₄⋊Dic₅	—	2	2	no	—
50	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
51	#683	—	2	2	no	—
52	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
53	C₂×#168	C₂²×C₅	1	1	yes	characteristic
54	#499	—	5	1	no	Sylow
55	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₅²	#135	1	2	yes	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	4	4	no	—
61	C₂×C₄×C₅	—	5	2	no	—
62	C₂×C₄×C₅	—	10	2	no	—
63	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
65	C₂×Dic₅	—	2	2	no	—
66	C₂×Dic₅	C₅×D₄	1	2	yes	—
67	C₅×C₈	—	2	4	no	—
68	C₅×C₈	—	5	2	no	—
69	C₅×C₈	Dic₁₀	1	2	yes	—
70	C₅×C₈	C₅×Q₈	1	2	yes	—
71	#168	C₅×Q₈	1	2	yes	—
72	#125	—	5	2	no	—
73	#107	—	5	2	no	—
74	#106	—	5	1	no	—
75	#106	C₅×D₅	1	1	yes	characteristic
76	C₄×C₈	—	5	1	no	—
77	C₅²	#499	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
80	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
81	C₄×C₅	—	2	2	no	—
82	C₄×C₅	—	2	2	no	—
83	C₄×C₅	—	4	2	no	—
84	C₄×C₅	—	4	2	no	—
85	C₄×C₅	—	8	4	no	—
86	C₄×C₅	—	10	2	no	—
87	C₄×C₅	—	20	2	no	—
88	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
89	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
90	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
91	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
92	Dic₅	—	2	2	no	—
93	Dic₅	—	4	2	no	—
94	C₂×C₈	—	5	1	no	—
95	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
96	C₄⋊C₄	—	5	2	no	—
97	C₄⋊C₄	—	10	2	no	—
98	C₄⋊C₄	—	10	2	no	—
99	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
100	C₄²	—	5	1	no	—
101	C₂×C₅	—	2	2	no	—
102	C₂×C₅	—	2	4	no	—
103	C₂×C₅	#4034	1	1	yes	characteristic
104	C₂×C₅	#4062	1	2	yes	—
105	C₂×C₅	C₅×C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₅	C₅×#135	1	2	yes	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	5	2	no	—
109	C₂×C₄	—	10	2	no	—
110	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
111	C₈	—	5	2	no	—
112	C₈	C₅×Dic₁₀	1	2	yes	—
113	C₅	—	2	2	no	—
114	C₅	#65147	1	1	yes	characteristic, direct product factor
115	C₅	C₅×#499	1	1	yes	characteristic, complemented
116	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	10	2	no	—
119	C₄	—	20	2	no	—
120	C₄	C₂×C₅×Dic₁₀	1	1	yes	characteristic
121	C₄	C₅×Q₈×D₅	1	1	yes	characteristic
122	C₂	C₅×#4034	1	1	yes	characteristic
123	C₂	C₅×#4062	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#4036

C₂

yes

maximal

C₅²×#106

C₂

yes

maximal, characteristic

C₅×#3949

C₂

yes

maximal, characteristic

C₅×C₈×Dic₅

C₂

yes

maximal, characteristic

C₅×#3939

C₂

yes

maximal

C₂×C₅²×C₈

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

C₅×C₄⋊Dic₅

—

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

—

C₄×C₅×Dic₅

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

C₅

yes

maximal, characteristic, direct product factor

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

D₄

yes

—

C₅²×C₈

Q₈

yes

—

C₅×#168

Q₈

yes

—

C₅×#125

—

#4036

C₂×C₅

yes

—

—

—

—

D₅

yes

characteristic

C₅×#106

C₂×C₅

yes

characteristic

C₄×C₅×C₈

—

#3949

C₂×C₅

yes

characteristic

C₈×Dic₅

C₂×C₅

yes

characteristic

#3939

C₂×C₅

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

characteristic

C₂×C₅×C₈

C₂²×C₅

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂²×C₅

yes

—

C₄²×C₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂²×C₅

yes

—

#683

—

C₄×Dic₅

C₂²×C₅

yes

characteristic

C₂×#168

C₂²×C₅

yes

characteristic

#499

—

Sylow

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂×C₅²

#135

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₅×D₄

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

Dic₁₀

yes

—

C₅×C₈

C₅×Q₈

yes

—

#168

C₅×Q₈

yes

—

—

—

—

C₅×D₅

yes

characteristic

C₄×C₈

—

C₅²

#499

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

characteristic

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Dic₁₀

yes

characteristic

C₄×C₅

Q₈×D₅

yes

characteristic

C₄×C₅

C₂×C₅×Q₈

yes

characteristic

C₄×C₅

C₂×C₅×Q₈

yes

characteristic

Dic₅

—

Dic₅

—

C₂×C₈

—

C₂×C₈

C₂×C₅×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅×D₅

yes

—

100

C₄²

—

101

C₂×C₅

—

102

C₂×C₅

—

103

C₂×C₅

#4034

yes

characteristic

104

C₂×C₅

#4062

yes

—

105

C₂×C₅

C₅×C₄⋊Q₈

yes

characteristic

106

C₂×C₅

C₅×#135

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×C₅×D₅

yes

characteristic

111

C₈

—

112

C₈

C₅×Dic₁₀

yes

—

113

C₅

—

114

C₅

#65147

yes

characteristic, direct product factor

115

C₅

C₅×#499

yes

characteristic, complemented

116

C₂²

C₅×D₄×D₅

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×C₅×Dic₁₀

yes

characteristic

121

C₄

C₅×Q₈×D₅

yes

characteristic

122

C₂

C₅×#4034

yes

characteristic

123

C₂

C₅×#4062

yes

—

	subgroup	G/H
center	C₂²×C₅	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×#106	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₅×#106	D₅
hypocenter	C₅	C₅×#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#106

subgroup

G/H

center

C₂²×C₅

D₄×D₅

commutator, cosocle

C₂×C₄×C₅

C₂³×C₅

Frattini

C₂×C₄

C₂²×C₅×D₅

Fitting

C₅²×#106

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#106

N	H ≅ G/N	type
14. #65147	C₅	direct (N × H)
77. C₅²	#499	semidirect
114. C₅	#65147	direct (N × H)
115. C₅	C₅×#499	semidirect

H ≅ G/N

type

14. #65147

C₅

direct (N × H)

77. C₅²

#499

semidirect

114. C₅

#65147

direct (N × H)

115. C₅

C₅×#499

semidirect

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

subgroup	index
C₅×#4036	2
C₅²×#106	2
C₅×#3949	2
C₅×C₈×Dic₅	2
C₅×#3939	2
C₅×#499	5
#65147	5

subgroup

index

C₅²×#106

C₅×C₈×Dic₅

Subgroups — C₅×#65147 #13102119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65147 #13102119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65147 #13102119