Group #13103513 — subgroups

Subgroups — #13103513

Order 1600.

201 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 74 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	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6032	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#170	C₂	1	1	yes	maximal, characteristic
4	#11716760	C₂	1	2	yes	maximal
5	#11716752	C₂	1	2	yes	maximal
6	C₂×C₅×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×#6032	C₂²	1	2	yes	—
8	C₂×C₅×#170	C₂²	1	1	yes	characteristic
9	C₂×C₅×#170	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	C₄	1	1	yes	characteristic
11	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	#87780	C₄	1	2	yes	—
13	Dic₅²	C₂²	1	2	yes	—
14	#65788	—	5	1	no	maximal
15	#64968	—	5	1	no	maximal
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×#1039	C₂×C₄	1	2	yes	—
18	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
20	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
21	C₅×Dic₁₀	—	2	2	no	—
22	#6032	—	2	2	no	—
23	C₅×#170	—	2	1	no	—
24	C₅×#170	Q₈	1	2	yes	complemented
25	Q₈×Dic₅	—	5	1	no	—
26	C₄×Dic₁₀	—	5	1	no	—
27	#4004	—	5	2	no	—
28	#4002	—	5	2	no	—
29	C₄⋊#170	—	5	1	no	—
30	C₄×#170	D₅	1	1	yes	characteristic
31	C₄×C₅×C₈	—	5	1	no	—
32	#3947	—	5	2	no	—
33	#3946	—	5	2	no	—
34	C₄⋊#168	—	5	1	no	—
35	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅²	M₁₆	1	2	yes	—
37	#1039	—	2	2	no	—
38	C₅×Dic₅	—	2	2	no	—
39	C₅×Dic₅	—	2	1	no	—
40	C₅×Dic₅	C₂×Q₈	1	1	yes	characteristic
41	C₅×Dic₅	#41	1	1	yes	characteristic
42	C₂×C₅×Q₈	—	5	1	no	—
43	C₂×Dic₁₀	AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×#170	—	5	2	no	—
45	C₂×#170	C₂×D₅	1	1	yes	characteristic
46	C₂×#170	C₂×D₅	1	1	yes	characteristic
47	C₂×C₅×C₈	—	5	1	no	—
48	C₂×C₅×C₈	—	5	1	no	—
49	C₄²×C₅	—	5	1	no	—
50	C₄⋊Dic₅	—	5	1	no	—
51	C₄⋊Dic₅	—	5	1	no	—
52	C₄⋊Dic₅	—	5	2	no	—
53	C₄⋊Dic₅	—	20	1	no	—
54	#683	—	5	2	no	—
55	C₄×Dic₅	—	5	2	no	—
56	C₄×Dic₅	—	5	2	no	—
57	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
58	C₂×#168	—	5	2	no	—
59	#446	—	25	1	no	Sylow
60	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₅²	#131	1	1	yes	characteristic
63	C₅×Q₈	—	10	2	no	—
64	C₂×C₄×C₅	—	4	1	no	—
65	C₂×C₄×C₅	—	5	1	no	—
66	C₂×C₄×C₅	—	5	2	no	—
67	C₂×C₄×C₅	—	5	1	no	—
68	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂×Dic₅	—	5	2	no	—
71	C₂×Dic₅	—	5	2	no	—
72	C₂×Dic₅	—	20	2	no	—
73	C₂×Dic₅	C₄×D₅	1	1	yes	characteristic
74	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
75	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
76	Dic₁₀	—	2	2	no	—
77	#170	—	2	1	no	—
78	#170	—	10	2	no	—
79	#170	Dic₁₀	1	2	yes	complemented
80	C₅×C₈	—	5	2	no	—
81	C₅×C₈	—	10	1	no	—
82	#168	—	10	2	no	—
83	C₄×Q₈	—	25	1	no	—
84	C₄⋊C₈	—	25	1	no	—
85	C₄⋊C₈	—	25	2	no	—
86	#97	—	25	2	no	—
87	C₄×C₈	—	5	1	no	—
88	C₅²	#446	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₅	—	4	1	no	—
90	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
91	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₄×C₅	—	4	2	no	—
93	C₄×C₅	—	5	1	no	—
94	C₄×C₅	—	5	1	no	—
95	C₄×C₅	—	10	1	no	—
96	C₄×C₅	—	10	2	no	—
97	C₄×C₅	#676	1	2	yes	—
98	C₄×C₅	#704	1	2	yes	—
99	Dic₅	—	2	2	no	—
100	Dic₅	—	2	1	no	—
101	Dic₅	—	10	2	no	—
102	Dic₅	—	10	2	no	—
103	Dic₅	—	40	2	no	—
104	Dic₅	C₂×Dic₁₀	1	1	yes	characteristic
105	Dic₅	#709	1	1	yes	characteristic
106	C₂×Q₈	—	5	1	no	—
107	C₂×C₈	—	5	1	no	—
108	C₂×C₈	—	5	1	no	—
109	C₂×C₈	—	25	2	no	—
110	C₄⋊C₄	—	25	2	no	—
111	C₄⋊C₄	—	25	1	no	—
112	C₄²	—	5	1	no	—
113	C₄²	—	25	2	no	—
114	C₂×C₅	—	4	1	no	—
115	C₂×C₅	—	4	1	no	—
116	C₂×C₅	—	4	1	no	—
117	C₂×C₅	C₄×Dic₁₀	1	1	yes	characteristic
118	C₂×C₅	C₂×#676	1	1	yes	characteristic
119	C₂×C₅	#4047	1	1	yes	characteristic
120	C₂×C₅	#4133	1	1	yes	characteristic
121	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₂×C₅	C₂×#704	1	1	yes	characteristic
123	Q₈	—	10	2	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	—	5	1	no	—
126	C₂×C₄	—	5	2	no	—
127	C₂×C₄	—	25	2	no	—
128	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₈	—	5	2	no	—
130	C₈	—	10	1	no	—
131	C₈	—	50	2	no	—
132	C₅	—	4	1	no	—
133	C₅	#64968	1	1	yes	characteristic, complemented
134	C₅	#65788	1	1	yes	characteristic, complemented
135	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₄	—	5	1	no	—
137	C₄	—	5	1	no	—
138	C₄	—	10	1	no	—
139	C₄	—	10	2	no	—
140	C₄	—	50	2	no	—
141	C₄	#87828	1	2	yes	—
142	C₂	AGL(1,𝔽₅)×Dic₁₀	1	1	yes	characteristic
143	C₂	#11717330	1	1	yes	characteristic
144	C₂	C₂×#87828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Dic₅×Dic₁₀

C₂

yes

maximal, characteristic

C₄⋊#6032

C₂

yes

maximal, characteristic

C₄×C₅×#170

C₂

yes

maximal, characteristic

#11716760

C₂

yes

maximal

#11716752

C₂

yes

maximal

C₂×C₅×Dic₁₀

C₄

yes

characteristic

C₂×#6032

C₂²

yes

—

C₂×C₅×#170

C₂²

yes

characteristic

C₂×C₅×#170

C₂²

yes

characteristic

C₄⋊#1039

C₄

yes

characteristic

C₄×C₅×Dic₅

C₂²

yes

characteristic

#87780

C₄

yes

—

Dic₅²

C₂²

yes

—

—

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂³

yes

characteristic

C₅×Dic₁₀

—

#6032

—

C₅×#170

—

C₅×#170

Q₈

yes

complemented

Q₈×Dic₅

—

C₄×Dic₁₀

—

—

—

—

D₅

yes

characteristic

C₄×C₅×C₈

—

#3947

—

#3946

—

C₄⋊#168

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

M₁₆

yes

—

#1039

—

C₅×Dic₅

—

C₅×Dic₅

—

C₅×Dic₅

C₂×Q₈

yes

characteristic

C₅×Dic₅

#41

yes

characteristic

C₂×C₅×Q₈

—

C₂×Dic₁₀

yes

—

C₂×D₅

yes

characteristic

C₂×#170

C₂×D₅

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₅

yes

—

—

C₂×C₅²

C₄×Q₈

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

#131

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₄×D₅

yes

characteristic

C₂×Dic₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

C₂²×D₅

yes

characteristic

Dic₁₀

—

#170

—

#170

—

#170

Dic₁₀

yes

complemented

C₅×C₈

—

C₅×C₈

—

#168

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₅²

#446

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

characteristic

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#676

yes

—

C₄×C₅

#704

yes

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

C₂×Dic₁₀

yes

characteristic

105

Dic₅

#709

yes

characteristic

106

C₂×Q₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂×C₅

—

115

C₂×C₅

—

116

C₂×C₅

—

117

C₂×C₅

C₄×Dic₁₀

yes

characteristic

118

C₂×C₅

C₂×#676

yes

characteristic

119

C₂×C₅

#4047

yes

characteristic

120

C₂×C₅

#4133

yes

characteristic

121

C₂×C₅

Q₈×AGL(1,𝔽₅)

yes

characteristic

122

C₂×C₅

C₂×#704

yes

characteristic

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₅×AGL(1,𝔽₅)

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₅

—

133

C₅

#64968

yes

characteristic, complemented

134

C₅

#65788

yes

characteristic, complemented

135

C₂²

C₂×D₅×AGL(1,𝔽₅)

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#87828

yes

—

142

C₂

AGL(1,𝔽₅)×Dic₁₀

yes

characteristic

143

C₂

#11717330

yes

characteristic

144

C₂

C₂×#87828

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₄×C₅²	C₂×C₄
cosocle	C₂×C₅×Dic₅	C₂³
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#446
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₂²

C₂×D₅×AGL(1,𝔽₅)

commutator, socle, prime radical

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₄×C₅²

C₂×C₄

cosocle

C₂×C₅×Dic₅

C₂³

C₂×C₄

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
24. C₅×#170	Q₈	semidirect
79. #170	Dic₁₀	semidirect
88. C₅²	#446	semidirect
133. C₅	#64968	semidirect
134. C₅	#65788	semidirect

H ≅ G/N

type

24. C₅×#170

Q₈

semidirect

79. #170

Dic₁₀

semidirect

88. C₅²

#446

semidirect

133. C₅

#64968

semidirect

134. C₅

#65788

semidirect

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

subgroup	index
Dic₅×Dic₁₀	2
C₄⋊#6032	2
C₄×C₅×#170	2
#11716760	2
#11716752	2
#65788	5
#64968	5

subgroup

index

Dic₅×Dic₁₀

C₄×C₅×#170

Subgroups — #13103513

Conjugacy classes of subgroups

Complements

Maximal subgroups