Group #13023002 — subgroups

Subgroups — #13023002

Order 1280.

282 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 132 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	#10603825	C₂	1	1	yes	maximal, characteristic
2	#10603823	C₂	1	1	yes	maximal, characteristic
3	#10598840	C₂	1	1	yes	maximal, characteristic
4	#10598515	C₂	1	4	yes	maximal, complemented
5	C₂×#3990	C₄	1	2	yes	—
6	C₂×D₅⋊C₁₆	C₄	1	2	yes	—
7	#64974	C₄	1	2	yes	—
8	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
9	#64886	C₄	1	2	yes	—
10	#64885	C₄	1	2	yes	—
11	#64859	C₂²	1	2	yes	—
12	#64859	C₂²	1	2	yes	—
13	C₅×C₈²	C₄	1	1	yes	characteristic
14	#64674	C₂²	1	2	yes	—
15	C₈×#168	C₄	1	1	yes	characteristic
16	#11405	—	5	1	no	maximal, Sylow
17	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
19	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×#674	C₂×C₄	1	4	yes	—
21	C₂×#674	C₂×C₄	1	4	yes	—
22	#3990	—	2	4	no	—
23	D₅⋊C₁₆	C₈	1	8	yes	complemented
24	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	#3952	—	2	2	no	—
27	#3946	—	2	2	no	—
28	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₄×#168	C₂×C₄	1	2	yes	—
31	C₄×#168	C₂³	1	1	yes	characteristic
32	C₄×M₃₂	—	5	1	no	—
33	C₂×#346	—	5	1	no	—
34	#1450	—	5	1	no	—
35	#1313	—	5	4	no	—
36	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
37	C₂×C₄×D₅	M₁₆	1	2	yes	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
40	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
41	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅	C₄²	1	1	yes	characteristic
43	C₄×Dic₅	M₁₆	1	2	yes	—
44	C₂×#168	—	2	2	no	—
45	C₂×#168	C₂²×C₄	1	1	yes	characteristic
46	C₂×#168	C₂²×C₄	1	1	yes	characteristic
47	C₈×D₅	C₂×C₈	1	4	yes	—
48	C₈×D₅	M₁₆	1	4	yes	—
49	#674	—	2	4	no	—
50	#674	C₂×C₈	1	8	yes	—
51	C₂×M₃₂	—	5	2	no	—
52	C₂²×C₁₆	—	5	2	no	—
53	#431	—	5	2	no	—
54	C₂×C₄×C₈	—	5	1	no	—
55	#346	—	5	2	no	—
56	#346	—	5	2	no	—
57	#346	—	5	2	no	—
58	C₄×C₁₆	—	5	2	no	—
59	#322	—	5	2	no	—
60	C₈²	—	5	1	no	—
61	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂²×D₅	#97	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
65	C₂×Dic₅	#97	1	2	yes	—
66	C₂×Dic₅	#97	1	1	yes	characteristic
67	C₄×D₅	—	4	2	no	—
68	C₄×D₅	C₄×C₈	1	2	yes	—
69	C₄×D₅	#97	1	2	yes	—
70	C₅×C₈	—	4	2	no	—
71	C₅×C₈	C₄×C₈	1	2	yes	—
72	C₅×C₈	#97	1	2	yes	—
73	#168	—	4	2	no	—
74	#168	C₂²×C₈	1	2	yes	—
75	#168	C₂×M₁₆	1	2	yes	—
76	C₂²×C₈	—	5	1	no	—
77	C₂²×C₈	—	5	1	no	—
78	C₂×C₄²	—	5	1	no	—
79	M₃₂	—	10	4	no	—
80	C₂×C₁₆	—	5	4	no	—
81	C₂×C₁₆	—	5	8	no	—
82	C₂×C₁₆	—	5	4	no	—
83	C₄⋊C₈	—	10	2	no	—
84	C₂²⋊C₈	—	10	2	no	—
85	C₄×C₈	—	5	1	no	—
86	C₄×C₈	—	5	1	no	—
87	C₄×C₈	—	5	1	no	—
88	C₄×C₈	—	5	2	no	—
89	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
90	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₂²×C₅	C₂×#97	1	1	yes	characteristic
92	C₂×D₅	—	2	2	no	—
93	C₂×D₅	#322	1	2	yes	—
94	C₄×C₅	—	2	2	no	—
95	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₅	C₂×#97	1	1	yes	characteristic
97	Dic₅	—	2	2	no	—
98	Dic₅	#322	1	2	yes	—
99	C₂²×C₄	—	5	1	no	—
100	C₂²×C₄	—	5	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	5	1	no	—
103	C₂×C₈	—	5	1	no	—
104	C₂×C₈	—	5	4	no	—
105	C₂×C₈	—	5	4	no	—
106	C₂×C₈	—	10	2	no	—
107	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₄²	—	5	2	no	—
110	C₄²	—	5	1	no	—
111	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₁₆	—	5	8	no	—
113	C₁₆	—	10	4	no	—
114	C₂×C₅	C₂×#322	1	1	yes	characteristic
115	C₂×C₅	#2110	1	2	yes	—
116	D₅	—	2	2	no	—
117	C₂³	—	5	1	no	—
118	C₂×C₄	—	5	1	no	—
119	C₂×C₄	—	5	2	no	—
120	C₂×C₄	—	5	2	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	20	2	no	—
123	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
124	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₈	—	5	2	no	—
127	C₈	—	5	2	no	—
128	C₈	—	20	2	no	—
129	C₈	C₈×AGL(1,𝔽₅)	1	2	yes	—
130	C₈	#3992	1	2	yes	—
131	C₅	#11405	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	5	2	no	—
133	C₂²	—	10	2	no	—
134	C₂²	#65676	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	5	2	no	—
137	C₄	—	10	2	no	—
138	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₄	C₂×#3992	1	1	yes	characteristic
140	C₂	—	10	2	no	—
141	C₂	#10603806	1	1	yes	characteristic
142	C₂	#10603832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603825

C₂

yes

maximal, characteristic

#10603823

C₂

yes

maximal, characteristic

#10598840

C₂

yes

maximal, characteristic

#10598515

C₂

yes

maximal, complemented

C₂×#3990

C₄

yes

—

C₂×D₅⋊C₁₆

C₄

yes

—

#64974

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64886

C₄

yes

—

#64885

C₄

yes

—

#64859

C₂²

yes

—

#64859

C₂²

yes

—

C₅×C₈²

C₄

yes

characteristic

#64674

C₂²

yes

—

C₈×#168

C₄

yes

characteristic

#11405

—

maximal, Sylow

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₄²×D₅

C₂×C₄

yes

characteristic

C₂×#674

C₂×C₄

yes

—

C₂×#674

C₂×C₄

yes

—

#3990

—

D₅⋊C₁₆

C₈

yes

complemented

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂³

yes

characteristic

C₄×M₃₂

—

C₂×#346

—

#1450

—

#1313

—

C₂×C₄×D₅

C₄²

yes

characteristic

C₂×C₄×D₅

M₁₆

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄²

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₄²

yes

characteristic

C₄×Dic₅

M₁₆

yes

—

C₂×#168

—

C₂×#168

C₂²×C₄

yes

characteristic

C₂×#168

C₂²×C₄

yes

characteristic

C₈×D₅

C₂×C₈

yes

—

C₈×D₅

M₁₆

yes

—

#674

—

#674

C₂×C₈

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

#346

—

#346

—

#346

—

C₄×C₁₆

—

#322

—

C₈²

—

C₈²

yes

C₂²×D₅

yes

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

#97

yes

—

C₂×Dic₅

#97

yes

characteristic

C₄×D₅

—

C₄×D₅

C₄×C₈

yes

—

C₄×D₅

#97

yes

—

C₅×C₈

—

C₅×C₈

C₄×C₈

yes

—

C₅×C₈

#97

yes

—

#168

—

#168

C₂²×C₈

yes

—

#168

C₂×M₁₆

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×C₅

C₂×#97

yes

characteristic

C₂×D₅

—

C₂×D₅

#322

yes

—

C₄×C₅

—

C₄×C₅

C₂×C₄×C₈

yes

characteristic

C₄×C₅

C₂×#97

yes

characteristic

Dic₅

—

Dic₅

#322

yes

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

108

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₂×C₅

C₂×#322

yes

characteristic

115

C₂×C₅

#2110

yes

—

116

D₅

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₅⋊M₁₆

yes

—

124

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

C₈×AGL(1,𝔽₅)

yes

—

130

C₈

#3992

yes

—

131

C₅

#11405

yes

characteristic, complemented, Sylow

132

C₂²

—

133

C₂²

—

134

C₂²

#65676

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

139

C₄

C₂×#3992

yes

characteristic

140

C₂

—

141

C₂

#10603806

yes

characteristic

142

C₂

#10603832

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂²×D₅	#97
cosocle	C₄×#168	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#11405
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

C₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₂×C₄×C₈

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

socle

C₂²×C₅

C₂×#97

C₂²×D₅

C₂³

C₈²

C₅

G ▷ C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₈ ◁ C₈²

N	H ≅ G/N	type
4. #10598515	C₂	semidirect
23. D₅⋊C₁₆	C₈	semidirect
131. C₅	#11405	semidirect

H ≅ G/N

type

4. #10598515

C₂

semidirect

23. D₅⋊C₁₆

C₈

semidirect

131. C₅

#11405

semidirect

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

subgroup	index
#10603825	2
#10603823	2
#10598840	2
#10598515	2
#11405	5

subgroup

index

Subgroups — #13023002

Conjugacy classes of subgroups

Complements

Maximal subgroups