Group #13025359 — subgroups

Subgroups — #13025359

Order 1280.

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

normal subgroups only

This group has 210 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	#10604841	C₂	1	2	yes	maximal
2	Dic₅×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#10598809	C₂	1	2	yes	maximal
4	#10598644	C₂	1	1	yes	maximal, characteristic
5	C₁₆×#170	C₂	1	1	yes	maximal, characteristic
6	#65788	—	2	2	no	—
7	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic
8	Q₁₆×Dic₅	C₂²	1	2	yes	—
9	Q₈⋊#170	—	2	2	no	—
10	#64885	C₂²	1	1	yes	characteristic
11	#64882	C₂²	1	2	yes	—
12	C₈×#170	C₂²	1	1	yes	characteristic
13	#64784	C₄	1	2	yes	—
14	#64724	C₄	1	1	yes	characteristic
15	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
16	#12083	—	5	1	no	maximal, Sylow
17	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
18	Q₈×Dic₅	—	2	2	no	—
19	#4004	—	4	2	no	—
20	#4002	—	4	2	no	—
21	C₄⋊#170	—	2	2	no	—
22	C₄×#170	D₄	1	1	yes	characteristic
23	C₄×#170	D₄	1	1	yes	characteristic
24	C₂×#674	—	2	1	no	—
25	C₅×Q₃₂	—	2	2	no	—
26	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
27	#3967	—	2	2	no	—
28	#3950	C₂×C₄	1	2	yes	—
29	C₈×Dic₅	C₂³	1	1	yes	characteristic
30	C₂×#672	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₃₂	—	5	1	no	—
32	#1591	—	5	2	no	—
33	#1371	—	5	1	no	—
34	#1332	—	5	2	no	—
35	C₈×C₁₆	—	5	1	no	—
36	C₂×C₅×Q₈	—	2	2	no	—
37	C₂×#170	—	2	1	no	—
38	C₂×#170	—	4	2	no	—
39	C₂×#170	D₈	1	1	yes	characteristic
40	C₂×#170	D₈	1	1	yes	characteristic
41	C₅×Q₁₆	—	2	2	no	—
42	C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊Dic₅	—	2	2	no	—
45	C₄⋊Dic₅	—	4	2	no	—
46	C₄×Dic₅	—	4	2	no	—
47	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
48	C₂×#168	#41	1	1	yes	characteristic
49	#674	—	2	2	no	—
50	C₅×C₁₆	M₁₆	1	2	yes	—
51	#672	—	2	1	no	—
52	C₂×Q₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
53	#446	—	10	2	no	—
54	C₄×Q₁₆	—	5	2	no	—
55	#366	—	5	1	no	—
56	#358	—	5	2	no	—
57	C₄×C₁₆	—	5	1	no	—
58	C₄×C₁₆	—	5	1	no	—
59	#335	—	5	2	no	—
60	Q₈⋊C₈	—	10	2	no	—
61	C₈²	—	5	1	no	—
62	C₅×Q₈	—	4	2	no	—
63	C₅×Q₈	—	4	2	no	—
64	C₂×C₄×C₅	—	4	2	no	—
65	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
66	C₂×Dic₅	—	4	2	no	—
67	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
68	C₂×Dic₅	#135	1	1	yes	characteristic
69	#170	—	2	2	no	—
70	#170	—	2	1	no	—
71	#170	—	8	2	no	—
72	#170	Q₃₂	1	2	yes	complemented
73	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
74	C₅×C₈	#131	1	1	yes	characteristic
75	#168	—	2	1	no	—
76	C₂×Q₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
77	C₄×Q₈	—	10	2	no	—
78	Q₃₂	—	2	2	no	—
79	C₂×C₁₆	—	5	1	no	—
80	C₂×C₁₆	—	10	1	no	—
81	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	#107	—	5	2	no	—
83	C₄⋊C₈	—	10	2	no	—
84	C₄⋊C₈	—	20	2	no	—
85	#103	—	10	2	no	—
86	#97	—	20	2	no	—
87	C₄×C₈	—	5	1	no	—
88	C₄×C₈	—	5	1	no	—
89	C₄×C₈	—	5	1	no	—
90	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
91	C₄×C₅	—	8	2	no	—
92	C₄×C₅	#436	1	1	yes	characteristic
93	C₄×C₅	#443	1	1	yes	characteristic
94	Dic₅	—	2	1	no	—
95	Dic₅	—	8	2	no	—
96	Dic₅	C₂×Q₃₂	1	1	yes	characteristic
97	Dic₅	#508	1	1	yes	characteristic
98	C₂×Q₈	—	2	2	no	—
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	C₂×C₈	—	5	1	no	—
102	C₂×C₈	—	5	1	no	—
103	C₂×C₈	—	5	1	no	—
104	C₂×C₈	—	10	1	no	—
105	C₂×C₈	—	20	2	no	—
106	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₄⋊C₄	—	10	2	no	—
108	C₄⋊C₄	—	20	2	no	—
109	C₄²	—	5	1	no	—
110	C₄²	—	20	2	no	—
111	C₁₆	—	10	2	no	—
112	C₁₆	—	10	1	no	—
113	C₁₆	#704	1	2	yes	—
114	C₂×C₅	#1589	1	1	yes	characteristic
115	C₂×C₅	C₄×Q₃₂	1	1	yes	characteristic
116	C₂×C₅	#2178	1	1	yes	characteristic
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	5	1	no	—
121	C₂×C₄	—	5	1	no	—
122	C₂×C₄	—	20	2	no	—
123	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₈	—	5	2	no	—
125	C₈	—	10	1	no	—
126	C₈	—	10	2	no	—
127	C₈	—	10	1	no	—
128	C₈	—	40	2	no	—
129	C₈	#4131	1	1	yes	characteristic
130	C₈	C₂×#704	1	1	yes	characteristic
131	C₅	#12083	1	1	yes	characteristic, complemented, Sylow
132	C₂²	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₄	—	5	1	no	—
134	C₄	—	5	1	no	—
135	C₄	—	8	2	no	—
136	C₄	—	10	1	no	—
137	C₄	—	40	2	no	—
138	C₄	#65732	1	1	yes	characteristic
139	C₄	#65774	1	1	yes	characteristic
140	C₂	AGL(1,𝔽₅)×Q₃₂	1	1	yes	characteristic
141	C₂	#10604136	1	1	yes	characteristic
142	C₂	#10604795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604841

C₂

yes

maximal

Dic₅×Q₃₂

C₂

yes

maximal, characteristic

#10598809

C₂

yes

maximal

#10598644

C₂

yes

maximal, characteristic

C₁₆×#170

C₂

yes

maximal, characteristic

#65788

—

C₂×C₅×Q₃₂

C₄

yes

characteristic

Q₁₆×Dic₅

C₂²

yes

—

Q₈⋊#170

—

#64885

C₂²

yes

characteristic

#64882

C₂²

yes

—

C₈×#170

C₂²

yes

characteristic

#64784

C₄

yes

—

#64724

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

#12083

—

maximal, Sylow

C₂×C₅×Q₁₆

C₂×C₄

yes

—

Q₈×Dic₅

—

—

—

—

D₄

yes

characteristic

C₄×#170

D₄

yes

characteristic

C₂×#674

—

C₅×Q₃₂

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

#3967

—

#3950

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₄×Q₃₂

—

#1591

—

#1371

—

#1332

—

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×#170

—

C₂×#170

—

C₂×#170

D₈

yes

characteristic

C₂×#170

D₈

yes

characteristic

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₄

yes

yes

—

C₅×C₁₆

M₁₆

yes

—

#672

—

C₂×Q₃₂

AGL(1,𝔽₅)

yes

characteristic

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂×D₈

yes

C₂×Dic₅

yes

—

—

—

Q₃₂

yes

complemented

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

C₂×Q₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×AGL(1,𝔽₅)

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₄×D₈

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

Dic₅

C₂×Q₃₂

yes

characteristic

Dic₅

#508

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#704

yes

—

114

C₂×C₅

#1589

yes

characteristic

115

C₂×C₅

C₄×Q₃₂

yes

characteristic

116

C₂×C₅

#2178

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#4131

yes

characteristic

130

C₈

C₂×#704

yes

characteristic

131

C₅

#12083

yes

characteristic, complemented, Sylow

132

C₂²

D₈×AGL(1,𝔽₅)

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#65732

yes

characteristic

139

C₄

#65774

yes

characteristic

140

C₂

AGL(1,𝔽₅)×Q₃₂

yes

characteristic

141

C₂

#10604136

yes

characteristic

142

C₂

#10604795

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×AGL(1,𝔽₅)
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×Q₃₂	C₄
socle, prime radical	C₂²×C₅	C₄×D₈
cosocle	C₈×Dic₅	C₂³
hypercenter	C₂×Q₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#12083
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×AGL(1,𝔽₅)

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×Q₃₂

C₄

socle, prime radical

C₂²×C₅

C₄×D₈

cosocle

C₈×Dic₅

C₂³

C₂×Q₃₂

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
72. #170	Q₃₂	semidirect
131. C₅	#12083	semidirect

H ≅ G/N

type

72. #170

Q₃₂

semidirect

131. C₅

#12083

semidirect

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

subgroup	index
#10604841	2
Dic₅×Q₃₂	2
#10598809	2
#10598644	2
C₁₆×#170	2
#12083	5

subgroup

index

Dic₅×Q₃₂

C₁₆×#170

Subgroups — #13025359

Conjugacy classes of subgroups

Complements

Maximal subgroups