Group #10604839 — subgroups

Subgroups — #10604839

Order 640.

152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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	#65788	C₂	1	1	yes	maximal, characteristic
2	Q₈×#170	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#170	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#170	C₂	1	1	yes	maximal, characteristic
6	#64884	C₂	1	1	yes	maximal, characteristic
7	#170⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×Q₁₆	C₄	1	1	yes	characteristic
9	Q₈×Dic₅	C₂²	1	1	yes	characteristic
10	Q₈×Dic₅	C₂²	1	1	yes	characteristic
11	#4004	—	2	1	no	—
12	#4002	—	2	1	no	—
13	C₄⋊#170	—	2	1	no	—
14	C₄⋊#170	C₂²	1	1	yes	characteristic
15	C₄⋊#170	C₂²	1	1	yes	characteristic
16	C₄×#170	—	2	1	no	—
17	C₄×#170	C₂²	1	1	yes	characteristic
18	C₄×#170	C₂²	1	1	yes	characteristic
19	#3967	C₄	1	1	yes	characteristic
20	#3967	C₄	1	1	yes	characteristic
21	#3950	C₄	1	1	yes	characteristic
22	C₈×Dic₅	C₂²	1	1	yes	characteristic
23	#1584	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×#170	—	2	1	no	—
27	C₂×#170	—	2	1	no	—
28	C₂×#170	—	2	1	no	—
29	C₂×#170	D₄	1	1	yes	characteristic
30	C₂×#170	D₄	1	1	yes	characteristic
31	C₅×Q₁₆	—	2	2	no	—
32	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₅	—	2	1	no	—
34	C₄⋊Dic₅	—	2	1	no	—
35	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₅	—	2	1	no	—
38	C₄×Dic₅	—	2	1	no	—
39	C₄×Dic₅	C₂³	1	1	yes	characteristic
40	C₂×#168	C₂×C₄	1	1	yes	characteristic
41	#446	—	5	1	no	—
42	C₈×Q₈	—	5	1	no	—
43	C₄×Q₁₆	—	5	1	no	—
44	#335	—	5	1	no	—
45	Q₈⋊C₈	—	5	1	no	—
46	Q₈⋊C₈	—	5	1	no	—
47	#322	—	5	1	no	—
48	C₅×Q₈	—	2	1	no	—
49	C₅×Q₈	—	2	1	no	—
50	C₅×Q₈	—	2	1	no	—
51	C₅×Q₈	M₁₆	1	2	yes	—
52	C₂×C₄×C₅	—	2	1	no	—
53	C₂×C₄×C₅	—	2	1	no	—
54	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₅	—	2	1	no	—
56	C₂×Dic₅	—	2	1	no	—
57	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅	#41	1	1	yes	characteristic
59	#170	—	2	1	no	—
60	#170	—	4	1	no	—
61	#170	—	4	1	no	—
62	#170	—	4	1	no	—
63	#170	Q₁₆	1	2	yes	complemented
64	C₅×C₈	—	2	1	no	—
65	#168	—	2	1	no	—
66	C₂×Q₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₄×Q₈	—	5	1	no	—
68	C₄×Q₈	—	5	1	no	—
69	#107	—	5	1	no	—
70	C₄⋊C₈	—	5	1	no	—
71	C₄⋊C₈	—	5	1	no	—
72	C₄⋊C₈	—	10	1	no	—
73	C₄⋊C₈	—	10	1	no	—
74	#103	—	5	1	no	—
75	#103	—	5	1	no	—
76	#97	—	10	1	no	—
77	C₄×C₈	—	5	1	no	—
78	C₄×C₈	—	5	1	no	—
79	C₄×C₈	—	5	1	no	—
80	C₄×C₈	—	10	1	no	—
81	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
82	C₄×C₅	—	2	2	no	—
83	C₄×C₅	—	4	1	no	—
84	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
85	C₄×C₅	#131	1	1	yes	characteristic
86	Dic₅	—	2	1	no	—
87	Dic₅	—	4	1	no	—
88	Dic₅	—	4	1	no	—
89	Dic₅	C₂×Q₁₆	1	1	yes	characteristic
90	Dic₅	#135	1	1	yes	characteristic
91	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	Q₁₆	—	2	2	no	—
94	C₂×C₈	—	5	1	no	—
95	C₂×C₈	—	5	1	no	—
96	C₂×C₈	—	5	1	no	—
97	C₂×C₈	—	10	1	no	—
98	C₂×C₈	—	10	1	no	—
99	C₂×C₈	—	10	1	no	—
100	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₄⋊C₄	—	5	1	no	—
102	C₄⋊C₄	—	5	1	no	—
103	C₄⋊C₄	—	10	1	no	—
104	C₄⋊C₄	—	10	1	no	—
105	C₄²	—	5	1	no	—
106	C₄²	—	10	1	no	—
107	C₄²	—	10	1	no	—
108	C₂×C₅	#435	1	1	yes	characteristic
109	C₂×C₅	C₄×Q₁₆	1	1	yes	characteristic
110	C₂×C₅	#444	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	#704	1	2	yes	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	5	1	no	—
118	C₂×C₄	—	5	1	no	—
119	C₂×C₄	—	10	1	no	—
120	C₂×C₄	—	10	1	no	—
121	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	5	2	no	—
124	C₈	—	10	1	no	—
125	C₈	—	10	1	no	—
126	C₈	—	20	1	no	—
127	C₈	—	20	1	no	—
128	C₈	—	20	1	no	—
129	C₅	#1584	1	1	yes	characteristic, complemented, Sylow
130	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	5	1	no	—
134	C₄	—	5	1	no	—
135	C₄	—	10	1	no	—
136	C₄	—	20	1	no	—
137	C₄	—	20	1	no	—
138	C₄	#4131	1	1	yes	characteristic
139	C₄	C₂×#704	1	1	yes	characteristic
140	C₂	Q₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂	#65741	1	1	yes	characteristic
142	C₂	#65773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65788

C₂

yes

maximal, characteristic

Q₈×#170

C₂

yes

maximal, characteristic

Q₁₆×Dic₅

C₂

yes

maximal, characteristic

Q₈⋊#170

C₂

yes

maximal, characteristic

Q₈⋊#170

C₂

yes

maximal, characteristic

#64884

C₂

yes

maximal, characteristic

#170⋊C₈

C₂

yes

maximal, characteristic

C₂×C₅×Q₁₆

C₄

yes

characteristic

Q₈×Dic₅

C₂²

yes

characteristic

Q₈×Dic₅

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#170

C₂²

yes

characteristic

C₄×#170

—

C₄×#170

C₂²

yes

characteristic

C₄×#170

C₂²

yes

characteristic

#3967

C₄

yes

characteristic

#3967

C₄

yes

characteristic

#3950

C₄

yes

characteristic

C₈×Dic₅

C₂²

yes

characteristic

#1584

—

maximal, Sylow

C₂×C₅×Q₈

C₂×C₄

yes

characteristic

C₂×C₅×Q₈

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#170

D₄

yes

characteristic

C₅×Q₁₆

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

characteristic

C₄⋊Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂³

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

#446

—

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

M₁₆

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×D₄

yes

C₂×Dic₅

yes

—

—

—

—

Q₁₆

yes

complemented

C₅×C₈

—

#168

—

C₂×Q₁₆

AGL(1,𝔽₅)

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₄×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×M₁₆

yes

characteristic

C₄×C₅

#131

yes

characteristic

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂×Q₁₆

yes

Dic₅

yes

C₂×Q₈

yes

C₂×Q₈

yes

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂×C₅

#435

yes

characteristic

109

C₂×C₅

C₄×Q₁₆

yes

characteristic

110

C₂×C₅

#444

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

#704

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₅

#1584

yes

characteristic, complemented, Sylow

130

C₂²

D₄×AGL(1,𝔽₅)

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#4131

yes

characteristic

139

C₄

C₂×#704

yes

characteristic

140

C₂

Q₁₆×AGL(1,𝔽₅)

yes

characteristic

141

C₂

#65741

yes

characteristic

142

C₂

#65773

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×Q₁₆	C₄
socle, prime radical	C₂²×C₅	C₄×D₄
cosocle	C₄×Dic₅	C₂³
hypercenter	C₂×Q₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#1584
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×Q₁₆

C₄

socle, prime radical

C₂²×C₅

C₄×D₄

cosocle

C₄×Dic₅

C₂³

C₂×Q₁₆

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
63. #170	Q₁₆	semidirect
129. C₅	#1584	semidirect

H ≅ G/N

type

63. #170

Q₁₆

semidirect

129. C₅

#1584

semidirect

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

subgroup	index
#65788	2
Q₈×#170	2
Q₁₆×Dic₅	2
Q₈⋊#170	2
#64884	2
#170⋊C₈	2
#1584	5

subgroup

index

Q₁₆×Dic₅

Subgroups — #10604839

Conjugacy classes of subgroups

Complements

Maximal subgroups