Group #20284483 — subgroups

Subgroups — C₂×#16810915 #20284483

Order 6784.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810915	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133150	C₂	1	2	yes	maximal
4	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133140	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	2	yes	—
9	C₄×Dic₁₀₆	—	2	2	no	—
10	C₄×Dic₁₀₆	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719919	—	2	1	no	—
13	C₂×C₄×Dic₅₃	—	2	1	no	—
14	#13133150	C₂²	1	8	yes	—
15	C₄⋊#88322	C₂²	1	4	yes	—
16	#13133140	C₂²	1	8	yes	—
17	C₂²×Q₈×C₅₃	—	2	1	no	—
18	C₄⋊Q₈×C₅₃	—	2	2	no	—
19	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	4	no	—
24	C₂×Q₈×C₅₃	—	4	4	no	—
25	C₂²×C₄×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	1	no	—
30	C₂²×Dic₅₃	—	2	1	no	—
31	C₂×Dic₁₀₆	—	2	1	no	—
32	C₂×Dic₁₀₆	—	2	2	no	—
33	C₂×Dic₁₀₆	Q₈	1	4	yes	complemented
34	C₂×Dic₁₀₆	C₂³	1	2	yes	—
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
38	C₄²×C₅₃	D₄	1	2	yes	—
39	C₄²×C₅₃	C₂³	1	2	yes	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	C₂³	1	2	yes	—
42	#11719919	—	2	4	no	—
43	C₄×Dic₅₃	—	2	4	no	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₂³	1	4	yes	—
46	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₅₃	—	4	8	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	4	4	no	—
51	C₂×C₄×C₅₃	Q₁₆	1	4	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₅₃	#41	1	2	yes	—
57	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	4	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	Dic₁₀₆	—	2	4	no	—
63	Dic₁₀₆	—	2	2	no	—
64	Dic₁₀₆	C₂×Q₈	1	4	yes	complemented
65	#88322	—	2	4	no	—
66	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
68	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	4	4	no	—
72	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
74	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
76	Dic₅₃	—	2	4	no	—
77	Dic₅₃	—	4	2	no	—
78	C₂×#477	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	#477	1	4	yes	complemented
80	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
84	C₂×C₄×Q₈	—	53	1	no	—
85	#477	—	53	8	no	—
86	C₂×#107	—	53	2	no	—
87	C₂×C₄⋊C₈	—	53	1	no	—
88	C₂×#103	—	53	2	no	—
89	C₅₃	C₂×#477	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	53	1	no	—
92	C₂²×C₈	—	53	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂×D₅₃	1	4	yes	—
95	C₄×Q₈	—	53	4	no	—
96	C₄×Q₈	—	106	2	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	53	1	no	—
99	C₂×C₄⋊C₄	—	106	1	no	—
100	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
101	C₂×C₄²	—	106	1	no	—
102	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
103	#107	—	53	8	no	—
104	C₄⋊C₈	—	53	4	no	—
105	#103	—	53	8	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	53	4	no	—
109	C₂×Q₈	—	53	2	no	—
110	C₂×Q₈	—	106	2	no	—
111	C₂×Q₈	—	106	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	106	1	no	—
114	C₂²×C₄	—	106	1	no	—
115	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	—	106	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	53	2	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	4	no	—
125	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
126	C₄²	—	106	4	no	—
127	C₄²	C₂²⋊D₅₃	1	2	yes	—
128	C₄²	C₂²×D₅₃	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	53	4	no	—
132	Q₈	—	106	4	no	—
133	Q₈	—	106	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	106	4	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	212	2	no	—
141	C₂×C₄	#11719925	1	4	yes	—
142	C₂×C₄	Q₈×D₅₃	1	2	yes	—
143	C₂×C₄	#11719949	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
147	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
148	C₈	—	106	4	no	—
149	C₂²	#13133193	1	2	yes	—
150	C₂²	C₂×#11719925	1	2	yes	—
151	C₂²	#13133243	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	4	4	no	—
155	C₄	—	106	4	no	—
156	C₄	—	212	2	no	—
157	C₄	#13133193	1	2	yes	—
158	C₄	C₂×#11719925	1	4	yes	—
159	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
160	C₄	C₂×#11719949	1	1	yes	characteristic
161	C₂	#16810915	1	4	yes	direct product factor
162	C₂	C₂×#13133193	1	1	yes	characteristic
163	C₂	C₂²×#11719925	1	1	yes	characteristic
164	C₂	C₂×#13133243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810915

C₂

yes

maximal, direct product factor

C₂×#13133150

C₂

yes

maximal

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

#13133150

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133140

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

complemented

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

complemented

#88322

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

#477

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₅₃

C₂×#477

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

100

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

101

C₂×C₄²

—

106

—

102

C₂×C₄²

C₂×D₅₃

yes

characteristic

103

#107

—

104

C₄⋊C₈

—

105

#103

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

106

—

111

C₂×Q₈

—

106

—

112

C₂²×C₄

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

—

106

—

115

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

116

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

117

C₂²×C₄

C₂²×D₅₃

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

C₂²×D₅₃

yes

—

126

C₄²

—

106

—

127

C₄²

C₂²⋊D₅₃

yes

—

128

C₄²

C₂²×D₅₃

yes

—

129

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

106

—

133

Q₈

—

106

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

#11719925

yes

—

142

C₂×C₄

Q₈×D₅₃

yes

—

143

C₂×C₄

#11719949

yes

—

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

147

C₂×C₄

C₂³×D₅₃

yes

characteristic

148

C₈

—

106

—

149

C₂²

#13133193

yes

—

150

C₂²

C₂×#11719925

yes

—

151

C₂²

#13133243

yes

—

152

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

106

—

156

C₄

—

212

—

157

C₄

#13133193

yes

—

158

C₄

C₂×#11719925

yes

—

159

C₄

C₂×Q₈×D₅₃

yes

characteristic

160

C₄

C₂×#11719949

yes

characteristic

161

C₂

#16810915

yes

direct product factor

162

C₂

C₂×#13133193

yes

characteristic

163

C₂

C₂²×#11719925

yes

characteristic

164

C₂

C₂×#13133243

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#477
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

C₂×#477

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16810915	C₂	direct (N × H)
33. C₂×Dic₁₀₆	Q₈	semidirect
64. Dic₁₀₆	C₂×Q₈	semidirect
79. C₂×C₅₃	#477	semidirect
89. C₅₃	C₂×#477	semidirect
161. C₂	#16810915	direct (N × H)

H ≅ G/N

type

2. #16810915

C₂

direct (N × H)

33. C₂×Dic₁₀₆

Q₈

semidirect

64. Dic₁₀₆

C₂×Q₈

semidirect

79. C₂×C₅₃

#477

semidirect

89. C₅₃

C₂×#477

semidirect

161. C₂

#16810915

direct (N × H)

Subgroups — C₂×#16810915 #20284483

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810915 #20284483

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810915 #20284483