Group #20284154 — subgroups

Subgroups — C₂×#16810834 #20284154

Order 6784.

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

normal subgroups only

This group has 480 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	C₂²×Dic₂₁₂	C₂	1	2	yes	maximal
2	C₂×#13133217	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
4	#16810834	C₂	1	8	yes	maximal, direct product factor
5	C₂×#13133142	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁₂	—	2	8	no	—
10	C₂×Dic₂₁₂	C₂²	1	8	yes	—
11	#13133217	—	2	2	no	—
12	#13133217	C₂²	1	4	yes	—
13	C₄×Dic₁₀₆	—	2	2	no	—
14	C₄×Dic₁₀₆	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₅₃	—	2	1	no	—
16	C₂×C₄⋊Dic₅₃	—	2	1	no	—
17	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#11719919	—	2	1	no	—
19	C₂×C₄×Dic₅₃	—	2	1	no	—
20	#13133142	C₂²	1	8	yes	—
21	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	—	2	1	no	—
29	C₂²×Dic₅₃	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	1	no	—
31	C₂×Dic₁₀₆	—	2	2	no	—
32	C₂×Dic₁₀₆	—	2	4	no	—
33	C₂×Dic₁₀₆	—	2	4	no	—
34	C₂×Dic₁₀₆	—	2	2	no	—
35	C₂×Dic₁₀₆	D₄	1	4	yes	—
36	C₂×Dic₁₀₆	C₂³	1	4	yes	—
37	C₂×Dic₁₀₆	C₂³	1	2	yes	—
38	C₂×C₈×C₅₃	—	2	4	no	—
39	C₂×C₈×C₅₃	C₂³	1	4	yes	—
40	C₄²×C₅₃	D₄	1	2	yes	—
41	C₄²×C₅₃	C₂³	1	2	yes	—
42	C₄⋊Dic₅₃	—	2	4	no	—
43	C₄⋊Dic₅₃	—	2	4	no	—
44	C₄⋊Dic₅₃	—	2	1	no	—
45	C₄⋊Dic₅₃	C₂³	1	2	yes	—
46	#11719919	—	2	4	no	—
47	C₄×Dic₅₃	—	2	4	no	—
48	Dic₂₁₂	—	2	16	no	—
49	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	Q₁₆	1	4	yes	complemented
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×D₄	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	4	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	—	4	4	no	—
63	C₂×Dic₅₃	—	4	2	no	—
64	Dic₁₀₆	—	2	4	no	—
65	Dic₁₀₆	—	2	2	no	—
66	Dic₁₀₆	—	2	4	no	—
67	Dic₁₀₆	—	2	4	no	—
68	Dic₁₀₆	—	4	4	no	—
69	Dic₁₀₆	C₂×D₄	1	4	yes	—
70	C₈×C₅₃	—	2	4	no	—
71	C₂²×C₅₃	#121	1	2	yes	—
72	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
73	C₂²×C₅₃	#137	1	2	yes	—
74	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	#121	1	2	yes	—
77	C₄×C₅₃	C₂×Q₁₆	1	4	yes	complemented
78	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
80	Dic₅₃	—	2	4	no	—
81	Dic₅₃	—	4	4	no	—
82	Dic₅₃	—	4	2	no	—
83	C₂×#462	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	#462	1	4	yes	complemented
85	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
86	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
87	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
88	C₂²×Q₁₆	—	53	2	no	—
89	C₂×C₄⋊Q₈	—	53	1	no	—
90	C₂×C₄×Q₈	—	53	1	no	—
91	#462	—	53	8	no	—
92	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
93	C₂×#103	—	53	2	no	—
94	C₅₃	C₂×#462	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	53	1	no	—
96	C₂²×Q₈	—	53	2	no	—
97	C₂×Q₁₆	—	53	8	no	—
98	C₂×Q₁₆	—	106	8	no	—
99	C₂²×C₈	C₂×D₅₃	1	2	yes	—
100	C₄⋊Q₈	—	53	4	no	—
101	C₄⋊Q₈	—	106	2	no	—
102	C₄×Q₈	—	53	4	no	—
103	C₄×Q₈	—	106	2	no	—
104	C₂×C₄⋊C₄	—	53	1	no	—
105	C₂×C₄⋊C₄	—	106	1	no	—
106	C₂×C₄⋊C₄	—	106	1	no	—
107	C₂×C₄⋊C₄	—	106	1	no	—
108	C₂×C₄²	—	106	1	no	—
109	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
111	#103	—	53	8	no	—
112	C₂×Q₈	—	53	4	no	—
113	C₂×Q₈	—	53	2	no	—
114	C₂×Q₈	—	53	4	no	—
115	C₂×Q₈	—	106	1	no	—
116	C₂×Q₈	—	106	2	no	—
117	C₂×Q₈	—	106	2	no	—
118	C₂×Q₈	—	106	4	no	—
119	C₂×Q₈	—	106	4	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	—	106	2	no	—
123	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
124	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	Q₁₆	—	106	16	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	C₂²×D₅₃	1	4	yes	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	—	106	1	no	—
131	C₄⋊C₄	—	106	4	no	—
132	C₄⋊C₄	—	106	4	no	—
133	C₄⋊C₄	—	106	4	no	—
134	C₄²	—	106	4	no	—
135	C₄²	D₂₁₂	1	2	yes	—
136	C₄²	C₂²×D₅₃	1	2	yes	—
137	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
138	Q₈	—	53	4	no	—
139	Q₈	—	106	4	no	—
140	Q₈	—	106	2	no	—
141	Q₈	—	106	4	no	—
142	Q₈	—	106	4	no	—
143	Q₈	—	212	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	106	4	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	4	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	Dic₂₁₂	1	4	yes	complemented
152	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
153	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
155	C₂×C₄	D₄×D₅₃	1	2	yes	—
156	C₂×C₄	#11719949	1	2	yes	—
157	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₂²	#13133192	1	2	yes	—
160	C₂²	C₂×Dic₂₁₂	1	2	yes	—
161	C₂²	#13133250	1	2	yes	—
162	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	106	4	no	—
165	C₄	—	212	4	no	—
166	C₄	—	212	2	no	—
167	C₄	#13133192	1	2	yes	—
168	C₄	C₂×Dic₂₁₂	1	4	yes	complemented
169	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
170	C₄	C₂×#11719949	1	1	yes	characteristic
171	C₂	#16810834	1	4	yes	direct product factor
172	C₂	C₂×#13133192	1	1	yes	characteristic
173	C₂	C₂²×Dic₂₁₂	1	1	yes	characteristic
174	C₂	C₂×#13133250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810834

C₂

yes

maximal, direct product factor

C₂×#13133142

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

#13133142

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×D₄

yes

—

C₈×C₅₃

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂×Q₁₆

yes

complemented

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×#137

yes

characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#462

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₂×#103

—

C₅₃

C₂×#462

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

106

—

102

C₄×Q₈

—

103

C₄×Q₈

—

106

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

—

106

C₂×C₄⋊C₄

—

106

—

107

C₂×C₄⋊C₄

—

106

—

108

C₂×C₄²

—

106

—

109

C₂×C₄²

C₂×D₅₃

yes

characteristic

110

C₄⋊C₈

C₂×D₅₃

yes

—

111

#103

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂×Q₈

—

106

—

117

C₂×Q₈

—

106

—

118

C₂×Q₈

—

106

—

119

C₂×Q₈

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

D₂₁₂

yes

characteristic

124

C₂²×C₄

D₂₁₂

yes

characteristic

125

C₂²×C₄

C₂²×D₅₃

yes

characteristic

126

Q₁₆

—

106

—

127

C₂×C₈

—

128

C₂×C₈

C₂²×D₅₃

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄²

—

106

—

135

C₄²

D₂₁₂

yes

—

136

C₄²

C₂²×D₅₃

yes

—

137

C₂³

C₂×D₂₁₂

yes

characteristic

138

Q₈

—

139

Q₈

—

106

—

140

Q₈

—

106

—

141

Q₈

—

106

—

142

Q₈

—

106

—

143

Q₈

—

212

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

212

—

150

C₂×C₄

—

212

—

151

C₂×C₄

Dic₂₁₂

yes

complemented

152

C₂×C₄

C₂×D₂₁₂

yes

—

153

C₂×C₄

C₂×D₂₁₂

yes

characteristic

154

C₂×C₄

C₂×D₂₁₂

yes

—

155

C₂×C₄

D₄×D₅₃

yes

—

156

C₂×C₄

#11719949

yes

—

157

C₂×C₄

C₂³×D₅₃

yes

characteristic

158

C₈

—

159

C₂²

#13133192

yes

—

160

C₂²

C₂×Dic₂₁₂

yes

—

161

C₂²

#13133250

yes

—

162

C₂²

C₂²×D₂₁₂

yes

characteristic

163

C₄

—

164

C₄

—

106

—

165

C₄

—

212

—

166

C₄

—

212

—

167

C₄

#13133192

yes

—

168

C₄

C₂×Dic₂₁₂

yes

complemented

169

C₄

C₂×D₄×D₅₃

yes

characteristic

170

C₄

C₂×#11719949

yes

characteristic

171

C₂

#16810834

yes

direct product factor

172

C₂

C₂×#13133192

yes

characteristic

173

C₂

C₂²×Dic₂₁₂

yes

characteristic

174

C₂

C₂×#13133250

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×#462
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

C₂×#462

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
4. #16810834	C₂	direct (N × H)
51. C₂×C₄×C₅₃	Q₁₆	semidirect
77. C₄×C₅₃	C₂×Q₁₆	semidirect
84. C₂×C₅₃	#462	semidirect
94. C₅₃	C₂×#462	semidirect
151. C₂×C₄	Dic₂₁₂	semidirect
168. C₄	C₂×Dic₂₁₂	semidirect
171. C₂	#16810834	direct (N × H)

H ≅ G/N

type

4. #16810834

C₂

direct (N × H)

51. C₂×C₄×C₅₃

Q₁₆

semidirect

77. C₄×C₅₃

C₂×Q₁₆

semidirect

84. C₂×C₅₃

#462

semidirect

94. C₅₃

C₂×#462

semidirect

151. C₂×C₄

Dic₂₁₂

semidirect

168. C₄

C₂×Dic₂₁₂

semidirect

171. C₂

#16810834

direct (N × H)

Subgroups — C₂×#16810834 #20284154

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810834 #20284154

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810834 #20284154