Group #20284457 — subgroups

Subgroups — C₂×#16810900 #20284457

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	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16810900	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133150	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133137	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₅₃	—	2	1	no	—
9	C₂²×#88322	C₂²	1	2	yes	—
10	C₄×D₂₁₂	—	2	2	no	—
11	C₄×D₂₁₂	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×#11719921	—	2	1	no	—
14	#13133150	C₂²	1	8	yes	complemented
15	C₄⋊#88322	C₂²	1	4	yes	complemented
16	#13133137	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	complemented
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	C₂³×D₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	4	4	no	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	1	no	—
31	C₂×D₂₁₂	—	2	1	no	—
32	C₂×D₂₁₂	—	2	2	no	—
33	C₂×D₂₁₂	Q₈	1	4	yes	complemented
34	C₂×D₂₁₂	C₂³	1	2	yes	—
35	C₂×C₄×D₅₃	—	2	4	no	—
36	C₂×C₄×D₅₃	—	4	4	no	—
37	C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₄²×C₅₃	D₄	1	2	yes	—
41	C₄²×C₅₃	C₂³	1	2	yes	—
42	#11719921	—	2	4	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	C₂³	1	2	yes	—
45	C₂×#88322	—	2	4	no	—
46	C₂×#88322	C₂³	1	4	yes	—
47	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×D₅₃	—	2	4	no	—
49	C₂²×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	—	4	2	no	—
51	C₂²×D₅₃	—	4	2	no	—
52	Q₈×C₅₃	—	4	8	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	D₈	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₅₃	#41	1	2	yes	—
62	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	4	2	no	—
65	D₂₁₂	—	2	4	no	—
66	D₂₁₂	—	2	2	no	—
67	D₂₁₂	C₂×Q₈	1	4	yes	complemented
68	C₄×D₅₃	—	4	8	no	—
69	#88322	—	2	4	no	—
70	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂²×C₅₃	C₂×D₈	1	2	yes	—
72	C₂²×C₅₃	#137	1	2	yes	—
73	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×D₅₃	—	2	4	no	—
75	C₂×D₅₃	—	4	2	no	—
76	C₂×D₅₃	—	4	2	no	—
77	C₂×D₅₃	—	4	8	no	—
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	4	4	no	—
80	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₅₃	C₂×D₈	1	4	yes	—
82	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
84	Dic₅₃	—	4	2	no	—
85	C₂×#474	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	#474	1	4	yes	complemented
87	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
90	D₅₃	—	4	4	no	—
91	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
92	C₂×C₄×D₄	—	53	1	no	—
93	#474	—	53	8	no	—
94	C₂×#107	—	53	2	no	—
95	C₂×C₄⋊C₈	—	53	1	no	—
96	C₂×#102	—	53	2	no	—
97	C₅₃	C₂×#474	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₈	—	2	1	no	—
99	C₂²×D₄	—	53	1	no	—
100	C₂³×C₄	—	106	1	no	—
101	C₂²×C₈	—	53	2	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂×D₅₃	1	4	yes	complemented
104	C₄×D₄	—	53	4	no	—
105	C₄×D₄	—	106	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	53	1	no	—
108	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
109	C₂×C₂²⋊C₄	—	106	1	no	—
110	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
111	#107	—	53	8	no	—
112	C₄⋊C₈	—	53	4	no	—
113	#102	—	53	8	no	—
114	C₂⁴	—	106	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	53	4	no	—
118	C₂×D₄	—	53	2	no	—
119	C₂×D₄	—	106	2	no	—
120	C₂×D₄	—	106	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	106	1	no	—
123	C₂²×C₄	—	106	4	no	—
124	C₂²×C₄	—	212	4	no	—
125	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₈	—	53	4	no	—
129	C₂×C₈	—	106	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	53	2	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
135	C₂²⋊C₄	—	106	4	no	—
136	C₄²	C₂²⋊D₅₃	1	2	yes	—
137	C₄²	C₂²×D₅₃	1	2	yes	—
138	C₂³	—	106	4	no	—
139	C₂³	—	106	2	no	—
140	C₂³	—	212	2	no	—
141	C₂³	—	212	2	no	—
142	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	Q₈	—	4	8	no	—
144	D₄	—	53	4	no	—
145	D₄	—	106	4	no	—
146	D₄	—	106	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	212	2	no	—
152	C₂×C₄	—	212	8	no	—
153	C₂×C₄	#11719922	1	4	yes	—
154	C₂×C₄	Q₈×D₅₃	1	2	yes	—
155	C₂×C₄	#11719949	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
159	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
160	C₈	—	106	4	no	—
161	C₂²	—	106	4	no	—
162	C₂²	—	212	2	no	—
163	C₂²	—	212	8	no	—
164	C₂²	—	212	2	no	—
165	C₂²	#13133193	1	2	yes	—
166	C₂²	C₂×#11719922	1	2	yes	—
167	C₂²	#13133248	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	212	2	no	—
172	C₄	#13133193	1	2	yes	—
173	C₄	C₂×#11719922	1	4	yes	—
174	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
175	C₄	C₂×#11719949	1	1	yes	characteristic
176	C₂	—	212	4	no	—
177	C₂	#16810900	1	4	yes	direct product factor
178	C₂	C₂×#13133193	1	1	yes	characteristic
179	C₂	C₂²×#11719922	1	1	yes	characteristic
180	C₂	C₂×#13133248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₁₂

C₂

yes

maximal, characteristic

#16810900

C₂

yes

maximal, direct product factor

C₂×#13133150

C₂

yes

maximal, complemented

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic, complemented

C₂×#13133137

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂²×C₄×D₅₃

—

C₂²×#88322

C₂²

yes

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719921

—

#13133150

C₂²

yes

complemented

C₄⋊#88322

C₂²

yes

complemented

#13133137

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

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

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

Q₈

yes

complemented

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

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

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

C₂×Q₈

yes

complemented

C₄×D₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₃

C₂²×D₈

yes

characteristic

C₂×C₅₃

C₂×#137

yes

characteristic

D₅₃

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

C₂×C₄×D₄

—

#474

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₅₃

C₂×#474

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

100

C₂³×C₄

—

106

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

C₂×D₅₃

yes

complemented

104

C₄×D₄

—

105

C₄×D₄

—

106

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

109

C₂×C₂²⋊C₄

—

106

—

110

C₂×C₄²

C₂×D₅₃

yes

characteristic

111

#107

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

106

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

106

—

120

C₂×D₄

—

106

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

—

212

—

125

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

127

C₂²×C₄

C₂²×D₅₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

106

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

C₂²×D₅₃

yes

—

135

C₂²⋊C₄

—

106

—

136

C₄²

C₂²⋊D₅₃

yes

—

137

C₄²

C₂²×D₅₃

yes

—

138

C₂³

—

106

—

139

C₂³

—

106

—

140

C₂³

—

212

—

141

C₂³

—

212

—

142

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

143

Q₈

—

144

D₄

—

145

D₄

—

106

—

146

D₄

—

106

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

#11719922

yes

—

154

C₂×C₄

Q₈×D₅₃

yes

—

155

C₂×C₄

#11719949

yes

—

156

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

157

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

159

C₂×C₄

C₂³×D₅₃

yes

characteristic

160

C₈

—

106

—

161

C₂²

—

106

—

162

C₂²

—

212

—

163

C₂²

—

212

—

164

C₂²

—

212

—

165

C₂²

#13133193

yes

—

166

C₂²

C₂×#11719922

yes

—

167

C₂²

#13133248

yes

—

168

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

212

—

172

C₄

#13133193

yes

—

173

C₄

C₂×#11719922

yes

—

174

C₄

C₂×Q₈×D₅₃

yes

characteristic

175

C₄

C₂×#11719949

yes

characteristic

176

C₂

—

212

—

177

C₂

#16810900

yes

direct product factor

178

C₂

C₂×#13133193

yes

characteristic

179

C₂

C₂²×#11719922

yes

characteristic

180

C₂

C₂×#13133248

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×#474
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×#474

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16810900	C₂	direct (N × H)
3. C₂×#13133150	C₂	semidirect
4. C₂×C₄⋊#88322	C₂	semidirect
6. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
14. #13133150	C₂²	semidirect
15. C₄⋊#88322	C₂²	semidirect
19. C₄⋊Q₈×C₅₃	C₂²	semidirect
33. C₂×D₂₁₂	Q₈	semidirect
67. D₂₁₂	C₂×Q₈	semidirect
86. C₂×C₅₃	#474	semidirect
91. C₂×C₄⋊Q₈	D₅₃	semidirect
97. C₅₃	C₂×#474	semidirect
103. C₄⋊Q₈	C₂×D₅₃	semidirect
177. C₂	#16810900	direct (N × H)

H ≅ G/N

type

2. #16810900

C₂

direct (N × H)

3. C₂×#13133150

C₂

semidirect

4. C₂×C₄⋊#88322

C₂

semidirect

6. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

14. #13133150

C₂²

semidirect

15. C₄⋊#88322

C₂²

semidirect

19. C₄⋊Q₈×C₅₃

C₂²

semidirect

33. C₂×D₂₁₂

Q₈

semidirect

67. D₂₁₂

C₂×Q₈

semidirect

86. C₂×C₅₃

#474

semidirect

91. C₂×C₄⋊Q₈

D₅₃

semidirect

97. C₅₃

C₂×#474

semidirect

103. C₄⋊Q₈

C₂×D₅₃

semidirect

177. C₂

#16810900

direct (N × H)

Subgroups — C₂×#16810900 #20284457

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810900 #20284457

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810900 #20284457