Group #20275962 — subgroups

Subgroups — #20275962

Order 6784.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133142	C₂	1	2	yes	maximal
3	C₂×#13133142	C₂	1	2	yes	maximal
4	#16810521	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	#13133217	—	2	4	no	—
8	C₂×C₄⋊Dic₅₃	—	2	1	no	—
9	C₂×C₄⋊Dic₅₃	—	2	1	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719919	—	2	1	no	—
13	#13133142	—	2	4	no	—
14	#13133142	—	2	4	no	—
15	#13133124	—	2	1	no	—
16	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	—	2	1	no	—
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	2	no	—
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₂×Dic₁₀₆	D₄	1	4	yes	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	4	no	—
35	C₄⋊Dic₅₃	—	2	4	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	D₄	1	2	yes	—
39	#11719919	—	4	2	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	4	no	—
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	4	2	no	—
54	C₂×Dic₅₃	—	4	2	no	—
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	4	1	no	—
57	C₂×Dic₅₃	—	4	1	no	—
58	C₂×Dic₅₃	—	4	1	no	—
59	C₂×Dic₅₃	—	4	1	no	—
60	Dic₁₀₆	—	2	4	no	—
61	Dic₁₀₆	—	2	4	no	—
62	Dic₁₀₆	—	4	2	no	—
63	Dic₁₀₆	—	4	2	no	—
64	C₈×C₅₃	—	4	2	no	—
65	C₂²×C₅₃	#120	1	1	yes	characteristic
66	C₂²×C₅₃	#121	1	1	yes	characteristic
67	C₂²×C₅₃	#124	1	1	yes	characteristic
68	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₅₃	#137	1	1	yes	characteristic
71	C₂²×C₅₃	#137	1	1	yes	characteristic
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#121	1	2	yes	—
75	C₄×C₅₃	#123	1	2	yes	—
76	Dic₅₃	—	4	4	no	—
77	Dic₅₃	—	4	2	no	—
78	Dic₅₃	—	8	1	no	—
79	#2043	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	#394	1	1	yes	characteristic
81	C₂×C₅₃	#448	1	1	yes	characteristic
82	C₂×C₅₃	#451	1	1	yes	characteristic
83	C₂×C₅₃	#461	1	1	yes	characteristic
84	C₂×C₅₃	#462	1	1	yes	characteristic
85	C₂×C₅₃	#487	1	1	yes	characteristic
86	C₂×C₅₃	#491	1	1	yes	characteristic
87	C₂×C₄⋊Q₈	—	53	1	no	—
88	C₂×#103	—	53	2	no	—
89	C₂×#103	—	53	2	no	—
90	#389	—	53	1	no	—
91	#336	D₅₃	1	1	yes	characteristic
92	C₅₃	#2043	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₈	—	53	2	no	—
94	C₂²×C₈	C₂×D₅₃	1	2	yes	—
95	C₄⋊Q₈	—	106	4	no	—
96	C₂×C₄⋊C₄	—	53	1	no	—
97	C₂×C₄⋊C₄	—	53	1	no	—
98	C₂×C₄⋊C₄	—	106	1	no	—
99	C₂×C₄⋊C₄	—	106	1	no	—
100	C₂×C₄⋊C₄	—	106	1	no	—
101	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
102	#103	—	106	4	no	—
103	#103	—	106	4	no	—
104	#95	—	106	1	no	—
105	C₂×Q₈	—	53	4	no	—
106	C₂×Q₈	—	106	2	no	—
107	C₂×Q₈	—	106	2	no	—
108	C₂×Q₈	—	106	2	no	—
109	C₂×Q₈	—	106	2	no	—
110	C₂×Q₈	—	106	2	no	—
111	C₂²×C₄	—	106	1	no	—
112	C₂²×C₄	—	106	2	no	—
113	C₂²×C₄	—	106	1	no	—
114	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
115	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄⋊C₄	—	53	2	no	—
121	C₄⋊C₄	—	106	4	no	—
122	C₄⋊C₄	—	106	4	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	2	no	—
125	C₄⋊C₄	—	212	2	no	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
128	Q₈	—	106	4	no	—
129	Q₈	—	106	4	no	—
130	Q₈	—	212	2	no	—
131	Q₈	—	212	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	—	212	1	no	—
140	C₂×C₄	—	212	1	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	—	212	2	no	—
143	C₂×C₄	—	212	1	no	—
144	C₂×C₄	—	212	1	no	—
145	C₂×C₄	#11719913	1	2	yes	—
146	C₂×C₄	Dic₂₁₂	1	2	yes	—
147	C₂×C₄	#11719945	1	2	yes	—
148	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
149	C₂×C₄	D₄×D₅₃	1	2	yes	—
150	C₂×C₄	#11719949	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#13133187	1	1	yes	characteristic
153	C₂²	#13133192	1	1	yes	characteristic
154	C₂²	#13133205	1	1	yes	characteristic
155	C₂²	C₂×#11719913	1	1	yes	characteristic
156	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
157	C₂²	#13133250	1	1	yes	characteristic
158	C₂²	#13133250	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	424	1	no	—
162	C₄	—	212	4	no	—
163	C₄	—	212	2	no	—
164	C₄	#13133190	1	2	yes	—
165	C₄	#13133202	1	2	yes	—
166	C₂	#16810541	1	1	yes	characteristic
167	C₂	#16810764	1	1	yes	characteristic
168	C₂	#16810781	1	1	yes	characteristic
169	C₂	#16810829	1	1	yes	characteristic
170	C₂	#16810834	1	1	yes	characteristic
171	C₂	#16810974	1	1	yes	characteristic
172	C₂	#16810992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal

C₂×#13133142

C₂

yes

maximal

#16810521

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

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×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#123

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2043

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

100

C₂×C₄⋊C₄

—

106

—

101

C₂×C₄²

C₂×D₅₃

yes

characteristic

102

#103

—

106

—

103

#103

—

106

—

104

#95

—

106

—

105

C₂×Q₈

—

106

C₂×Q₈

—

106

—

107

C₂×Q₈

—

106

—

108

C₂×Q₈

—

106

—

109

C₂×Q₈

—

106

—

110

C₂×Q₈

—

106

—

111

C₂²×C₄

—

106

—

112

C₂²×C₄

—

106

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

D₂₁₂

yes

characteristic

115

C₂²×C₄

D₂₁₂

yes

characteristic

116

C₂²×C₄

C₂²×D₅₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

106

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

212

—

126

C₄²

—

127

C₂³

C₂×D₂₁₂

yes

characteristic

128

Q₈

—

106

—

129

Q₈

—

106

—

130

Q₈

—

212

—

131

Q₈

—

212

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

212

—

139

C₂×C₄

—

212

—

140

C₂×C₄

—

212

—

141

C₂×C₄

—

212

—

142

C₂×C₄

—

212

—

143

C₂×C₄

—

212

—

144

C₂×C₄

—

212

—

145

C₂×C₄

#11719913

yes

—

146

C₂×C₄

Dic₂₁₂

yes

—

147

C₂×C₄

#11719945

yes

—

148

C₂×C₄

D₄×D₅₃

yes

characteristic

149

C₂×C₄

D₄×D₅₃

yes

—

150

C₂×C₄

#11719949

yes

characteristic

151

C₈

—

152

C₂²

#13133187

yes

characteristic

153

C₂²

#13133192

yes

characteristic

154

C₂²

#13133205

yes

characteristic

155

C₂²

C₂×#11719913

yes

characteristic

156

C₂²

C₂×Dic₂₁₂

yes

characteristic

157

C₂²

#13133250

yes

characteristic

158

C₂²

#13133250

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

424

—

162

C₄

—

212

—

163

C₄

—

212

—

164

C₄

#13133190

yes

—

165

C₄

#13133202

yes

—

166

C₂

#16810541

yes

characteristic

167

C₂

#16810764

yes

characteristic

168

C₂

#16810781

yes

characteristic

169

C₂

#16810829

yes

characteristic

170

C₂

#16810834

yes

characteristic

171

C₂

#16810974

yes

characteristic

172

C₂

#16810992

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#2043
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
92. C₅₃	#2043	semidirect

H ≅ G/N

type

92. C₅₃

#2043

semidirect

Subgroups — #20275962

Conjugacy classes of subgroups

Complements