Group #20274892 — subgroups

Subgroups — #20274892

Order 6784.

348 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂³×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	—	2	4	no	—
7	C₂²×#88322	—	2	2	no	—
8	C₂²×#88322	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
12	#13133146	—	2	4	no	—
13	#13133124	C₄	1	2	yes	—
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₅₃	—	2	2	no	—
18	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂²⋊Dic₅₃	—	2	2	no	—
32	C₂²⋊Dic₅₃	C₈	1	4	yes	complemented
33	C₄×Dic₅₃	—	2	2	no	—
34	C₄×Dic₅₃	—	2	2	no	—
35	C₂×#88322	—	2	8	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	D₄	1	4	yes	—
39	C₂³×C₅₃	—	2	2	no	—
40	C₂³×C₅₃	—	2	2	no	—
41	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂³×C₅₃	C₂×C₈	1	2	yes	—
43	C₂³×C₅₃	M₁₆	1	2	yes	—
44	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	2	yes	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	C₂×C₈	1	4	yes	complemented
60	C₈×C₅₃	—	4	1	no	—
61	C₈×C₅₃	—	4	1	no	—
62	#88322	—	2	4	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	—	2	4	no	—
66	C₂²×C₅₃	—	2	4	no	—
67	C₂²×C₅₃	C₄⋊C₈	1	4	yes	complemented
68	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₃	C₄×D₄	1	2	yes	—
70	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂²×C₅₃	#131	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#120	1	1	yes	characteristic
75	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
76	C₄×C₅₃	#123	1	1	yes	characteristic
77	Dic₅₃	—	2	4	no	—
78	Dic₅₃	—	4	2	no	—
79	#1850	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	4	no	—
81	C₂×C₅₃	#385	1	1	yes	characteristic
82	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₅₃	#424	1	1	yes	characteristic
84	C₂×C₅₃	C₈×D₄	1	2	yes	—
85	C₂×C₅₃	#435	1	2	yes	—
86	C₂³×C₈	—	53	1	no	—
87	C₂×C₄⋊C₈	—	53	2	no	—
88	C₂×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
89	C₄×C₂²⋊C₄	—	53	1	no	—
90	#336	—	53	2	no	—
91	C₅₃	#1850	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×C₈	—	53	2	no	—
94	C₂²×C₈	—	106	4	no	—
95	C₂²×C₈	—	106	2	no	—
96	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	—	53	1	no	—
99	C₂×C₂²⋊C₄	—	53	1	no	—
100	C₂×C₄²	—	53	2	no	—
101	C₄⋊C₈	—	106	4	no	—
102	C₂²⋊C₈	—	2	2	no	—
103	#95	—	53	2	no	—
104	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	—	53	2	no	—
109	C₂²×C₄	Dic₁₀₆	1	2	yes	—
110	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	106	8	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂²⋊C₄	—	53	4	no	—
122	C₂²⋊C₄	—	106	2	no	—
123	C₄²	—	106	2	no	—
124	C₄²	—	106	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₈×D₅₃	1	2	yes	—
128	C₂³	#11719912	1	2	yes	—
129	C₂³	#11719919	1	2	yes	—
130	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	53	4	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	#11719919	1	2	yes	—
142	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₃	1	2	yes	—
144	C₂×C₄	#11719947	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₈	—	4	1	no	—
147	C₈	—	4	1	no	—
148	C₈	—	106	4	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	#13133146	1	4	yes	complemented
154	C₂²	C₂×#11719919	1	1	yes	characteristic
155	C₂²	#13133177	1	2	yes	—
156	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
157	C₂²	C₂×#11719912	1	1	yes	characteristic
158	C₂²	#13133225	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	—	106	4	no	—
161	C₄	—	212	2	no	—
162	C₄	#13133185	1	1	yes	characteristic
163	C₄	#13133194	1	2	yes	—
164	C₄	#13133198	1	1	yes	characteristic
165	C₂	—	2	4	no	—
166	C₂	#16810495	1	1	yes	characteristic
167	C₂	C₂×#13133146	1	1	yes	characteristic
168	C₂	#16810651	1	1	yes	characteristic
169	C₂	#16810686	1	2	yes	—
170	C₂	#16810691	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#88322

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₂²×#88322

—

C₂²×#88322

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₄

yes

characteristic

C₂×C₂²⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133146

—

#13133124

C₄

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×C₈

yes

—

C₂³×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1850

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#385

yes

characteristic

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

#424

yes

characteristic

C₂×C₅₃

C₈×D₄

yes

—

C₂×C₅₃

#435

yes

—

C₂³×C₈

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

D₅₃

yes

characteristic

C₄×C₂²⋊C₄

—

#336

—

C₅₃

#1850

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

100

C₂×C₄²

—

101

C₄⋊C₈

—

106

—

102

C₂²⋊C₈

—

103

#95

—

104

C₂⁴

C₄×D₅₃

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

Dic₁₀₆

yes

—

110

C₂²×C₄

C₄×D₅₃

yes

characteristic

111

C₂²×C₄

C₂²⋊D₅₃

yes

—

112

C₂²×C₄

C₂²×D₅₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

106

—

123

C₄²

—

106

—

124

C₄²

—

106

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₈×D₅₃

yes

—

128

C₂³

#11719912

yes

—

129

C₂³

#11719919

yes

—

130

C₂³

C₂×C₄×D₅₃

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

106

—

141

C₂×C₄

#11719919

yes

—

142

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

143

C₂×C₄

D₄×D₅₃

yes

—

144

C₂×C₄

#11719947

yes

—

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

106

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#13133146

yes

complemented

154

C₂²

C₂×#11719919

yes

characteristic

155

C₂²

#13133177

yes

—

156

C₂²

C₂×C₈×D₅₃

yes

characteristic

157

C₂²

C₂×#11719912

yes

characteristic

158

C₂²

#13133225

yes

—

159

C₄

—

160

C₄

—

106

—

161

C₄

—

212

—

162

C₄

#13133185

yes

characteristic

163

C₄

#13133194

yes

—

164

C₄

#13133198

yes

characteristic

165

C₂

—

166

C₂

#16810495

yes

characteristic

167

C₂

C₂×#13133146

yes

characteristic

168

C₂

#16810651

yes

characteristic

169

C₂

#16810686

yes

—

170

C₂

#16810691

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₂²⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#1850
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₂²⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₂²⋊C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#13133146	C₂	semidirect
4. #16810377	C₂	semidirect
32. C₂²⋊Dic₅₃	C₈	semidirect
59. C₂×Dic₅₃	C₂×C₈	semidirect
67. C₂²×C₅₃	C₄⋊C₈	semidirect
91. C₅₃	#1850	semidirect
153. C₂²	#13133146	semidirect

H ≅ G/N

type

2. C₂×#13133146

C₂

semidirect

4. #16810377

C₂

semidirect

32. C₂²⋊Dic₅₃

C₈

semidirect

59. C₂×Dic₅₃

C₂×C₈

semidirect

67. C₂²×C₅₃

C₄⋊C₈

semidirect

91. C₅₃

#1850

semidirect

153. C₂²

#13133146

semidirect

Subgroups — #20274892

Conjugacy classes of subgroups

Complements