Group #20274749 — subgroups

Subgroups — #20274749

Order 6784.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810480	C₂	1	2	yes	maximal
4	#16810385	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719912	—	2	2	no	—
8	C₂×#11719912	—	2	2	no	—
9	C₂×#11719912	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#11719921	—	2	2	no	—
13	#13133149	—	2	2	no	—
14	#13133124	—	2	2	no	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	#107×C₅₃	—	2	2	no	—
18	C₂³×D₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	—	2	2	no	—
22	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	—	2	1	no	—
24	C₂×C₄×D₅₃	—	2	1	no	—
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×C₄×D₅₃	—	2	1	no	—
27	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅₃	Q₈	1	2	yes	—
29	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄⋊C₄×C₅₃	—	2	2	no	—
34	#11719921	—	4	4	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×C₄	1	2	yes	—
40	#11719912	—	2	8	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	1	no	—
43	C₂²×D₅₃	—	2	1	no	—
44	C₂²×D₅₃	—	2	2	no	—
45	C₂²×D₅₃	—	2	2	no	—
46	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	1	no	—
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	4	2	no	—
59	C₂×Dic₅₃	—	4	2	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
62	C₄×D₅₃	—	2	2	no	—
63	C₄×D₅₃	—	2	2	no	—
64	C₄×D₅₃	—	2	2	no	—
65	C₄×D₅₃	—	2	2	no	—
66	C₈×C₅₃	C₂²⋊C₄	1	4	yes	complemented
67	#88322	—	2	2	no	—
68	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₃	#121	1	1	yes	characteristic
70	C₂²×C₅₃	#121	1	1	yes	characteristic
71	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₅₃	#137	1	2	yes	—
73	C₂×D₅₃	—	2	2	no	—
74	C₂×D₅₃	—	2	4	no	—
75	C₂×D₅₃	—	4	2	no	—
76	C₂×D₅₃	—	4	1	no	—
77	C₂×D₅₃	—	4	1	no	—
78	C₄×C₅₃	—	8	2	no	—
79	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	#117	1	1	yes	characteristic
81	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
83	Dic₅₃	—	2	2	no	—
84	Dic₅₃	—	8	2	no	—
85	#1809	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	#380	1	1	yes	characteristic
87	C₂×C₅₃	#428	1	2	yes	—
88	C₂×C₅₃	#468	1	2	yes	—
89	C₂×C₅₃	#469	1	1	yes	characteristic
90	C₂×C₅₃	#470	1	1	yes	characteristic
91	D₅₃	—	4	2	no	—
92	C₂²×M₁₆	—	53	1	no	—
93	C₂×#107	D₅₃	1	1	yes	characteristic, complemented
94	C₂×#106	—	53	1	no	—
95	#380	—	53	2	no	—
96	#340	—	53	2	no	—
97	C₅₃	#1809	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	—	53	1	no	—
99	C₂×M₁₆	—	53	4	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂×M₁₆	—	106	2	no	—
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	53	2	no	—
105	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
106	C₂×C₂²⋊C₄	—	106	2	no	—
107	#107	—	2	2	no	—
108	#106	—	106	2	no	—
109	#95	—	106	2	no	—
110	C₂⁴	—	53	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	53	1	no	—
113	C₂²×C₄	—	53	2	no	—
114	C₂²×C₄	—	53	2	no	—
115	C₂²×C₄	—	106	1	no	—
116	C₂²×C₄	—	106	1	no	—
117	C₂²×C₄	—	106	1	no	—
118	C₂²×C₄	—	106	1	no	—
119	C₂²×C₄	—	106	2	no	—
120	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	M₁₆	—	106	8	no	—
122	C₂×C₈	—	53	2	no	—
123	C₂×C₈	—	106	1	no	—
124	C₂×C₈	—	106	1	no	—
125	C₂×C₈	C₄×D₅₃	1	2	yes	—
126	C₂×C₈	D₂₁₂	1	2	yes	—
127	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	106	2	no	—
131	C₄⋊C₄	—	106	2	no	—
132	C₂²⋊C₄	—	212	4	no	—
133	C₂³	—	53	2	no	—
134	C₂³	—	106	1	no	—
135	C₂³	—	106	1	no	—
136	C₂³	—	106	2	no	—
137	C₂³	—	106	2	no	—
138	C₂³	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	53	2	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	212	2	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	—	212	2	no	—
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	C₂×C₄	#11719947	1	1	yes	characteristic
155	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
156	C₂×C₄	#11719949	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₈	—	106	2	no	—
159	C₈	#11719921	1	4	yes	complemented
160	C₂²	—	106	4	no	—
161	C₂²	—	106	2	no	—
162	C₂²	—	212	2	no	—
163	C₂²	—	212	1	no	—
164	C₂²	—	212	1	no	—
165	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
166	C₂²	#13133188	1	1	yes	characteristic
167	C₂²	#13133192	1	1	yes	characteristic
168	C₂²	#13133241	1	2	yes	—
169	C₂²	#13133249	1	2	yes	—
170	C₄	—	8	2	no	—
171	C₄	—	106	2	no	—
172	C₄	—	424	2	no	—
173	C₄	C₂×#11719921	1	1	yes	characteristic
174	C₄	#13133175	1	1	yes	characteristic
175	C₄	#13133193	1	1	yes	characteristic
176	C₄	#13133197	1	1	yes	characteristic
177	C₂	—	212	2	no	—
178	C₂	#16810480	1	1	yes	characteristic
179	C₂	#16810664	1	2	yes	—
180	C₂	#16810870	1	2	yes	—
181	C₂	#16810873	1	1	yes	characteristic
182	C₂	#16810878	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal, characteristic

C₂×#13133149

C₂

yes

maximal, characteristic, complemented

#16810480

C₂

yes

maximal

#16810385

C₂

yes

maximal, complemented

C₂×#107×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

—

#13133149

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂³×D₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

#11719912

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

complemented

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#468

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×M₁₆

—

C₂×#107

D₅₃

yes

characteristic, complemented

C₂×#106

—

#380

—

#340

—

C₅₃

#1809

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

100

C₂×M₁₆

—

106

—

101

C₂×M₁₆

—

106

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₅₃

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

106

C₂×C₂²⋊C₄

—

106

—

107

#107

—

108

#106

—

106

—

109

#95

—

106

—

110

C₂⁴

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

106

—

116

C₂²×C₄

—

106

—

117

C₂²×C₄

—

106

—

118

C₂²×C₄

—

106

—

119

C₂²×C₄

—

106

—

120

C₂²×C₄

C₂²×D₅₃

yes

characteristic

121

M₁₆

—

106

—

122

C₂×C₈

—

123

C₂×C₈

—

106

—

124

C₂×C₈

—

106

—

125

C₂×C₈

C₄×D₅₃

yes

—

126

C₂×C₈

D₂₁₂

yes

—

127

C₂×C₈

C₂²⋊D₅₃

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₂²⋊C₄

—

212

—

133

C₂³

—

134

C₂³

—

106

—

135

C₂³

—

106

—

136

C₂³

—

106

—

137

C₂³

—

106

—

138

C₂³

D₄×D₅₃

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

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

—

212

—

152

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

153

C₂×C₄

C₂×D₂₁₂

yes

characteristic

154

C₂×C₄

#11719947

yes

characteristic

155

C₂×C₄

Q₈×D₅₃

yes

characteristic

156

C₂×C₄

#11719949

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

158

C₈

—

106

—

159

C₈

#11719921

yes

complemented

160

C₂²

—

106

—

161

C₂²

—

106

—

162

C₂²

—

212

—

163

C₂²

—

212

—

164

C₂²

—

212

—

165

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

166

C₂²

#13133188

yes

characteristic

167

C₂²

#13133192

yes

characteristic

168

C₂²

#13133241

yes

—

169

C₂²

#13133249

yes

—

170

C₄

—

171

C₄

—

106

—

172

C₄

—

424

—

173

C₄

C₂×#11719921

yes

characteristic

174

C₄

#13133175

yes

characteristic

175

C₄

#13133193

yes

characteristic

176

C₄

#13133197

yes

characteristic

177

C₂

—

212

—

178

C₂

#16810480

yes

characteristic

179

C₂

#16810664

yes

—

180

C₂

#16810870

yes

—

181

C₂

#16810873

yes

characteristic

182

C₂

#16810878

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1809
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#13133149	C₂	semidirect
4. #16810385	C₂	semidirect
5. C₂×#107×C₅₃	C₂	semidirect
9. C₂×#11719912	C₄	semidirect
66. C₈×C₅₃	C₂²⋊C₄	semidirect
93. C₂×#107	D₅₃	semidirect
97. C₅₃	#1809	semidirect
159. C₈	#11719921	semidirect

H ≅ G/N

type

2. C₂×#13133149

C₂

semidirect

4. #16810385

C₂

semidirect

5. C₂×#107×C₅₃

C₂

semidirect

9. C₂×#11719912

C₄

semidirect

66. C₈×C₅₃

C₂²⋊C₄

semidirect

93. C₂×#107

D₅₃

semidirect

97. C₅₃

#1809

semidirect

159. C₈

#11719921

semidirect

Subgroups — #20274749

Conjugacy classes of subgroups

Complements