Group #20273407 — subgroups

Subgroups — #20273407

Order 6784.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16811171	C₂	1	1	yes	maximal, characteristic
2	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719947	C₄	1	2	yes	—
9	C₂×C₈×D₅₃	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133204	C₄	1	2	yes	—
12	#13133188	—	2	1	no	—
13	#13133188	—	2	1	no	—
14	D₄×Dic₅₃	—	2	1	no	—
15	D₄×Dic₅₃	—	2	1	no	—
16	C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	#13133146	—	2	1	no	—
20	#13133129	—	2	1	no	—
21	#13133127	—	2	1	no	—
22	#127×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊D₅₃	—	2	2	no	—
29	C₂²×Dic₅₃	—	2	2	no	—
30	#11719947	—	4	4	no	—
31	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
34	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²⋊Dic₅₃	—	2	2	no	—
39	C₂²⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	D₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₈×D₅₃	—	4	2	no	—
48	C₂³×C₅₃	—	2	2	no	—
49	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	D₄×C₅₃	—	4	2	no	—
51	D₄×C₅₃	—	4	1	no	—
52	D₄×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₅₃	—	4	4	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	4	4	no	—
61	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄×D₅₃	—	2	2	no	—
65	C₄×D₅₃	—	2	1	no	—
66	C₄×D₅₃	—	2	1	no	—
67	C₄×D₅₃	—	2	1	no	—
68	C₄×D₅₃	—	2	1	no	—
69	Dic₁₀₆	—	4	2	no	—
70	C₈×C₅₃	—	4	1	no	—
71	C₈×C₅₃	—	4	1	no	—
72	#88322	—	4	1	no	—
73	#88322	—	4	1	no	—
74	C₂²×C₅₃	—	4	2	no	—
75	C₂²×C₅₃	—	4	2	no	—
76	C₂²×C₅₃	—	4	2	no	—
77	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×D₅₃	—	2	1	no	—
79	C₂×D₅₃	—	2	1	no	—
80	C₂×D₅₃	#100	1	2	yes	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	#135	1	1	yes	characteristic
83	C₄×C₅₃	#135	1	1	yes	characteristic
84	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₅₃	#137	1	1	yes	characteristic
86	Dic₅₃	—	2	1	no	—
87	Dic₅₃	—	2	1	no	—
88	Dic₅₃	—	2	1	no	—
89	Dic₅₃	—	4	2	no	—
90	#1551	—	53	1	no	maximal, Sylow
91	C₂×C₅₃	—	8	2	no	—
92	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
93	C₂×C₅₃	#416	1	1	yes	characteristic
94	C₂×C₅₃	#417	1	1	yes	characteristic
95	D₅₃	—	2	2	no	—
96	#532	—	53	1	no	—
97	#432	—	53	1	no	—
98	C₂×C₄⋊C₈	—	53	1	no	—
99	#333	—	53	1	no	—
100	#333	—	53	1	no	—
101	#331	—	53	1	no	—
102	#331	D₅₃	1	1	yes	characteristic, complemented
103	C₅₃	#1551	1	1	yes	characteristic, complemented, Sylow
104	C₂×#41	—	53	2	no	—
105	C₂²×C₈	—	106	1	no	—
106	C₄⋊Q₈	—	53	1	no	—
107	#127	C₂×D₅₃	1	1	yes	characteristic
108	#124	—	53	2	no	—
109	#121	—	106	1	no	—
110	#121	—	106	1	no	—
111	C₄×D₄	—	106	1	no	—
112	C₄×D₄	—	106	1	no	—
113	C₂×C₄²	—	53	1	no	—
114	C₄⋊C₈	—	53	1	no	—
115	C₄⋊C₈	—	53	1	no	—
116	C₄⋊C₈	—	106	1	no	—
117	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₂²⋊C₈	—	106	1	no	—
120	#97	—	106	1	no	—
121	#41	—	212	4	no	—
122	C₂×Q₈	—	53	2	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	106	2	no	—
126	C₂×D₄	C₄×D₅₃	1	2	yes	—
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	53	1	no	—
130	C₂²×C₄	—	106	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	106	1	no	—
134	C₂×C₈	—	106	1	no	—
135	C₂×C₈	—	212	2	no	—
136	C₄⋊C₄	—	106	1	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₂²⋊C₄	—	106	2	no	—
139	C₂²⋊C₄	—	106	2	no	—
140	C₄²	—	53	1	no	—
141	C₄²	—	53	1	no	—
142	C₄²	—	53	1	no	—
143	C₄²	C₂²×D₅₃	1	1	yes	characteristic
144	C₂³	—	2	2	no	—
145	C₂³	—	53	1	no	—
146	Q₈	—	212	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	212	4	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	—	106	1	no	—
159	C₂×C₄	—	106	1	no	—
160	C₂×C₄	—	106	1	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	212	4	no	—
163	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	212	1	no	—
169	C₈	—	212	1	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	53	2	no	—
174	C₂²	—	106	1	no	—
175	C₂²	—	106	1	no	—
176	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	106	1	no	—
179	C₄	—	106	1	no	—
180	C₄	—	106	1	no	—
181	C₄	—	212	2	no	—
182	C₄	#13133236	1	1	yes	characteristic
183	C₄	#13133238	1	1	yes	characteristic
184	C₄	#13133241	1	1	yes	characteristic
185	C₄	#13133246	1	1	yes	characteristic
186	C₂	—	8	2	no	—
187	C₂	—	106	2	no	—
188	C₂	#100×D₅₃	1	1	yes	characteristic
189	C₂	#16810616	1	1	yes	characteristic
190	C₂	#16810621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811171

C₂

yes

maximal, characteristic

#16810679

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

C₅₃×#331

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

C₄

yes

—

C₂×C₈×D₅₃

—

#13133217

C₂²

yes

characteristic

#13133204

C₄

yes

—

#13133188

—

#13133188

—

D₄×Dic₅₃

—

D₄×Dic₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

#127×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719947

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₈×D₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#100

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1551

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#532

—

#432

—

C₂×C₄⋊C₈

—

#333

—

100

#333

—

101

#331

—

102

#331

D₅₃

yes

characteristic, complemented

103

C₅₃

#1551

yes

characteristic, complemented, Sylow

104

C₂×#41

—

105

C₂²×C₈

—

106

—

106

C₄⋊Q₈

—

107

#127

C₂×D₅₃

yes

characteristic

108

#124

—

109

#121

—

106

—

110

#121

—

106

—

111

C₄×D₄

—

106

—

112

C₄×D₄

—

106

—

113

C₂×C₄²

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

106

—

117

C₄⋊C₈

C₂×D₅₃

yes

characteristic

118

C₄⋊C₈

C₂×D₅₃

yes

characteristic

119

C₂²⋊C₈

—

106

—

120

#97

—

106

—

121

#41

—

212

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

106

—

126

C₂×D₄

C₄×D₅₃

yes

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

106

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

106

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

212

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₂²⋊C₄

—

106

—

139

C₂²⋊C₄

—

106

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₂²×D₅₃

yes

characteristic

144

C₂³

—

145

C₂³

—

146

Q₈

—

212

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

212

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

212

—

163

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

164

C₂×C₄

D₄×D₅₃

yes

characteristic

165

C₂×C₄

D₄×D₅₃

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

—

212

—

169

C₈

—

212

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

106

—

175

C₂²

—

106

—

176

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

177

C₄

—

178

C₄

—

106

—

179

C₄

—

106

—

180

C₄

—

106

—

181

C₄

—

212

—

182

C₄

#13133236

yes

characteristic

183

C₄

#13133238

yes

characteristic

184

C₄

#13133241

yes

characteristic

185

C₄

#13133246

yes

characteristic

186

C₂

—

187

C₂

—

106

—

188

C₂

#100×D₅₃

yes

characteristic

189

C₂

#16810616

yes

characteristic

190

C₂

#16810621

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#331	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#331	D₅₃
hypocenter	C₅₃	#1551
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#331

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16810679	C₂	semidirect
3. C₄⋊C₈×D₅₃	C₂	semidirect
4. #16810371	C₂	semidirect
5. #16810371	C₂	semidirect
6. #16810367	C₂	semidirect
7. C₅₃×#331	C₂	semidirect
102. #331	D₅₃	semidirect
103. C₅₃	#1551	semidirect

H ≅ G/N

type

2. #16810679

C₂

semidirect

3. C₄⋊C₈×D₅₃

C₂

semidirect

4. #16810371

C₂

semidirect

5. #16810371

C₂

semidirect

6. #16810367

C₂

semidirect

7. C₅₃×#331

C₂

semidirect

102. #331

D₅₃

semidirect

103. C₅₃

#1551

semidirect

Subgroups — #20273407

Conjugacy classes of subgroups

Complements