Group #27303629 — subgroups

Subgroups — #27303629

Order 13568.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810436	C₂	1	2	yes	maximal, complemented
3	#20275626	C₂	1	2	yes	maximal, complemented
4	#20272809	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810628	—	2	2	no	—
9	C₂×#13133134	—	2	2	no	—
10	#16810578	C₂²	1	1	yes	characteristic
11	#16810441	—	2	2	no	—
12	#16810436	—	2	4	no	—
13	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₅₃	—	2	4	no	—
15	C₂×D₁₆×C₅₃	—	2	2	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	2	no	—
18	C₂×#11719917	D₄	1	1	yes	characteristic
19	#13133176	D₄	1	1	yes	characteristic
20	#13133157	—	2	2	no	—
21	C₂×#11719908	—	2	1	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133138	—	4	4	no	—
25	#13133134	—	4	4	no	—
26	#13133127	—	2	1	no	—
27	C₈×Dic₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	2	no	—
32	C₂×D₈×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	D₄	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	D₁₆×C₅₃	—	2	8	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
39	C₂⁴×C₅₃	—	4	2	no	—
40	C₂×D₄×C₅₃	—	2	4	no	—
41	C₂×D₄×C₅₃	—	4	2	no	—
42	C₂×D₄×C₅₃	—	4	2	no	—
43	C₂×D₄×C₅₃	—	4	2	no	—
44	C₂×D₄×C₅₃	—	4	2	no	—
45	C₂×D₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	D₈×C₅₃	—	2	4	no	—
48	D₈×C₅₃	—	2	4	no	—
49	D₈×C₅₃	—	4	2	no	—
50	D₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₂²⋊Dic₅₃	—	2	1	no	—
56	C₄⋊Dic₅₃	D₈	1	2	yes	—
57	C₄×Dic₅₃	—	2	1	no	—
58	#11719917	—	2	2	no	—
59	#11719917	—	4	2	no	—
60	#11719917	—	4	2	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	4	2	no	—
63	C₁₆×C₅₃	—	2	2	no	—
64	#11719908	—	4	1	no	—
65	C₂³×C₅₃	—	4	4	no	—
66	C₂³×C₅₃	—	8	2	no	—
67	C₂³×C₅₃	—	8	2	no	—
68	C₂³×C₅₃	—	16	4	no	—
69	C₂³×C₅₃	#135	1	1	yes	characteristic
70	D₄×C₅₃	—	4	4	no	—
71	D₄×C₅₃	—	4	4	no	—
72	D₄×C₅₃	—	8	2	no	—
73	D₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₈×C₅₃	#120	1	1	yes	characteristic
80	C₈×C₅₃	#123	1	2	yes	—
81	C₈×C₅₃	#124	1	1	yes	characteristic
82	#88322	—	4	1	no	—
83	#88322	—	8	2	no	—
84	#13439	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	8	4	no	—
87	C₂²×C₅₃	—	16	4	no	—
88	C₂²×C₅₃	—	16	4	no	—
89	C₂²×C₅₃	—	16	4	no	—
90	C₂²×C₅₃	#463	1	2	yes	—
91	C₂²×C₅₃	#466	1	1	yes	characteristic
92	C₄×C₅₃	#394	1	1	yes	characteristic
93	C₄×C₅₃	#459	1	2	yes	—
94	C₄×C₅₃	#469	1	1	yes	characteristic
95	Dic₅₃	—	4	2	no	—
96	C₂²×D₁₆	D₅₃	1	1	yes	characteristic
97	#2151	—	53	1	no	—
98	C₂×#361	—	53	2	no	—
99	#1978	—	53	2	no	—
100	#1381	—	53	1	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	—	16	4	no	—
103	C₂×C₅₃	#2054	1	1	yes	characteristic
104	C₂×C₅₃	#2221	1	2	yes	—
105	C₂²×D₈	C₂×D₅₃	1	2	yes	—
106	C₂×D₁₆	—	2	4	no	—
107	C₂×D₁₆	—	2	2	no	—
108	C₂×M₃₂	—	53	1	no	—
109	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
110	C₂×#108	—	53	2	no	—
111	#418	—	106	2	no	—
112	C₂×#100	—	106	2	no	—
113	#405	—	53	1	no	—
114	C₄⋊C₁₆	—	106	2	no	—
115	#361	—	106	4	no	—
116	C₅₃	#13439	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	C₂²⋊D₅₃	1	4	yes	—
124	C₂×M₁₆	—	53	1	no	—
125	C₂×M₁₆	—	106	2	no	—
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#117	—	53	1	no	—
128	D₁₆	—	2	8	no	—
129	M₃₂	—	106	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	106	1	no	—
132	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
133	#108	—	106	2	no	—
134	#108	—	106	2	no	—
135	#102	—	212	4	no	—
136	#100	—	212	4	no	—
137	#97	—	106	1	no	—
138	C₄×C₈	—	106	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	106	2	no	—
152	M₁₆	—	212	2	no	—
153	M₁₆	—	212	2	no	—
154	C₂×C₈	—	106	1	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	#11719947	1	1	yes	characteristic
157	C₂×C₈	#11719947	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
159	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	—	53	2	no	—
161	C₂²⋊C₄	—	106	1	no	—
162	C₄²	—	106	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	212	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#13133236	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	#13133188	1	1	yes	characteristic
176	C₂×C₄	#13133189	1	2	yes	—
177	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
178	C₂×C₄	#13133241	1	2	yes	—
179	C₈	—	424	2	no	—
180	C₈	—	212	1	no	—
181	C₈	#13133186	1	1	yes	characteristic
182	C₈	#13133198	1	2	yes	—
183	C₈	#13133204	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#16810836	1	2	yes	—
190	C₂²	#16810856	1	1	yes	characteristic
191	C₄	—	212	2	no	—
192	C₄	#16810537	1	1	yes	characteristic
193	C₄	#16810816	1	2	yes	—
194	C₄	#16810873	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	#20276023	1	1	yes	characteristic
198	C₂	#20276832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276473

C₂

yes

maximal, characteristic, complemented

C₂×#16810436

C₂

yes

maximal, complemented

#20275626

C₂

yes

maximal, complemented

#20272809

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

#16810628

—

C₂×#13133134

—

#16810578

C₂²

yes

characteristic

#16810441

—

#16810436

—

C₂²×D₈×C₅₃

C₂²

yes

—

C₂×D₁₆×C₅₃

—

C₂×D₁₆×C₅₃

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂×#11719917

D₄

yes

characteristic

#13133176

D₄

yes

—

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂²×D₄×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

D₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

D₈

yes

—

C₄×Dic₅₃

—

—

—

—

—

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#135

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

#120

yes

characteristic

C₈×C₅₃

#123

yes

—

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#463

yes

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#469

yes

characteristic

Dic₅₃

—

C₂²×D₁₆

D₅₃

yes

—

—

—

100

#1381

—

101

C₂×C₅₃

—

102

C₂×C₅₃

—

103

C₂×C₅₃

#2054

yes

characteristic

104

C₂×C₅₃

#2221

yes

—

105

C₂²×D₈

C₂×D₅₃

yes

—

106

C₂×D₁₆

—

107

C₂×D₁₆

—

108

C₂×M₃₂

—

109

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

110

C₂×#108

—

111

#418

—

106

—

112

C₂×#100

—

106

—

113

#405

—

114

C₄⋊C₁₆

—

106

—

115

#361

—

106

—

116

C₅₃

#13439

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

C₂²⋊D₅₃

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

106

—

126

C₂²×C₈

C₂²×D₅₃

yes

characteristic

127

#117

—

128

D₁₆

—

129

M₃₂

—

106

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

106

—

132

C₂×C₁₆

C₂²⋊D₅₃

yes

—

133

#108

—

106

—

134

#108

—

106

—

135

#102

—

212

—

136

#100

—

212

—

137

#97

—

106

—

138

C₄×C₈

—

106

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

D₄×D₅₃

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

106

—

152

M₁₆

—

212

—

153

M₁₆

—

212

—

154

C₂×C₈

—

106

—

155

C₂×C₈

—

212

—

156

C₂×C₈

#11719947

yes

characteristic

157

C₂×C₈

#11719947

yes

—

158

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

159

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

106

—

162

C₄²

—

106

—

163

C₁₆

—

164

C₁₆

—

212

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#13133236

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

106

—

175

C₂×C₄

#13133188

yes

characteristic

176

C₂×C₄

#13133189

yes

—

177

C₂×C₄

D₈×D₅₃

yes

characteristic

178

C₂×C₄

#13133241

yes

—

179

C₈

—

424

—

180

C₈

—

212

—

181

C₈

#13133186

yes

characteristic

182

C₈

#13133198

yes

—

183

C₈

#13133204

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#16810836

yes

—

190

C₂²

#16810856

yes

characteristic

191

C₄

—

212

—

192

C₄

#16810537

yes

characteristic

193

C₄

#16810816

yes

—

194

C₄

#16810873

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#20276023

yes

characteristic

198

C₂

#20276832

yes

—

	subgroup	G/H
center	C₂²	#16810856
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂²×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#466
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	#13439
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

subgroup

G/H

center

C₂²

#16810856

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting, prime radical

C₂²×D₁₆×C₅₃

C₂

socle

C₂²×C₅₃

#466

hypercenter

C₂²×D₁₆

D₅₃

hypocenter

C₅₃

#13439

LCS

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

N	H ≅ G/N	type
1. #20276473	C₂	semidirect
2. C₂×#16810436	C₂	semidirect
3. #20275626	C₂	semidirect
4. #20272809	C₂	semidirect
116. C₅₃	#13439	semidirect

H ≅ G/N

type

1. #20276473

C₂

semidirect

2. C₂×#16810436

C₂

semidirect

3. #20275626

C₂

semidirect

4. #20272809

C₂

semidirect

116. C₅₃

#13439

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276473	2
C₂×#16810436	2
#20275626	2
#20272809	2
C₂²×D₁₆×C₅₃	2
#13439	53

subgroup

index

C₂²×D₁₆×C₅₃

Subgroups — #27303629

Conjugacy classes of subgroups

Complements

Maximal subgroups