Group #27303632 — subgroups

Subgroups — #27303632

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	C₂²×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810437	C₂	1	2	yes	maximal, complemented
3	#20275625	C₂	1	2	yes	maximal, complemented
4	#20272809	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₂₄	C₂²	1	2	yes	—
7	C₂×D₈₄₈	—	2	4	no	—
8	C₂×D₈₄₈	—	2	2	no	—
9	C₂×#13133153	C₂²	1	2	yes	—
10	#16810627	—	2	2	no	—
11	C₂×#13133133	—	2	2	no	—
12	#16810437	—	2	4	no	—
13	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×D₂₁₂	—	2	2	no	—
18	C₂×D₄₂₄	—	2	2	no	—
19	C₂×D₄₂₄	—	2	2	no	—
20	C₂×D₄₂₄	—	2	2	no	—
21	C₂×D₄₂₄	—	2	2	no	—
22	C₂×D₄₂₄	—	2	2	no	—
23	C₂×D₄₂₄	D₄	1	4	yes	—
24	C₂×#11719917	—	2	2	no	—
25	D₈₄₈	—	2	8	no	—
26	#13133153	—	2	2	no	—
27	#13133153	—	2	2	no	—
28	#13133137	—	4	4	no	—
29	#13133133	—	4	4	no	—
30	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#117×C₅₃	D₄	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	2	no	—
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	—	2	1	no	—
36	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
37	#97×C₅₃	—	2	1	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₂³×D₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×D₂₁₂	—	2	4	no	—
42	C₂×D₂₁₂	—	4	2	no	—
43	C₂×D₂₁₂	—	4	2	no	—
44	C₂×D₂₁₂	—	4	2	no	—
45	C₂×D₂₁₂	—	4	2	no	—
46	C₂×D₂₁₂	—	4	2	no	—
47	M₁₆×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
54	C₂²⋊C₄×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	#11719917	—	4	2	no	—
57	#11719917	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	D₄₂₄	—	2	4	no	—
60	D₄₂₄	—	2	4	no	—
61	D₄₂₄	—	4	2	no	—
62	D₄₂₄	—	4	2	no	—
63	C₁₆×C₅₃	—	2	2	no	—
64	C₁₆×C₅₃	—	4	1	no	—
65	C₂³×C₅₃	#135	1	1	yes	characteristic
66	C₂²×D₅₃	—	4	4	no	—
67	C₂²×D₅₃	—	8	2	no	—
68	C₂²×D₅₃	—	8	2	no	—
69	C₂²×D₅₃	—	16	4	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#121	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
75	D₂₁₂	—	4	4	no	—
76	D₂₁₂	—	4	4	no	—
77	D₂₁₂	—	8	2	no	—
78	D₂₁₂	—	8	2	no	—
79	C₈×C₅₃	—	4	1	no	—
80	C₈×C₅₃	#120	1	1	yes	characteristic
81	C₈×C₅₃	#123	1	2	yes	—
82	C₈×C₅₃	#124	1	1	yes	characteristic
83	#88322	—	8	2	no	—
84	#13439	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	#463	1	2	yes	—
87	C₂²×C₅₃	#466	1	1	yes	characteristic
88	C₂×D₅₃	—	8	4	no	—
89	C₂×D₅₃	—	16	4	no	—
90	C₂×D₅₃	—	16	4	no	—
91	C₂×D₅₃	—	16	4	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	#394	1	1	yes	characteristic
94	C₄×C₅₃	#459	1	2	yes	—
95	C₄×C₅₃	#469	1	1	yes	characteristic
96	C₂²×D₁₆	—	53	1	no	—
97	#2151	D₅₃	1	1	yes	characteristic, complemented
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	#2054	1	1	yes	characteristic
103	C₂×C₅₃	#2221	1	2	yes	—
104	D₅₃	—	16	4	no	—
105	C₂²×D₈	—	53	2	no	—
106	C₂×D₁₆	—	106	4	no	—
107	C₂×D₁₆	—	106	2	no	—
108	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	2	no	—
115	#361	—	106	4	no	—
116	C₅₃	#13439	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	106	2	no	—
118	C₂×D₈	—	53	4	no	—
119	C₂×D₈	—	106	2	no	—
120	C₂×D₈	—	106	2	no	—
121	C₂×D₈	—	106	2	no	—
122	C₂×D₈	—	106	2	no	—
123	C₂×D₈	—	106	2	no	—
124	C₂×M₁₆	—	106	2	no	—
125	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#117	D₂₁₂	1	1	yes	characteristic
128	D₁₆	—	106	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	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	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	212	2	no	—
140	C₂×D₄	—	106	4	no	—
141	C₂×D₄	—	212	2	no	—
142	C₂×D₄	—	212	2	no	—
143	C₂×D₄	—	212	2	no	—
144	C₂×D₄	—	212	2	no	—
145	C₂×D₄	—	212	2	no	—
146	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	D₈	—	106	4	no	—
148	D₈	—	106	4	no	—
149	D₈	—	212	2	no	—
150	D₈	—	212	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	212	2	no	—
153	M₁₆	—	212	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	#11719945	1	1	yes	characteristic
157	C₂×C₈	D₄×D₅₃	1	2	yes	—
158	C₂×C₈	#11719949	1	2	yes	—
159	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	D₄₂₄	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	848	4	no	—
166	C₂³	—	424	2	no	—
167	C₂³	—	424	2	no	—
168	C₂³	—	212	4	no	—
169	C₂³	#13133239	1	1	yes	characteristic
170	D₄	—	424	2	no	—
171	D₄	—	424	2	no	—
172	D₄	—	212	4	no	—
173	D₄	—	212	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#13133191	1	1	yes	characteristic
176	C₂×C₄	#13133192	1	2	yes	—
177	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
178	C₂×C₄	#13133245	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	424	2	no	—
181	C₈	#13133185	1	1	yes	characteristic
182	C₈	#13133202	1	2	yes	—
183	C₈	#13133206	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	848	4	no	—
186	C₂²	—	848	4	no	—
187	C₂²	—	848	4	no	—
188	C₂²	—	424	4	no	—
189	C₂²	#16810839	1	2	yes	—
190	C₂²	#16810859	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	#16810540	1	1	yes	characteristic
193	C₄	#16810819	1	2	yes	—
194	C₄	#16810876	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	848	4	no	—
197	C₂	#20276026	1	1	yes	characteristic
198	C₂	#20276835	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈₄₈

C₂

yes

maximal, characteristic

C₂×#16810437

C₂

yes

maximal, complemented

#20275625

C₂

yes

maximal, complemented

#20272809

C₂

yes

maximal, characteristic, complemented

C₅₃×#2151

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₂₄

C₂²

yes

—

C₂×D₈₄₈

—

C₂×D₈₄₈

—

C₂×#13133153

C₂²

yes

—

#16810627

—

C₂×#13133133

—

#16810437

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#405

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂²×D₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

D₄

yes

—

C₂×#11719917

—

D₈₄₈

—

—

—

—

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

—

C₄⋊C₄×C₅₃

D₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#135

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×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

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

#120

yes

characteristic

C₈×C₅₃

#123

yes

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#463

yes

—

C₂²×C₅₃

#466

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#394

yes

characteristic

C₄×C₅₃

#459

yes

—

C₄×C₅₃

#469

yes

characteristic

C₂²×D₁₆

—

#2151

D₅₃

yes

characteristic, complemented

C₂×#361

—

#1978

—

100

#1381

—

101

C₂×C₅₃

—

102

C₂×C₅₃

#2054

yes

characteristic

103

C₂×C₅₃

#2221

yes

—

104

D₅₃

—

105

C₂²×D₈

—

106

C₂×D₁₆

—

106

—

107

C₂×D₁₆

—

106

—

108

C₂×M₃₂

C₂×D₅₃

yes

characteristic

109

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

110

C₂×#108

—

111

#418

—

106

—

112

C₂×#100

—

106

—

113

#405

C₂×D₅₃

yes

characteristic

114

C₄⋊C₁₆

—

115

#361

—

106

—

116

C₅₃

#13439

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

106

—

118

C₂×D₈

—

119

C₂×D₈

—

106

—

120

C₂×D₈

—

106

—

121

C₂×D₈

—

106

—

122

C₂×D₈

—

106

—

123

C₂×D₈

—

106

—

124

C₂×M₁₆

—

106

—

125

C₂×M₁₆

D₂₁₂

yes

characteristic

126

C₂²×C₈

C₂²×D₅₃

yes

characteristic

127

#117

D₂₁₂

yes

characteristic

128

D₁₆

—

106

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂²⋊D₅₃

yes

—

133

#108

—

106

—

134

#108

—

106

—

135

#102

—

212

—

136

#100

—

212

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

212

—

140

C₂×D₄

—

106

—

141

C₂×D₄

—

212

—

142

C₂×D₄

—

212

—

143

C₂×D₄

—

212

—

144

C₂×D₄

—

212

—

145

C₂×D₄

—

212

—

146

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

147

D₈

—

106

—

148

D₈

—

106

—

149

D₈

—

212

—

150

D₈

—

212

—

151

M₁₆

—

152

M₁₆

—

212

—

153

M₁₆

—

212

—

154

C₂×C₈

—

155

C₂×C₈

—

212

—

156

C₂×C₈

#11719945

yes

characteristic

157

C₂×C₈

D₄×D₅₃

yes

—

158

C₂×C₈

#11719949

yes

—

159

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

160

C₄⋊C₄

D₄₂₄

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

848

—

166

C₂³

—

424

—

167

C₂³

—

424

—

168

C₂³

—

212

—

169

C₂³

#13133239

yes

characteristic

170

D₄

—

424

—

171

D₄

—

424

—

172

D₄

—

212

—

173

D₄

—

212

—

174

C₂×C₄

—

175

C₂×C₄

#13133191

yes

characteristic

176

C₂×C₄

#13133192

yes

—

177

C₂×C₄

C₂×D₄₂₄

yes

characteristic

178

C₂×C₄

#13133245

yes

—

179

C₈

—

180

C₈

—

424

—

181

C₈

#13133185

yes

characteristic

182

C₈

#13133202

yes

—

183

C₈

#13133206

yes

characteristic

184

C₂²

—

185

C₂²

—

848

—

186

C₂²

—

848

—

187

C₂²

—

848

—

188

C₂²

—

424

—

189

C₂²

#16810839

yes

—

190

C₂²

#16810859

yes

characteristic

191

C₄

—

192

C₄

#16810540

yes

characteristic

193

C₄

#16810819

yes

—

194

C₄

#16810876

yes

characteristic

195

C₂

—

196

C₂

—

848

—

197

C₂

#20276026

yes

characteristic

198

C₂

#20276835

yes

—

	subgroup	G/H
center	C₂²	#16810859
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2151	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂²×D₈₄₈	C₂
hypercenter	#2151	D₅₃
hypocenter	C₅₃	#13439
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#16810859

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2151

C₂

socle

C₂²×C₅₃

#466

prime radical

C₂²×D₈₄₈

C₂

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. C₂×#16810437	C₂	semidirect
3. #20275625	C₂	semidirect
4. #20272809	C₂	semidirect
5. C₅₃×#2151	C₂	semidirect
97. #2151	D₅₃	semidirect
116. C₅₃	#13439	semidirect

H ≅ G/N

type

2. C₂×#16810437

C₂

semidirect

3. #20275625

C₂

semidirect

4. #20272809

C₂

semidirect

5. C₅₃×#2151

C₂

semidirect

97. #2151

D₅₃

semidirect

116. C₅₃

#13439

semidirect

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

subgroup	index
C₂²×D₈₄₈	2
C₂×#16810437	2
#20275625	2
#20272809	2
C₅₃×#2151	2
#13439	53

subgroup

index

C₂²×D₈₄₈

C₅₃×#2151

Subgroups — #27303632

Conjugacy classes of subgroups

Complements

Maximal subgroups