Group #28302172 — subgroups

Subgroups — #28302172

Order 13568.

358 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₄×#13133288	C₂	1	1	yes	maximal, characteristic
2	#20289917	C₂	1	2	yes	maximal
3	#20289909	C₂	1	1	yes	maximal, characteristic
4	#20289546	C₂	1	2	yes	maximal
5	#20273460	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133288	C₄	1	2	yes	—
7	C₄×#11719936	C₂²	1	2	yes	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	#16811499	C₄	1	2	yes	—
11	#88324⋊C₈	—	2	2	no	—
12	#16811464	C₄	1	2	yes	—
13	#16811464	C₄	1	2	yes	—
14	C₈×#88324	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810375	C₄	1	2	yes	—
18	#16810374	C₄	1	1	yes	characteristic
19	C₅₃×#334	C₄	1	1	yes	characteristic
20	C₂×#11719936	C₂×C₄	1	4	yes	—
21	C₂×D₅₃⋊C₈	—	2	1	no	—
22	C₂×D₅₃⋊C₈	Q₈	1	2	yes	—
23	#13133288	C₈	1	8	yes	complemented
24	#13133272	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	C₂×C₄	1	2	yes	—
29	C₄×#88324	D₄	1	2	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133129	—	2	2	no	—
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	—	2	2	no	—
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	4	no	—
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	C₄⋊C₄	1	4	yes	—
47	#11719936	—	2	4	no	—
48	#11719936	C₂×C₈	1	8	yes	—
49	D₅₃⋊C₈	—	4	2	no	—
50	D₅₃⋊C₈	—	4	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅₃	#41	1	1	yes	characteristic
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₄²	1	1	yes	characteristic
61	C₈×D₅₃	—	2	2	no	—
62	C₈×D₅₃	C₂×C₈	1	4	yes	—
63	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×Dic₅₃	#117	1	1	yes	characteristic
69	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
70	C₄×D₅₃	—	2	2	no	—
71	C₄×D₅₃	—	2	2	no	—
72	C₄×D₅₃	C₂²×C₈	1	2	yes	—
73	C₄×D₅₃	#131	1	2	yes	—
74	#88324	—	2	4	no	—
75	#88324	—	2	8	no	—
76	#88324	—	4	2	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	—	4	2	no	—
79	C₈×C₅₃	C₄×C₈	1	2	yes	—
80	#88322	—	2	1	no	—
81	#88322	—	4	2	no	—
82	#88322	C₄×C₈	1	2	yes	—
83	#8294	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂×D₅₃	—	2	1	no	—
86	C₂×D₅₃	—	2	1	no	—
87	C₂×D₅₃	C₈×Q₈	1	2	yes	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₃	#405	1	1	yes	characteristic
91	C₄×C₅₃	#444	1	2	yes	—
92	Dic₅₃	—	2	1	no	—
93	Dic₅₃	#430	1	2	yes	—
94	Dic₅₃	C₈×D₄	1	2	yes	—
95	C₄×#108	—	53	1	no	—
96	#1564	—	53	1	no	—
97	C₈×M₁₆	—	53	2	no	—
98	C₂×#322	—	53	1	no	—
99	#1278	—	53	2	no	—
100	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₃	#1773	1	1	yes	characteristic
102	C₂×C₅₃	#1778	1	1	yes	characteristic
103	D₅₃	—	2	2	no	—
104	#431	—	53	2	no	—
105	C₂×#108	—	53	2	no	—
106	C₄×M₁₆	—	53	2	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	#341	—	53	2	no	—
111	#335	—	53	2	no	—
112	#334	—	53	1	no	—
113	#334	#6157	1	1	yes	characteristic
114	#322	—	53	2	no	—
115	#322	—	53	2	no	—
116	#322	—	106	2	no	—
117	C₈²	—	106	2	no	—
118	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	53	4	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	2	no	—
122	C₂²×C₈	—	53	1	no	—
123	C₂²×C₈	—	106	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	#108	—	53	8	no	—
126	C₄⋊C₈	—	53	2	no	—
127	C₄⋊C₈	C₂×#6157	1	2	yes	—
128	C₂²⋊C₈	—	106	2	no	—
129	#97	—	106	2	no	—
130	C₄×C₈	—	53	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	—	53	1	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	106	2	no	—
137	C₄×C₈	—	106	2	no	—
138	C₄×C₈	—	106	1	no	—
139	C₄×C₈	C₂×#6157	1	1	yes	characteristic
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	M₁₆	—	53	8	no	—
144	M₁₆	—	106	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	4	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	4	no	—
152	C₂×C₈	—	106	2	no	—
153	C₂×C₈	—	106	2	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	C₄×#6157	1	1	yes	characteristic
157	C₂×C₈	C₄×#6157	1	1	yes	characteristic
158	C₄²	—	53	1	no	—
159	C₄²	—	53	1	no	—
160	C₄²	—	53	1	no	—
161	C₄²	C₂²×#6157	1	1	yes	characteristic
162	C₂³	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	53	1	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	106	2	no	—
169	C₂×C₄	—	106	2	no	—
170	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
171	C₂×C₄	D₄×#6157	1	1	yes	characteristic
172	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	4	2	no	—
175	C₈	—	53	2	no	—
176	C₈	—	106	4	no	—
177	C₈	—	106	8	no	—
178	C₈	—	106	1	no	—
179	C₈	—	212	2	no	—
180	C₈	—	212	2	no	—
181	C₈	C₈×#6157	1	2	yes	—
182	C₂²	—	53	2	no	—
183	C₂²	—	106	1	no	—
184	C₂²	—	106	1	no	—
185	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	53	2	no	—
188	C₄	—	53	2	no	—
189	C₄	—	106	1	no	—
190	C₄	C₂×C₈×#6157	1	1	yes	characteristic
191	C₄	#16811589	1	1	yes	characteristic
192	C₄	#16811668	1	2	yes	—
193	C₂	—	106	2	no	—
194	C₂	#20290508	1	1	yes	characteristic
195	C₂	#20290523	1	1	yes	characteristic
196	C₂	#20290536	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133288

C₂

yes

maximal, characteristic

#20289917

C₂

yes

maximal

#20289909

C₂

yes

maximal, characteristic

#20289546

C₂

yes

maximal

#20273460

C₂

yes

maximal, characteristic

C₂×#13133288

C₄

yes

—

C₄×#11719936

C₂²

yes

—

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

C₄×D₅₃⋊C₈

C₂²

yes

characteristic

#16811499

C₄

yes

—

#88324⋊C₈

—

#16811464

C₄

yes

—

#16811464

C₄

yes

—

C₈×#88324

—

#16810675

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

C₄

yes

—

#16810374

C₄

yes

characteristic

C₅₃×#334

C₄

yes

characteristic

C₂×#11719936

C₂×C₄

yes

—

C₂×D₅₃⋊C₈

—

C₂×D₅₃⋊C₈

Q₈

yes

—

#13133288

C₈

yes

—

—

—

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₄×#88324

D₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133129

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

C₄⋊C₄

yes

—

#11719936

—

#11719936

C₂×C₈

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

C₄×Dic₅₃

yes

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₂²×D₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#117

yes

characteristic

C₂×Dic₅₃

C₄×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²×C₈

yes

—

C₄×D₅₃

#131

yes

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

#88322

—

#88322

—

#88322

C₄×C₈

yes

—

#8294

—

maximal, Sylow

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₈×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄×C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

#430

yes

—

Dic₅₃

C₈×D₄

yes

—

C₄×#108

—

#1564

—

C₈×M₁₆

—

C₂×#322

—

#1278

—

100

C₂×C₅₃

C₈×C₄⋊C₄

yes

characteristic

101

C₂×C₅₃

#1773

yes

characteristic

102

C₂×C₅₃

#1778

yes

characteristic

103

D₅₃

—

104

#431

—

105

C₂×#108

—

106

C₄×M₁₆

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#341

—

111

#335

—

112

#334

—

113

yes

114

—

115

#322

—

116

#322

—

106

—

117

C₈²

—

106

—

118

C₅₃

#8294

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

106

—

124

C₂×C₄²

—

125

#108

—

126

C₄⋊C₈

—

127

C₄⋊C₈

C₂×#6157

yes

—

128

C₂²⋊C₈

—

106

—

129

#97

—

106

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

106

—

137

C₄×C₈

—

106

—

138

C₄×C₈

—

106

—

139

C₄×C₈

C₂×#6157

yes

characteristic

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

M₁₆

—

144

M₁₆

—

106

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

106

—

151

C₂×C₈

—

106

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

212

—

155

C₂×C₈

—

212

—

156

C₂×C₈

C₄×#6157

yes

characteristic

157

C₂×C₈

C₄×#6157

yes

characteristic

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

C₂²×#6157

yes

characteristic

162

C₂³

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

106

—

169

C₂×C₄

—

106

—

170

C₂×C₄

C₂×C₄×#6157

yes

characteristic

171

C₂×C₄

D₄×#6157

yes

characteristic

172

C₂×C₄

Q₈×#6157

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

106

—

177

C₈

—

106

—

178

C₈

—

106

—

179

C₈

—

212

—

180

C₈

—

212

—

181

C₈

C₈×#6157

yes

—

182

C₂²

—

183

C₂²

—

106

—

184

C₂²

—

106

—

185

C₂²

C₄⋊C₄×#6157

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

106

—

190

C₄

C₂×C₈×#6157

yes

characteristic

191

C₄

#16811589

yes

characteristic

192

C₄

#16811668

yes

—

193

C₂

—

106

—

194

C₂

#20290508

yes

characteristic

195

C₂

#20290523

yes

characteristic

196

C₂

#20290536

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#334	#6157
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#334

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂²×D₅₃

C₄×Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
23. #13133288	C₈	semidirect
118. C₅₃	#8294	semidirect

H ≅ G/N

type

23. #13133288

C₈

semidirect

118. C₅₃

#8294

semidirect

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

subgroup	index
C₄×#13133288	2
#20289917	2
#20289909	2
#20289546	2
#20273460	2
#8294	53

subgroup

index

Subgroups — #28302172

Conjugacy classes of subgroups

Complements

Maximal subgroups