Group #28303521 — subgroups

Subgroups — #28303521

Order 13568.

310 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 120 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	#20290262	C₂	1	2	yes	maximal
2	#20289555	C₂	1	2	yes	maximal
3	#20289539	C₂	1	2	yes	maximal
4	#20274401	C₂	1	1	yes	maximal, characteristic
5	#16811639	C₂²	1	4	yes	—
6	#16811486	—	2	2	no	—
7	#16811484	—	2	2	no	—
8	C₂×#13133128	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×#13133127	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810469	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₄⋊#88324	—	2	4	no	—
16	C₂²⋊#88324	—	2	4	no	—
17	C₄×#88324	—	2	4	no	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
22	#13133128	—	2	2	no	—
23	#13133127	—	2	2	no	—
24	#13133127	—	2	2	no	—
25	C₈×Dic₅₃	C₈	1	4	yes	complemented
26	C₈×Dic₅₃	D₄	1	1	yes	characteristic
27	C₈×Dic₅₃	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
30	#13133124	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	2	1	no	—
38	C₂²×Dic₅₃	#41	1	2	yes	—
39	C₂×#88324	—	4	4	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	M₁₆	1	2	yes	—
49	C₄×Dic₅₃	—	2	1	no	—
50	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
51	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄×Dic₅₃	#41	1	2	yes	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	—	2	1	no	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₂×C₈	1	2	yes	—
61	C₂³×C₅₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	4	2	no	—
72	C₂×Dic₅₃	—	4	2	no	—
73	C₂×Dic₅₃	#124	1	2	yes	—
74	C₂×Dic₅₃	#125	1	2	yes	—
75	#88324	—	8	4	no	—
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
78	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
79	#88322	—	2	1	no	—
80	#88322	—	2	1	no	—
81	#88322	—	2	2	no	—
82	#8780	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	#382	1	1	yes	characteristic
85	C₂²×C₅₃	#431	1	1	yes	characteristic
86	C₂²×C₅₃	#432	1	1	yes	characteristic
87	C₄×C₅₃	—	4	2	no	—
88	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₅₃	#423	1	1	yes	characteristic
90	C₄×C₅₃	#431	1	1	yes	characteristic
91	C₄×C₅₃	#432	1	1	yes	characteristic
92	Dic₅₃	—	2	2	no	—
93	Dic₅₃	—	2	2	no	—
94	Dic₅₃	—	4	2	no	—
95	#1726	—	53	1	no	—
96	#1564	—	53	2	no	—
97	#1281	—	53	2	no	—
98	#1274	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1831	1	1	yes	characteristic
102	C₂×C₅₃	#1833	1	2	yes	—
103	#431	—	53	4	no	—
104	C₂×#97	—	53	1	no	—
105	C₂×#97	—	53	1	no	—
106	C₂×#97	—	53	1	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	#6157	1	1	yes	characteristic
110	#376	—	53	1	no	—
111	#335	—	106	2	no	—
112	#334	—	106	2	no	—
113	C₅₃	#8780	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
117	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
118	C₂×C₄²	—	53	1	no	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
121	C₄⋊C₈	—	106	4	no	—
122	C₂²⋊C₈	—	106	4	no	—
123	#97	—	106	2	no	—
124	#97	—	106	2	no	—
125	#97	—	106	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	53	1	no	—
128	C₄×C₈	—	53	1	no	—
129	C₄×C₈	—	53	1	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	53	4	no	—
132	C₄×C₈	—	106	4	no	—
133	#95	—	106	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	—	106	1	no	—
137	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	106	1	no	—
144	C₂×C₈	—	106	1	no	—
145	C₂×C₈	—	106	1	no	—
146	C₂×C₈	—	106	1	no	—
147	C₂×C₈	—	212	4	no	—
148	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
149	C₂×C₈	#11719936	1	2	yes	—
150	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
151	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
152	C₂×C₈	#11719940	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	53	1	no	—
155	C₄²	—	53	1	no	—
156	C₄²	—	53	2	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	106	1	no	—
159	C₄²	#11719936	1	2	yes	—
160	C₂³	#13133296	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	53	2	no	—
164	C₂×C₄	—	53	2	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	—	212	2	no	—
167	C₂×C₄	—	212	2	no	—
168	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
169	C₂×C₄	#13133296	1	1	yes	characteristic
170	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	106	1	no	—
176	C₈	—	106	1	no	—
177	C₈	—	106	2	no	—
178	C₈	—	424	4	no	—
179	C₈	#13133284	1	2	yes	—
180	C₈	#13133284	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#16811567	1	1	yes	characteristic
183	C₂²	#16811642	1	1	yes	characteristic
184	C₂²	#16811644	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	106	2	no	—
187	C₄	—	106	2	no	—
188	C₄	—	212	2	no	—
189	C₄	C₂×#13133284	1	1	yes	characteristic
190	C₄	#16811627	1	1	yes	characteristic
191	C₄	#16811642	1	1	yes	characteristic
192	C₄	#16811644	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#20290632	1	1	yes	characteristic
196	C₂	#20290634	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290262

C₂

yes

maximal

#20289555

C₂

yes

maximal

#20289539

C₂

yes

maximal

#20274401

C₂

yes

maximal, characteristic

#16811639

C₂²

yes

—

#16811486

—

#16811484

—

C₂×#13133128

C₄

yes

characteristic

C₂×#13133127

C₄

yes

characteristic

C₂×#13133127

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810469

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

—

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂³

yes

—

—

—

C₈×Dic₅₃

C₈

yes

complemented

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

#41

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

#41

yes

—

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#124

yes

—

C₂×Dic₅₃

#125

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

C₂×C₅₃

—

100

C₂×C₅₃

—

101

C₂×C₅₃

#1831

yes

characteristic

102

C₂×C₅₃

#1833

yes

—

103

#431

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×#97

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

#6157

yes

characteristic

110

#376

—

111

#335

—

106

—

112

#334

—

106

—

113

C₅₃

#8780

yes

characteristic, complemented, Sylow

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂×#6157

yes

characteristic

117

C₂²×C₈

C₂×#6157

yes

characteristic

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×#6157

yes

characteristic

121

C₄⋊C₈

—

106

—

122

C₂²⋊C₈

—

106

—

123

#97

—

106

—

124

#97

—

106

—

125

#97

—

106

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

106

—

133

#95

—

106

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

106

—

137

C₂²×C₄

C₂²×#6157

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

212

—

148

C₂×C₈

D₅₃⋊C₈

yes

—

149

C₂×C₈

#11719936

yes

—

150

C₂×C₈

C₄⋊#6157

yes

characteristic

151

C₂×C₈

C₄⋊#6157

yes

characteristic

152

C₂×C₈

#11719940

yes

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

106

—

159

C₄²

#11719936

yes

—

160

C₂³

#13133296

yes

characteristic

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

106

—

166

C₂×C₄

—

212

—

167

C₂×C₄

—

212

—

168

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

169

C₂×C₄

#13133296

yes

characteristic

170

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

171

C₂×C₄

C₂×#11719940

yes

characteristic

172

C₂×C₄

C₂×#11719936

yes

characteristic

173

C₂×C₄

C₂×#11719936

yes

characteristic

174

C₈

—

175

C₈

—

106

—

176

C₈

—

106

—

177

C₈

—

106

—

178

C₈

—

424

—

179

C₈

#13133284

yes

—

180

C₈

#13133284

yes

—

181

C₂²

—

182

C₂²

#16811567

yes

characteristic

183

C₂²

#16811642

yes

characteristic

184

C₂²

#16811644

yes

characteristic

185

C₄

—

186

C₄

—

106

—

187

C₄

—

106

—

188

C₄

—

212

—

189

C₄

C₂×#13133284

yes

characteristic

190

C₄

#16811627

yes

characteristic

191

C₄

#16811642

yes

characteristic

192

C₄

#16811644

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

#20290632

yes

characteristic

196

C₂

#20290634

yes

—

	subgroup	G/H
center	C₂×C₄	#13133296
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8780
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

#13133296

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×Dic₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
25. C₈×Dic₅₃	C₈	semidirect
113. C₅₃	#8780	semidirect

H ≅ G/N

type

25. C₈×Dic₅₃

C₈

semidirect

113. C₅₃

#8780

semidirect

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

subgroup	index
#20290262	2
#20289555	2
#20289539	2
#20274401	2
#8780	53

subgroup

index

Subgroups — #28303521

Conjugacy classes of subgroups

Complements

Maximal subgroups