Group #28298604 — subgroups

Subgroups — #28298604

Order 13568.

198 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 68 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	#20290275	C₂	1	1	yes	maximal, characteristic
2	#20290269	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810441	C₂	1	1	yes	maximal, characteristic
4	#16811644	—	2	1	no	—
5	#16811626	—	2	1	no	—
6	#16811487	C₄	1	2	yes	—
7	#16811485	C₄	1	2	yes	—
8	#16811484	—	2	1	no	—
9	#16811484	—	2	1	no	—
10	C₂²×#11719908	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810441	—	2	1	no	—
13	#16810441	—	2	1	no	—
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719940	—	4	1	no	—
16	#13133284	—	4	1	no	—
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₂²⋊#88324	—	4	1	no	—
20	#13133272	—	4	1	no	—
21	C₂²×#88322	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×#11719908	—	2	1	no	—
24	C₂×#11719908	—	4	1	no	—
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	1	no	—
31	C₂×C₁₆×C₅₃	—	4	1	no	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
34	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
35	#11719940	—	8	1	no	—
36	#11719940	—	8	1	no	—
37	C₂×#88324	—	4	2	no	—
38	C₂×#88324	—	4	2	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	D₈	1	1	yes	characteristic
46	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
47	C₂×#88322	—	4	1	no	—
48	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆×C₅₃	—	4	1	no	—
51	#11719908	—	4	1	no	—
52	C₂³×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	#102	1	1	yes	characteristic
61	C₂×Dic₅₃	#102	1	1	yes	characteristic
62	C₂×Dic₅₃	#103	1	1	yes	characteristic
63	C₂×Dic₅₃	#103	1	1	yes	characteristic
64	#88324	—	8	2	no	—
65	#88324	—	8	2	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	#100	1	1	yes	characteristic
68	C₈×C₅₃	#101	1	1	yes	characteristic
69	#88322	—	2	1	no	—
70	#88322	—	2	1	no	—
71	#7243	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	#340	1	1	yes	characteristic
75	C₂²×C₅₃	#365	1	1	yes	characteristic
76	C₂²×C₅₃	#365	1	1	yes	characteristic
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	#338	1	1	yes	characteristic
79	C₄×C₅₃	#343	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	2	1	no	—
83	Dic₅₃	#359	1	2	yes	—
84	C₂×C₄⋊C₁₆	—	53	1	no	—
85	#1568	—	53	1	no	—
86	#1566	—	53	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1292	1	1	yes	characteristic
90	C₂×C₅₃	#1382	1	1	yes	characteristic
91	C₂×C₅₃	#1383	1	1	yes	characteristic
92	C₂²×C₁₆	—	53	1	no	—
93	C₂²×C₁₆	#6157	1	1	yes	characteristic
94	#432	—	106	1	no	—
95	#423	—	106	1	no	—
96	C₂×C₄×C₈	—	53	1	no	—
97	C₄⋊C₁₆	—	106	1	no	—
98	C₄⋊C₁₆	—	106	1	no	—
99	#335	—	53	2	no	—
100	#334	—	53	2	no	—
101	#334	—	106	1	no	—
102	#334	—	106	1	no	—
103	C₅₃	#7243	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	212	1	no	—
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
107	C₂×C₄²	—	53	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	C₂×C₁₆	—	106	1	no	—
111	C₂×C₁₆	—	212	1	no	—
112	C₄⋊C₈	—	106	2	no	—
113	C₄⋊C₈	—	106	2	no	—
114	C₄⋊C₈	—	212	1	no	—
115	C₂²⋊C₈	—	212	1	no	—
116	#97	—	212	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
124	M₁₆	—	424	1	no	—
125	M₁₆	—	424	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	212	1	no	—
131	C₂×C₈	—	212	2	no	—
132	C₂×C₈	—	212	2	no	—
133	C₂×C₈	C₄×#6157	1	1	yes	characteristic
134	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₁₆	—	4	1	no	—
140	C₁₆	—	212	1	no	—
141	C₂³	#13133285	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	#13133286	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	106	1	no	—
155	C₈	—	106	1	no	—
156	C₈	—	424	2	no	—
157	C₈	—	424	2	no	—
158	C₈	#13133277	1	1	yes	characteristic
159	C₈	#13133278	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	#16811497	1	1	yes	characteristic
163	C₂²	#16811547	1	1	yes	characteristic
164	C₂²	#16811548	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	53	2	no	—
167	C₄	—	106	1	no	—
168	C₄	—	106	1	no	—
169	C₄	—	106	1	no	—
170	C₄	#16811492	1	1	yes	characteristic
171	C₄	#16811503	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#20289585	1	1	yes	characteristic
175	C₂	#20289777	1	1	yes	characteristic
176	C₂	#20289780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290275

C₂

yes

maximal, characteristic

#20290269

C₂

yes

maximal, characteristic

C₂×#16810441

C₂

yes

maximal, characteristic

#16811644

—

#16811626

—

#16811487

C₄

yes

—

#16811485

C₄

yes

—

#16811484

—

#16811484

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810441

—

#16810441

—

C₂²×C₁₆×C₅₃

C₄

yes

—

—

—

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

—

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#359

yes

—

C₂×C₄⋊C₁₆

—

#1568

—

#1566

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

#432

—

106

—

#423

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₄⋊C₁₆

—

106

—

#335

—

100

#334

—

101

#334

—

106

—

102

#334

—

106

—

103

C₅₃

#7243

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

—

212

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#6157

yes

characteristic

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

106

—

111

C₂×C₁₆

—

212

—

112

C₄⋊C₈

—

106

—

113

C₄⋊C₈

—

106

—

114

C₄⋊C₈

—

212

—

115

C₂²⋊C₈

—

212

—

116

#97

—

212

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₄⋊#6157

yes

characteristic

124

M₁₆

—

424

—

125

M₁₆

—

424

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

212

—

131

C₂×C₈

—

212

—

132

C₂×C₈

—

212

—

133

C₂×C₈

C₄×#6157

yes

characteristic

134

C₂×C₈

C₂²⋊#6157

yes

characteristic

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₁₆

—

140

C₁₆

—

212

—

141

C₂³

#13133285

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

152

C₂×C₄

#13133286

yes

characteristic

153

C₈

—

154

C₈

—

106

—

155

C₈

—

106

—

156

C₈

—

424

—

157

C₈

—

424

—

158

C₈

#13133277

yes

characteristic

159

C₈

#13133278

yes

characteristic

160

C₂²

—

161

C₂²

—

162

C₂²

#16811497

yes

characteristic

163

C₂²

#16811547

yes

characteristic

164

C₂²

#16811548

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

106

—

168

C₄

—

106

—

169

C₄

—

106

—

170

C₄

#16811492

yes

characteristic

171

C₄

#16811503

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#20289585

yes

characteristic

175

C₂

#20289777

yes

characteristic

176

C₂

#20289780

yes

characteristic

	subgroup	G/H
center	C₂²	#16811497
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7243
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#16811497

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₈×Dic₅₃

C₂²

C₂²×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
103. C₅₃	#7243	semidirect

H ≅ G/N

type

103. C₅₃

#7243

semidirect

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

subgroup	index
#20290275	2
#20290269	2
C₂×#16810441	2
#7243	53

subgroup

index

Subgroups — #28298604

Conjugacy classes of subgroups

Complements

Maximal subgroups