Group #28298309 — subgroups

Subgroups — #28298309

Order 13568.

242 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 98 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×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#335	C₂	1	1	yes	maximal, characteristic
3	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
5	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
6	#16811509	—	2	1	no	—
7	C₄×#11719910	—	2	1	no	—
8	#16811487	—	2	1	no	—
9	#16811486	C₄	1	2	yes	—
10	C₄⋊C₈×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic
13	#16810375	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic
15	C₂×D₅₃⋊C₈	—	4	1	no	—
16	D₅₃⋊C₁₆	—	2	2	no	—
17	D₅₃⋊C₁₆	C₈	1	4	yes	complemented
18	C₂×#11719910	—	2	2	no	—
19	#13133284	—	4	1	no	—
20	C₄⋊#88324	—	2	2	no	—
21	C₂×C₈×D₅₃	—	4	1	no	—
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
24	C₄²×D₅₃	D₄	1	1	yes	characteristic
25	C₄⋊#88322	—	2	1	no	—
26	#13133146	—	4	1	no	—
27	C₄×#88322	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
34	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×#88324	—	4	2	no	—
36	D₅₃⋊C₈	—	8	1	no	—
37	D₅₃⋊C₈	—	8	1	no	—
38	C₂×C₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×#88322	—	4	1	no	—
46	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₅₃	—	2	2	no	—
49	C₈×D₅₃	—	8	1	no	—
50	C₈×D₅₃	—	8	1	no	—
51	C₈×D₅₃	C₂×C₈	1	2	yes	—
52	C₈×D₅₃	M₁₆	1	2	yes	—
53	#11719910	—	2	4	no	—
54	C₂²×D₅₃	#107	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
58	C₂×Dic₅₃	#102	1	1	yes	characteristic
59	C₂×Dic₅₃	#103	1	1	yes	characteristic
60	C₂×Dic₅₃	#106	1	1	yes	characteristic
61	C₄×D₅₃	—	2	2	no	—
62	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
63	C₄×D₅₃	#108	1	2	yes	—
64	C₄×D₅₃	D₁₆	1	1	yes	characteristic
65	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
66	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
67	C₄×D₅₃	Q₃₂	1	1	yes	characteristic
68	#88324	—	8	2	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	8	1	no	—
71	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
72	C₈×C₅₃	#97	1	1	yes	characteristic
73	#88322	—	2	1	no	—
74	#88322	—	8	1	no	—
75	#88322	C₂²⋊C₈	1	2	yes	—
76	#7154	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#340	1	1	yes	characteristic
78	C₂×D₅₃	#335	1	2	yes	—
79	C₂×D₅₃	#366	1	1	yes	characteristic
80	C₂×D₅₃	#367	1	1	yes	characteristic
81	C₂×D₅₃	#368	1	2	yes	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	#336	1	1	yes	characteristic
84	C₄×C₅₃	#341	1	1	yes	characteristic
85	C₄×C₅₃	#357	1	1	yes	characteristic
86	C₄×C₅₃	#358	1	1	yes	characteristic
87	Dic₅₃	—	2	1	no	—
88	Dic₅₃	#334	1	2	yes	—
89	Dic₅₃	#357	1	1	yes	characteristic
90	Dic₅₃	#358	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	53	1	no	—
92	C₂×#335	—	53	1	no	—
93	C₂×#335	—	53	1	no	—
94	C₂×C₅₃	#1277	1	1	yes	characteristic
95	C₂×C₅₃	#1380	1	1	yes	characteristic
96	C₂×C₅₃	#1381	1	1	yes	characteristic
97	D₅₃	#1371	1	2	yes	—
98	D₅₃	#1372	1	2	yes	—
99	C₂²×C₁₆	—	53	2	no	—
100	C₂×C₄⋊C₈	—	106	1	no	—
101	C₂×C₄⋊C₈	—	106	1	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₄×C₁₆	—	106	1	no	—
104	C₄×C₁₆	—	106	1	no	—
105	#335	—	53	2	no	—
106	#335	—	53	1	no	—
107	#335	—	106	1	no	—
108	#335	—	106	1	no	—
109	#335	#6157	1	1	yes	characteristic
110	C₅₃	#7154	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	—	53	1	no	—
113	C₂²×C₈	—	212	1	no	—
114	C₂²×C₈	—	212	1	no	—
115	C₂×C₄²	—	53	1	no	—
116	C₂×C₁₆	—	53	4	no	—
117	C₂×C₁₆	—	106	2	no	—
118	C₂×C₁₆	—	106	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	—	106	2	no	—
121	C₄⋊C₈	—	106	1	no	—
122	C₄⋊C₈	—	212	1	no	—
123	C₄⋊C₈	—	212	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₄×C₈	C₂×#6157	1	1	yes	characteristic
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	53	1	no	—
130	C₂²×C₄	—	53	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	—	424	1	no	—
138	C₂×C₈	—	424	1	no	—
139	C₂×C₈	—	424	1	no	—
140	C₂×C₈	—	424	1	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	—	212	1	no	—
143	C₂×C₈	C₄×#6157	1	1	yes	characteristic
144	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
145	C₄²	—	53	1	no	—
146	C₄²	—	53	1	no	—
147	C₄²	—	53	1	no	—
148	C₄²	C₂²⋊#6157	1	1	yes	characteristic
149	C₁₆	—	106	4	no	—
150	C₂³	—	53	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	53	1	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	53	2	no	—
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#13133279	1	1	yes	characteristic
163	C₂×C₄	#13133280	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	8	1	no	—
166	C₈	—	53	2	no	—
167	C₈	—	106	1	no	—
168	C₈	—	424	2	no	—
169	C₈	—	424	1	no	—
170	C₈	C₈×#6157	1	1	yes	characteristic
171	C₈	#13133271	1	1	yes	characteristic
172	C₂²	—	53	2	no	—
173	C₂²	—	53	1	no	—
174	C₂²	—	53	1	no	—
175	C₂²	—	53	2	no	—
176	C₂²	#16811496	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	53	2	no	—
179	C₄	—	53	1	no	—
180	C₄	—	53	1	no	—
181	C₄	—	106	1	no	—
182	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
183	C₄	#16811498	1	1	yes	characteristic
184	C₄	#16811535	1	1	yes	characteristic
185	C₄	#16811536	1	1	yes	characteristic
186	C₂	—	53	2	no	—
187	C₂	—	53	2	no	—
188	C₂	#20289544	1	1	yes	characteristic
189	C₂	#20289772	1	1	yes	characteristic
190	C₂	#20289774	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

D₅₃⋊#335

C₂

yes

maximal, characteristic

#335×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

D₅₃⋊(C₄⋊C₈)

—

—

—

—

C₄

yes

—

C₄⋊C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810375

—

C₅₃×#335

C₄

yes

characteristic

C₂×D₅₃⋊C₈

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

C₈

yes

—

—

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

yes

characteristic

C₂×#88324

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#11719910

—

C₂²×D₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₄×D₅₃

—

C₄×D₅₃

C₄⋊C₈

yes

—

C₄×D₅₃

#108

yes

—

C₄×D₅₃

D₁₆

yes

characteristic

C₄×D₅₃

SD₃₂

yes

characteristic

C₄×D₅₃

SD₃₂

yes

characteristic

C₄×D₅₃

Q₃₂

yes

characteristic

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

—

—

C₂²⋊C₈

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

yes

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#334

yes

—

Dic₅₃

yes

Dic₅₃

yes

C₂×C₄×C₁₆

—

C₂×#335

—

C₂×#335

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

D₅₃

#1372

yes

—

C₂²×C₁₆

—

100

C₂×C₄⋊C₈

—

106

—

101

C₂×C₄⋊C₈

—

106

—

102

C₂×C₄×C₈

—

103

C₄×C₁₆

—

106

—

104

C₄×C₁₆

—

106

—

105

#335

—

106

#335

—

107

#335

—

106

—

108

#335

—

106

—

109

#335

#6157

yes

characteristic

110

C₅₃

#7154

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

212

—

114

C₂²×C₈

—

212

—

115

C₂×C₄²

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

106

—

118

C₂×C₁₆

—

106

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₄⋊C₈

—

106

—

122

C₄⋊C₈

—

212

—

123

C₄⋊C₈

—

212

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

C₂×#6157

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

106

—

137

C₂×C₈

—

424

—

138

C₂×C₈

—

424

—

139

C₂×C₈

—

424

—

140

C₂×C₈

—

424

—

141

C₂×C₈

—

212

—

142

C₂×C₈

—

212

—

143

C₂×C₈

C₄×#6157

yes

characteristic

144

C₂×C₈

C₄⋊#6157

yes

characteristic

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²⋊#6157

yes

characteristic

149

C₁₆

—

106

—

150

C₂³

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

106

—

161

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

162

C₂×C₄

#13133279

yes

characteristic

163

C₂×C₄

#13133280

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

106

—

168

C₈

—

424

—

169

C₈

—

424

—

170

C₈

C₈×#6157

yes

characteristic

171

C₈

#13133271

yes

characteristic

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16811496

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

106

—

182

C₄

(C₂×#6157)⋊C₈

yes

characteristic

183

C₄

#16811498

yes

characteristic

184

C₄

#16811535

yes

characteristic

185

C₄

#16811536

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#20289544

yes

characteristic

189

C₂

#20289772

yes

characteristic

190

C₂

#20289774

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#107
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7154
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#13133279

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
17. D₅₃⋊C₁₆	C₈	semidirect
110. C₅₃	#7154	semidirect

H ≅ G/N

type

17. D₅₃⋊C₁₆

C₈

semidirect

110. C₅₃

#7154

semidirect

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

subgroup	index
C₄×D₅₃⋊C₁₆	2
D₅₃⋊#335	2
#335×D₅₃	2
#7154	53

subgroup

index

C₄×D₅₃⋊C₁₆

D₅₃⋊#335

#335×D₅₃

#7154

Subgroups — #28298309

Conjugacy classes of subgroups

Complements

Maximal subgroups