Group #28311390 — subgroups

Subgroups — #28311390

Order 13568.

342 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 116 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	#20291049	C₂	1	2	yes	maximal
2	#20289760	C₂	1	2	yes	maximal
3	#20289759	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₂²	1	4	yes	—
6	#16811544	—	2	4	no	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	#13133291	—	2	4	no	—
15	#13133291	—	2	4	no	—
16	C₂×#11719910	—	2	4	no	—
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#88322	—	2	2	no	—
23	C₈×Dic₅₃	—	2	2	no	—
24	C₈×Dic₅₃	—	2	2	no	—
25	C₈×Dic₅₃	C₈	1	2	yes	—
26	C₈×Dic₅₃	C₈	1	2	yes	—
27	C₈×Dic₅₃	D₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	1	no	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	—	2	1	no	—
36	C₂²×Dic₅₃	M₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₄²×C₅₃	M₁₆	1	2	yes	—
45	C₄×Dic₅₃	—	2	8	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
48	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
49	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₅₃	M₁₆	1	2	yes	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	C₂×C₈	1	2	yes	—
54	C₂×#88322	C₂×D₄	1	1	yes	characteristic
55	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
56	C₂×#88322	#41	1	2	yes	—
57	C₂×#88322	#41	1	2	yes	—
58	#11719910	—	4	4	no	—
59	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	4	no	—
71	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
72	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	—	2	2	no	—
75	#88322	—	2	2	no	—
76	#88322	#124	1	2	yes	—
77	#88322	#125	1	2	yes	—
78	#12883	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₂²×C₅₃	#432	1	1	yes	characteristic
82	C₂²×C₅₃	#432	1	1	yes	characteristic
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	#382	1	1	yes	characteristic
85	C₄×C₅₃	#423	1	1	yes	characteristic
86	C₄×C₅₃	#431	1	1	yes	characteristic
87	C₄×C₅₃	#431	1	1	yes	characteristic
88	Dic₅₃	—	2	4	no	—
89	Dic₅₃	—	2	2	no	—
90	Dic₅₃	#364	1	4	yes	—
91	#2150	—	53	2	no	—
92	C₄²×C₈	—	53	1	no	—
93	#1376	—	53	2	no	—
94	#1376	—	53	2	no	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	#1831	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#364	1	2	yes	—
99	C₂×M₃₂	—	53	4	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	C₂×C₄×C₈	—	53	1	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	#6157	1	1	yes	characteristic
106	C₄³	—	53	1	no	—
107	C₄⋊C₁₆	—	106	4	no	—
108	C₅₃	#12883	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	—	53	1	no	—
111	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
112	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
113	C₂×C₄²	—	53	1	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	—	106	2	no	—
116	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
117	M₃₂	—	106	4	no	—
118	M₃₂	—	106	4	no	—
119	C₂×C₁₆	—	106	4	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	2	no	—
123	C₄×C₈	—	53	2	no	—
124	C₄×C₈	—	53	2	no	—
125	C₄×C₈	—	106	2	no	—
126	C₄×C₈	—	106	2	no	—
127	C₄×C₈	—	106	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	—	106	1	no	—
131	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	53	2	no	—
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	106	1	no	—
142	C₂×C₈	—	106	1	no	—
143	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
144	C₂×C₈	C₄⋊#6157	1	2	yes	—
145	C₄²	—	2	1	no	—
146	C₄²	—	53	2	no	—
147	C₄²	—	53	1	no	—
148	C₄²	—	53	1	no	—
149	C₄²	—	53	2	no	—
150	C₄²	—	106	8	no	—
151	C₄²	—	106	1	no	—
152	C₄²	#11719940	1	2	yes	—
153	C₁₆	—	212	4	no	—
154	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	53	2	no	—
159	C₂×C₄	—	106	4	no	—
160	C₂×C₄	—	106	4	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
163	C₂×C₄	#13133296	1	1	yes	characteristic
164	C₂×C₄	#13133296	1	1	yes	characteristic
165	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
167	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	53	2	no	—
171	C₈	—	53	2	no	—
172	C₈	—	106	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
175	C₂²	#16811643	1	1	yes	characteristic
176	C₂²	#16811646	1	1	yes	characteristic
177	C₄	—	2	4	no	—
178	C₄	—	53	4	no	—
179	C₄	—	106	4	no	—
180	C₄	—	106	2	no	—
181	C₄	#16811567	1	1	yes	characteristic
182	C₄	#16811626	1	1	yes	characteristic
183	C₄	#16811640	1	1	yes	characteristic
184	C₄	#16811641	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#20290631	1	1	yes	characteristic
188	C₂	D₅₃⋊#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291049

C₂

yes

maximal

#20289760

C₂

yes

maximal

#20289759

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#13133291

C₂²

yes

—

#16811544

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₄²×Dic₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

—

—

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×#88322

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

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

M₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂×D₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₂×#88322

#41

yes

—

C₂×#88322

#41

yes

—

#11719910

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

#124

yes

—

#88322

#125

yes

—

#12883

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

#1376

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1831

yes

characteristic

C₂×C₅₃

C₂×#364

yes

—

C₂×M₃₂

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

—

105

C₂×C₄×C₈

#6157

yes

characteristic

106

C₄³

—

107

C₄⋊C₁₆

—

106

—

108

C₅₃

#12883

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×#6157

yes

characteristic

112

C₂²×C₈

C₂×#6157

yes

characteristic

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

106

—

116

C₂×C₄²

C₂×#6157

yes

characteristic

117

M₃₂

—

106

—

118

M₃₂

—

106

—

119

C₂×C₁₆

—

106

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

106

—

126

C₄×C₈

—

106

—

127

C₄×C₈

—

106

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

106

—

131

C₂²×C₄

C₂²×#6157

yes

characteristic

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

D₅₃⋊C₈

yes

—

144

C₂×C₈

C₄⋊#6157

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

106

—

151

C₄²

—

106

—

152

C₄²

#11719940

yes

—

153

C₁₆

—

212

—

154

C₂³

D₅₃⋊M₁₆

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

163

C₂×C₄

#13133296

yes

characteristic

164

C₂×C₄

#13133296

yes

characteristic

165

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

166

C₂×C₄

C₂×#11719940

yes

characteristic

167

C₂×C₄

D₅₃⋊M₁₆

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

106

—

173

C₂²

—

174

C₂²

D₅₃⋊(C₄⋊C₈)

yes

characteristic

175

C₂²

#16811643

yes

characteristic

176

C₂²

#16811646

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

106

—

180

C₄

—

106

—

181

C₄

#16811567

yes

characteristic

182

C₄

#16811626

yes

characteristic

183

C₄

#16811640

yes

characteristic

184

C₄

#16811641

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#20290631

yes

characteristic

188

C₂

D₅₃⋊#364

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12883
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×#88322

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
108. C₅₃	#12883	semidirect

H ≅ G/N

type

108. C₅₃

#12883

semidirect

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

subgroup	index
#20291049	2
#20289760	2
#20289759	2
C₄×C₈×Dic₅₃	2
#12883	53

subgroup

index

C₄×C₈×Dic₅₃

Subgroups — #28311390

Conjugacy classes of subgroups

Complements

Maximal subgroups