Group #27282716 — subgroups

Subgroups — #27282716

Order 13568.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20273465	C₂	1	2	yes	maximal
3	#20273007	C₂	1	1	yes	maximal, characteristic
4	#20272565	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810676	C₂²	1	1	yes	characteristic
8	C₂×#13133153	C₄	1	2	yes	—
9	C₄×#11719917	C₂²	1	2	yes	—
10	#16810388	C₄	1	1	yes	characteristic
11	#16810388	C₄	1	1	yes	characteristic
12	#16810375	—	2	2	no	—
13	#16810373	—	2	2	no	—
14	#16810345	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	2	no	—
18	C₂×#11719917	C₂×C₄	1	2	yes	—
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	#13133153	—	2	4	no	—
21	C₄⋊#88322	—	2	1	no	—
22	C₄⋊#88322	D₄	1	1	yes	characteristic
23	C₄⋊#88322	D₄	1	1	yes	characteristic
24	C₂²⋊#88322	—	2	1	no	—
25	C₂²⋊#88322	Q₈	1	2	yes	—
26	#13133128	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	1	no	—
29	C₄×#88322	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	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	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	8	2	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	1	yes	characteristic
51	#11719917	—	2	2	no	—
52	#11719917	—	2	2	no	—
53	#11719917	—	2	2	no	—
54	#11719917	M₁₆	1	4	yes	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₅₃	#131	1	2	yes	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	—	8	2	no	—
75	#88322	—	2	4	no	—
76	#88322	—	4	2	no	—
77	#88322	—	4	2	no	—
78	#88322	—	4	1	no	—
79	#88322	—	4	1	no	—
80	#9610	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	#389	1	1	yes	characteristic
84	C₂²×C₅₃	#446	1	2	yes	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	#423	1	1	yes	characteristic
89	C₄×C₅₃	#424	1	1	yes	characteristic
90	C₄×C₅₃	#435	1	2	yes	—
91	C₄×C₅₃	#472	1	1	yes	characteristic
92	C₄×C₅₃	#473	1	1	yes	characteristic
93	C₄×#108	—	53	1	no	—
94	#1565	—	53	2	no	—
95	C₂×#335	D₅₃	1	1	yes	characteristic
96	#1452	—	53	1	no	—
97	#1283	—	53	2	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	#1940	1	1	yes	characteristic
100	C₂×C₅₃	#1952	1	1	yes	characteristic
101	C₂×C₅₃	#1953	1	1	yes	characteristic
102	#431	—	53	1	no	—
103	#431	—	53	1	no	—
104	C₂×#108	—	53	2	no	—
105	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
106	C₄×M₁₆	—	53	2	no	—
107	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	#341	—	53	1	no	—
109	#341	—	53	1	no	—
110	#335	—	2	2	no	—
111	#335	—	106	2	no	—
112	#334	—	106	2	no	—
113	#322	—	106	2	no	—
114	C₅₃	#9610	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	53	2	no	—
116	C₂×M₁₆	—	53	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	#108	—	106	4	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	53	1	no	—
125	C₄⋊C₈	—	53	1	no	—
126	C₄⋊C₈	—	106	1	no	—
127	C₂²⋊C₈	—	53	2	no	—
128	C₂²⋊C₈	—	106	1	no	—
129	#97	—	106	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	1	no	—
132	C₄×C₈	—	106	1	no	—
133	C₄×C₈	Dic₁₀₆	1	2	yes	—
134	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
135	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	#11719947	1	1	yes	characteristic
137	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
138	M₁₆	—	53	4	no	—
139	M₁₆	—	106	2	no	—
140	M₁₆	—	106	2	no	—
141	M₁₆	—	106	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	—	106	1	no	—
149	C₂×C₈	—	106	1	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	#11719919	1	2	yes	—
152	C₂×C₈	#11719919	1	2	yes	—
153	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
154	C₄²	D₄×D₅₃	1	1	yes	characteristic
155	C₄²	#11719949	1	1	yes	characteristic
156	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂³	#13133182	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
162	C₂×C₄	#13133179	1	1	yes	characteristic
163	C₂×C₄	#13133188	1	1	yes	characteristic
164	C₂×C₄	#13133193	1	1	yes	characteristic
165	C₂×C₄	#13133208	1	1	yes	characteristic
166	C₂×C₄	#13133216	1	1	yes	characteristic
167	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
168	C₂×C₄	#13133223	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	8	2	no	—
172	C₈	—	106	4	no	—
173	C₈	—	212	2	no	—
174	C₈	—	212	1	no	—
175	C₈	—	212	1	no	—
176	C₈	—	212	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#16810519	1	1	yes	characteristic
180	C₂²	#16810752	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#16810646	1	1	yes	characteristic
185	C₄	#16810649	1	1	yes	characteristic
186	C₄	#16810693	1	2	yes	—
187	C₄	#16810890	1	1	yes	characteristic
188	C₄	#16810895	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#20275436	1	1	yes	characteristic
191	C₂	#20275498	1	1	yes	characteristic
192	C₂	#20275503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133153

C₂

yes

maximal, characteristic

#20273465

C₂

yes

maximal

#20273007

C₂

yes

maximal, characteristic

#20272565

C₂

yes

maximal

C₂×C₅₃×#335

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810676

C₂²

yes

characteristic

C₂×#13133153

C₄

yes

—

C₄×#11719917

C₂²

yes

—

#16810388

C₄

yes

characteristic

#16810388

C₄

yes

—

—

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₂×#11719917

C₂×C₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

#13133153

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

characteristic

C₄⋊#88322

D₄

yes

characteristic

C₂²⋊#88322

—

C₂²⋊#88322

Q₈

yes

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

—

—

—

M₁₆

yes

—

—

—

—

—

—

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

characteristic

C₂²×C₅₃

#446

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

#1452

—

#1283

—

C₂×C₅₃

—

C₂×C₅₃

#1940

yes

characteristic

100

C₂×C₅₃

#1952

yes

characteristic

101

C₂×C₅₃

#1953

yes

characteristic

102

#431

—

103

#431

—

104

C₂×#108

—

105

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

106

C₄×M₁₆

—

107

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

108

#341

—

109

#341

—

110

#335

—

111

#335

—

106

—

112

#334

—

106

—

113

#322

—

106

—

114

C₅₃

#9610

yes

characteristic, complemented, Sylow

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₅₃

yes

characteristic

119

C₂²×C₈

C₄×D₅₃

yes

characteristic

120

C₂×C₄²

C₂²×D₅₃

yes

characteristic

121

#108

—

106

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

106

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

106

—

129

#97

—

106

—

130

C₄×C₈

—

106

—

131

C₄×C₈

—

106

—

132

C₄×C₈

—

106

—

133

C₄×C₈

Dic₁₀₆

yes

—

134

C₄×C₈

C₂²⋊D₅₃

yes

—

135

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂²×C₄

#11719947

yes

characteristic

137

C₂²×C₄

Q₈×D₅₃

yes

characteristic

138

M₁₆

—

139

M₁₆

—

106

—

140

M₁₆

—

106

—

141

M₁₆

—

106

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

106

—

148

C₂×C₈

—

106

—

149

C₂×C₈

—

106

—

150

C₂×C₈

—

106

—

151

C₂×C₈

#11719919

yes

—

152

C₂×C₈

#11719919

yes

—

153

C₄²

C₂×Dic₁₀₆

yes

characteristic

154

C₄²

D₄×D₅₃

yes

characteristic

155

C₄²

#11719949

yes

characteristic

156

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

157

C₂³

#13133182

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×#11719919

yes

characteristic

162

C₂×C₄

#13133179

yes

characteristic

163

C₂×C₄

#13133188

yes

characteristic

164

C₂×C₄

#13133193

yes

characteristic

165

C₂×C₄

#13133208

yes

characteristic

166

C₂×C₄

#13133216

yes

characteristic

167

C₂×C₄

M₁₆×D₅₃

yes

—

168

C₂×C₄

#13133223

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

106

—

173

C₈

—

212

—

174

C₈

—

212

—

175

C₈

—

212

—

176

C₈

—

212

—

177

C₂²

—

178

C₂²

—

179

C₂²

#16810519

yes

characteristic

180

C₂²

#16810752

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#16810646

yes

characteristic

185

C₄

#16810649

yes

characteristic

186

C₄

#16810693

yes

—

187

C₄

#16810890

yes

characteristic

188

C₄

#16810895

yes

characteristic

189

C₂

—

190

C₂

#20275436

yes

characteristic

191

C₂

#20275498

yes

characteristic

192

C₂

#20275503

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9610
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

subgroup

G/H

center

C₂×C₄

#13133188

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#335

C₂

socle

C₂²×C₅₃

#389

prime radical

C₂³×C₅₃

C₄×Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

N	H ≅ G/N	type
114. C₅₃	#9610	semidirect

H ≅ G/N

type

114. C₅₃

#9610

semidirect

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

subgroup	index
C₄×#13133153	2
#20273465	2
#20273007	2
#20272565	2
C₂×C₅₃×#335	2
#9610	53

subgroup

index

C₂×C₅₃×#335

Subgroups — #27282716

Conjugacy classes of subgroups

Complements

Maximal subgroups