Group #27300427 — subgroups

Subgroups — #27300427

Order 13568.

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

normal subgroups only

This group has 366 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	#20283326	C₂	1	1	yes	maximal, characteristic
2	#20276329	C₂	1	1	yes	maximal, characteristic
3	#20272831	C₂	1	2	yes	maximal, complemented
4	#20272810	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1812	C₂	1	1	yes	maximal, characteristic
6	#16811390	—	2	1	no	—
7	C₂×#13133157	C₂²	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	#16810672	—	2	1	no	—
10	#16810672	C₄	1	4	yes	complemented
11	C₂×#13133153	—	2	2	no	—
12	C₂×#13133153	C₂²	1	2	yes	—
13	C₂²⋊#11719908	—	2	2	no	—
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#427	C₂²	1	2	yes	—
16	C₅₃×#410	—	2	1	no	—
17	C₅₃×H(3,ℤ/4)	—	2	1	no	—
18	#13133224	—	4	4	no	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	2	no	—
21	#13133157	—	2	2	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂×#11719908	—	2	1	no	—
24	C₂×#11719908	—	2	2	no	—
25	#13133153	—	2	4	no	—
26	#13133153	—	2	2	no	—
27	#13133153	C₂×C₄	1	4	yes	—
28	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
29	#131×C₅₃	—	2	2	no	—
30	#131×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	#117×C₅₃	—	2	2	no	—
36	#107×C₅₃	—	2	2	no	—
37	#106×C₅₃	—	2	2	no	—
38	#99×C₅₃	—	4	2	no	—
39	#41×C₅₃	—	2	2	no	—
40	#41×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
43	C₂×D₄×C₅₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₅₃	—	2	2	no	—
47	M₁₆×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₂²⋊C₄×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	#11719917	—	2	4	no	—
59	#11719917	—	4	4	no	—
60	#11719917	—	4	2	no	—
61	C₂×#88322	—	4	4	no	—
62	C₂×#88322	—	4	2	no	—
63	#11719908	—	4	1	no	—
64	#11719908	—	4	1	no	—
65	C₂³×C₅₃	#103	1	2	yes	—
66	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₅₃	—	2	2	no	—
68	D₄×C₅₃	—	2	4	no	—
69	D₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	#103	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#123	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	#117	1	2	yes	—
80	C₈×C₅₃	#123	1	1	yes	characteristic
81	C₈×C₅₃	#123	1	1	yes	characteristic
82	#88322	—	4	4	no	—
83	#12877	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	—	4	1	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
88	C₂²×C₅₃	#481	1	2	yes	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	16	2	no	—
91	C₄×C₅₃	#381	1	1	yes	characteristic
92	C₄×C₅₃	#418	1	1	yes	characteristic
93	C₄×C₅₃	#484	1	2	yes	—
94	#2916	—	53	1	no	—
95	#2113	—	53	1	no	—
96	#1812	D₅₃	1	1	yes	characteristic
97	#1389	—	53	2	no	—
98	#1382	—	53	2	no	—
99	C₂×C₅₃	—	2	2	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	—	4	2	no	—
102	C₂×C₅₃	#1824	1	1	yes	characteristic
103	#568	—	106	1	no	—
104	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
105	C₂×M₃₂	—	53	1	no	—
106	C₂²×C₁₆	—	53	1	no	—
107	#430	—	53	4	no	—
108	#430	—	106	1	no	—
109	C₂×#108	—	53	2	no	—
110	C₂×#108	—	106	2	no	—
111	#427	C₂×D₅₃	1	2	yes	—
112	#410	—	2	1	no	—
113	C₂²⋊C₁₆	—	106	2	no	—
114	H(3,ℤ/4)	—	2	1	no	—
115	C₅₃	#12877	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	C₂²⋊D₅₃	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	212	4	no	—
120	C₂×M₁₆	—	106	2	no	—
121	C₂×M₁₆	—	106	4	no	—
122	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂²×C₈	D₂₁₂	1	2	yes	—
125	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	#117	—	2	2	no	—
127	M₃₂	—	106	2	no	—
128	C₂×C₁₆	—	106	1	no	—
129	C₂×C₁₆	—	106	1	no	—
130	C₂×C₁₆	—	106	2	no	—
131	#108	—	53	4	no	—
132	#108	—	106	2	no	—
133	#108	—	106	4	no	—
134	#107	—	2	2	no	—
135	#106	—	2	2	no	—
136	#99	—	4	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	#11719925	1	1	yes	characteristic
140	C₂×D₄	#11719923	1	2	yes	—
141	C₂×D₄	#11719925	1	1	yes	characteristic
142	C₂²×C₄	#11719921	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	106	4	no	—
147	M₁₆	—	212	4	no	—
148	M₁₆	—	212	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	212	4	no	—
152	C₂×C₈	—	212	2	no	—
153	C₂×C₈	#11719921	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
156	C₂×C₈	#11719947	1	2	yes	—
157	C₂×C₈	#11719947	1	2	yes	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₁₆	—	212	1	no	—
162	C₁₆	—	212	1	no	—
163	C₂³	#13133140	1	2	yes	—
164	C₂³	C₂×#11719923	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	#13133140	1	2	yes	—
172	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
173	C₂×C₄	#13133198	1	2	yes	—
174	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
175	C₂×C₄	#13133242	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	212	4	no	—
178	C₈	#13133173	1	2	yes	—
179	C₈	#13133201	1	1	yes	characteristic
180	C₈	#13133201	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#13133140	1	1	yes	characteristic
185	C₂²	#16810939	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	16	2	no	—
188	C₄	#16810484	1	1	yes	characteristic
189	C₄	#16810627	1	1	yes	characteristic
190	C₄	#16810956	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#20274803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283326

C₂

yes

maximal, characteristic

#20276329

C₂

yes

maximal, characteristic

#20272831

C₂

yes

maximal, complemented

#20272810

C₂

yes

maximal, complemented

C₅₃×#1812

C₂

yes

maximal, characteristic

#16811390

—

C₂×#13133157

C₂²

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

#16810672

—

#16810672

C₄

yes

complemented

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

C₂²⋊#11719908

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#427

C₂²

yes

—

C₅₃×#410

—

C₅₃×H(3,ℤ/4)

—

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

#107×C₅₃

—

#106×C₅₃

—

#99×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

Q₁₆

yes

characteristic

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂×D₄×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

—

—

—

C₂³×C₅₃

#103

yes

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#103

yes

characteristic

C₂²×C₅₃

#481

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2916

—

#2113

—

#1812

D₅₃

yes

characteristic

#1389

—

#1382

—

C₂×C₅₃

—

100

C₂×C₅₃

—

101

C₂×C₅₃

—

102

C₂×C₅₃

#1824

yes

characteristic

103

#568

—

106

—

104

C₂×#131

C₂×D₅₃

yes

characteristic

105

C₂×M₃₂

—

106

C₂²×C₁₆

—

107

#430

—

108

#430

—

106

—

109

C₂×#108

—

110

C₂×#108

—

106

—

111

#427

C₂×D₅₃

yes

—

112

#410

—

113

C₂²⋊C₁₆

—

106

—

114

H(3,ℤ/4)

—

115

C₅₃

#12877

yes

characteristic, complemented, Sylow

116

C₂×#41

C₂²⋊D₅₃

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

212

—

120

C₂×M₁₆

—

106

—

121

C₂×M₁₆

—

106

—

122

C₂×M₁₆

C₄×D₅₃

yes

—

123

C₂×M₁₆

C₂²⋊D₅₃

yes

characteristic

124

C₂²×C₈

D₂₁₂

yes

—

125

C₂²×C₈

C₂²×D₅₃

yes

characteristic

126

#117

—

127

M₃₂

—

106

—

128

C₂×C₁₆

—

106

—

129

C₂×C₁₆

—

106

—

130

C₂×C₁₆

—

106

—

131

#108

—

132

#108

—

106

—

133

#108

—

106

—

134

#107

—

135

#106

—

136

#99

—

137

#41

—

138

#41

—

139

C₂×Q₈

#11719925

yes

characteristic

140

C₂×D₄

#11719923

yes

—

141

C₂×D₄

#11719925

yes

characteristic

142

C₂²×C₄

#11719921

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

106

—

147

M₁₆

—

212

—

148

M₁₆

—

212

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

212

—

152

C₂×C₈

—

212

—

153

C₂×C₈

#11719921

yes

—

154

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

155

C₂×C₈

C₂×D₂₁₂

yes

characteristic

156

C₂×C₈

#11719947

yes

—

157

C₂×C₈

#11719947

yes

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₁₆

—

212

—

162

C₁₆

—

212

—

163

C₂³

#13133140

yes

—

164

C₂³

C₂×#11719923

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#13133140

yes

—

172

C₂×C₄

C₂×#11719921

yes

characteristic

173

C₂×C₄

#13133198

yes

—

174

C₂×C₄

C₂×#11719925

yes

characteristic

175

C₂×C₄

#13133242

yes

—

176

C₈

—

177

C₈

—

212

—

178

C₈

#13133173

yes

—

179

C₈

#13133201

yes

characteristic

180

C₈

#13133201

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#13133140

yes

characteristic

185

C₂²

#16810939

yes

—

186

C₄

—

187

C₄

—

188

C₄

#16810484

yes

characteristic

189

C₄

#16810627

yes

characteristic

190

C₄

#16810956

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#20274803

yes

characteristic

	subgroup	G/H
center	C₄	#16810627
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1812	C₂
socle	C₂×C₅₃	#1824
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1812	D₅₃
hypocenter	C₅₃	#12877
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#16810627

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1812

C₂

socle

C₂×C₅₃

#1824

prime radical

C₂×#41×C₅₃

D₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
3. #20272831	C₂	semidirect
4. #20272810	C₂	semidirect
10. #16810672	C₄	semidirect
115. C₅₃	#12877	semidirect

H ≅ G/N

type

3. #20272831

C₂

semidirect

4. #20272810

C₂

semidirect

10. #16810672

C₄

semidirect

115. C₅₃

#12877

semidirect

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

subgroup	index
#20283326	2
#20276329	2
#20272831	2
#20272810	2
C₅₃×#1812	2
#12877	53

subgroup

index

C₅₃×#1812

Subgroups — #27300427

Conjugacy classes of subgroups

Complements

Maximal subgroups