Group #27300275 — subgroups

Subgroups — #27300275

Order 13568.

374 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810452	C₂	1	2	yes	maximal, complemented
3	#20274753	C₂	1	1	yes	maximal, characteristic
4	#20272808	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	4	no	—
7	C₂×#13133157	C₄	1	4	yes	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810587	—	2	1	no	—
12	#16810452	—	2	4	no	—
13	#16810388	—	2	1	no	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#407	—	2	1	no	—
17	C₅₃×#341	—	2	1	no	—
18	C₂×#11719917	—	2	2	no	—
19	#13133157	—	2	8	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	C₂×#11719908	D₄	1	4	yes	—
24	#13133153	—	2	2	no	—
25	#13133153	—	2	2	no	—
26	C₂²⋊#88322	—	4	2	no	—
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	#108×C₅₃	—	2	2	no	—
36	#108×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	4	2	no	—
38	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	D₈	1	2	yes	—
43	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₅₃	—	4	2	no	—
46	M₁₆×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	2	yes	—
57	#11719917	—	4	2	no	—
58	#11719917	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	#11719908	—	2	2	no	—
61	#11719908	C₂²⋊C₄	1	4	yes	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	#107	1	2	yes	—
65	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	#107	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#121	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#135	1	2	yes	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	8	2	no	—
77	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₅₃	#117	1	1	yes	characteristic
79	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
80	#88322	—	8	2	no	—
81	#12838	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₂²×C₅₃	#467	1	2	yes	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	#380	1	1	yes	characteristic
90	C₄×C₅₃	#427	1	1	yes	characteristic
91	C₄×C₅₃	#466	1	2	yes	—
92	C₂²×M₃₂	—	53	1	no	—
93	C₂×#368	—	53	2	no	—
94	#1810	—	53	1	no	—
95	#1810	D₅₃	1	1	yes	characteristic
96	#1381	—	53	2	no	—
97	C₂×C₅₃	—	2	2	no	—
98	C₂×C₅₃	—	4	2	no	—
99	C₂×C₅₃	#1808	1	1	yes	characteristic
100	C₂×C₅₃	#2161	1	2	yes	—
101	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×M₃₂	—	53	4	no	—
103	C₂×M₃₂	—	106	4	no	—
104	C₂²×C₁₆	—	53	1	no	—
105	C₂²×C₁₆	—	53	1	no	—
106	C₂×#108	—	53	2	no	—
107	C₂×#108	C₂×D₅₃	1	2	yes	—
108	#407	—	2	1	no	—
109	#407	—	106	1	no	—
110	#368	—	106	4	no	—
111	#341	—	2	1	no	—
112	#341	—	106	1	no	—
113	C₅₃	#12838	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	—	106	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	Dic₁₀₆	1	2	yes	—
120	C₂²×C₈	C₄×D₅₃	1	2	yes	—
121	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	M₃₂	—	106	8	no	—
124	C₂×C₁₆	—	53	2	no	—
125	C₂×C₁₆	—	53	4	no	—
126	C₂×C₁₆	—	53	2	no	—
127	C₂×C₁₆	—	106	2	no	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#108	—	106	2	no	—
131	#108	—	106	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	—	212	2	no	—
134	C₂⁴	#11719925	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	#11719919	1	2	yes	—
138	C₂²×C₄	#11719922	1	2	yes	—
139	C₂²×C₄	#11719925	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	212	2	no	—
144	M₁₆	—	212	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	212	2	no	—
151	C₂×C₈	#11719919	1	2	yes	—
152	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
153	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	D₄×D₅₃	1	2	yes	—
155	C₂×C₈	#11719947	1	2	yes	—
156	C₁₆	—	53	4	no	—
157	C₁₆	—	106	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	4	2	no	—
160	C₂³	#13133150	1	2	yes	—
161	C₂³	C₂×#11719925	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	#13133150	1	2	yes	—
167	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
168	C₂×C₄	#13133189	1	2	yes	—
169	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
170	C₂×C₄	#13133237	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	424	2	no	—
174	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
175	C₈	#13133173	1	1	yes	characteristic
176	C₈	#13133194	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×#13133150	1	1	yes	characteristic
182	C₂²	#16810862	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	#16810481	1	1	yes	characteristic
185	C₄	#16810659	1	1	yes	characteristic
186	C₄	#16810857	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	#20274742	1	1	yes	characteristic
190	C₂	#20276509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13133157

C₂

yes

maximal, characteristic

C₂×#16810452

C₂

yes

maximal, complemented

#20274753

C₂

yes

maximal, characteristic

#20272808

C₂

yes

maximal, complemented

C₅₃×#1810

C₂

yes

maximal, characteristic

C₂×#13133157

—

C₂×#13133157

C₄

yes

—

C₂²×#11719908

C₂²

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

#16810587

—

#16810452

—

#16810388

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#407

—

C₅₃×#341

—

—

—

—

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

—

C₂×#11719908

D₄

yes

—

#13133153

—

#13133153

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#107

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

#88322

—

#12838

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#467

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂²×M₃₂

—

C₂×#368

—

#1810

—

#1810

D₅₃

yes

characteristic

#1381

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1808

yes

characteristic

100

C₂×C₅₃

#2161

yes

—

101

C₂³×C₈

C₂×D₅₃

yes

characteristic

102

C₂×M₃₂

—

103

C₂×M₃₂

—

106

—

104

C₂²×C₁₆

—

105

C₂²×C₁₆

—

106

C₂×#108

—

107

C₂×#108

C₂×D₅₃

yes

—

108

#407

—

109

#407

—

106

—

110

#368

—

106

—

111

#341

—

112

#341

—

106

—

113

C₅₃

#12838

yes

characteristic, complemented, Sylow

114

C₂³×C₄

C₂²⋊D₅₃

yes

characteristic

115

C₂×M₁₆

—

116

C₂×M₁₆

—

106

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

Dic₁₀₆

yes

—

120

C₂²×C₈

C₄×D₅₃

yes

—

121

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

122

C₂²×C₈

C₂²×D₅₃

yes

characteristic

123

M₃₂

—

106

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

106

—

128

#108

—

129

#108

—

130

#108

—

106

—

131

#108

—

106

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

212

—

134

C₂⁴

#11719925

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

#11719919

yes

—

138

C₂²×C₄

#11719922

yes

—

139

C₂²×C₄

#11719925

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

212

—

144

M₁₆

—

212

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

212

—

151

C₂×C₈

#11719919

yes

—

152

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

153

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

154

C₂×C₈

D₄×D₅₃

yes

—

155

C₂×C₈

#11719947

yes

—

156

C₁₆

—

157

C₁₆

—

106

—

158

C₂³

—

159

C₂³

—

160

C₂³

#13133150

yes

—

161

C₂³

C₂×#11719925

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#13133150

yes

—

167

C₂×C₄

C₂×#11719919

yes

characteristic

168

C₂×C₄

#13133189

yes

—

169

C₂×C₄

C₂×#11719922

yes

characteristic

170

C₂×C₄

#13133237

yes

—

171

C₈

—

172

C₈

—

173

C₈

—

424

—

174

C₈

C₂²⋊C₄×D₅₃

yes

characteristic

175

C₈

#13133173

yes

characteristic

176

C₈

#13133194

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×#13133150

yes

characteristic

182

C₂²

#16810862

yes

—

183

C₄

—

184

C₄

#16810481

yes

characteristic

185

C₄

#16810659

yes

characteristic

186

C₄

#16810857

yes

—

187

C₂

—

188

C₂

—

189

C₂

#20274742

yes

characteristic

190

C₂

#20276509

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1810	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1810	D₅₃
hypocenter	C₅₃	#12838
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1810

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂⁴×C₅₃

Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#16810452	C₂	semidirect
4. #20272808	C₂	semidirect
113. C₅₃	#12838	semidirect

H ≅ G/N

type

2. C₂×#16810452

C₂

semidirect

4. #20272808

C₂

semidirect

113. C₅₃

#12838

semidirect

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

subgroup	index
C₂²×#13133157	2
C₂×#16810452	2
#20274753	2
#20272808	2
C₅₃×#1810	2
#12838	53

subgroup

index

C₅₃×#1810

Subgroups — #27300275

Conjugacy classes of subgroups

Complements

Maximal subgroups