Group #27300272 — subgroups

Subgroups — #27300272

Order 13568.

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

normal subgroups only

This group has 394 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	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810452	C₂	1	2	yes	maximal, complemented
3	#20274750	C₂	1	2	yes	maximal
4	#20272809	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₅₃	—	2	4	no	—
9	C₂×C₁₆×D₅₃	C₄	1	4	yes	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810586	—	2	2	no	—
12	#16810452	—	2	4	no	—
13	#16810387	—	2	2	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
17	C₂×#11719917	—	2	2	no	—
18	C₂²×#88322	D₄	1	1	yes	characteristic
19	C₂×C₈×D₅₃	—	2	2	no	—
20	C₂×C₈×D₅₃	—	2	2	no	—
21	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₅₃	Q₈	1	2	yes	—
23	C₂×#11719908	C₂×C₄	1	2	yes	—
24	C₂×#11719908	D₄	1	4	yes	—
25	C₁₆×D₅₃	—	2	8	no	—
26	#13133153	—	2	2	no	—
27	#13133153	—	2	2	no	—
28	#13133129	—	4	4	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	#108×C₅₃	—	2	2	no	—
34	#108×C₅₃	—	2	2	no	—
35	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₅₃	—	2	2	no	—
39	C₂×C₄×D₅₃	—	2	2	no	—
40	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₅₃	D₈	1	2	yes	—
42	M₁₆×C₅₃	—	4	2	no	—
43	M₁₆×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	C₄⋊C₄	1	2	yes	—
54	C₈×D₅₃	—	2	2	no	—
55	C₈×D₅₃	—	2	2	no	—
56	C₈×D₅₃	—	2	4	no	—
57	C₁₆×C₅₃	—	2	2	no	—
58	#11719908	C₂²⋊C₄	1	4	yes	—
59	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	—	2	4	no	—
62	C₂²×D₅₃	#107	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#121	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	2	yes	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	#107	1	2	yes	—
69	C₄×D₅₃	—	2	2	no	—
70	C₄×D₅₃	—	2	2	no	—
71	C₄×D₅₃	—	2	4	no	—
72	C₈×C₅₃	—	8	2	no	—
73	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₅₃	#117	1	1	yes	characteristic
75	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
76	#88322	—	2	2	no	—
77	#88322	—	8	2	no	—
78	#12837	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
81	C₂²×C₅₃	#467	1	2	yes	—
82	C₂×D₅₃	—	2	8	no	—
83	C₂×D₅₃	—	2	2	no	—
84	C₂×D₅₃	—	2	2	no	—
85	C₂×D₅₃	#368	1	4	yes	complemented
86	C₄×C₅₃	#380	1	1	yes	characteristic
87	C₄×C₅₃	#427	1	1	yes	characteristic
88	C₄×C₅₃	#466	1	2	yes	—
89	Dic₅₃	—	2	2	no	—
90	C₂³×C₁₆	—	53	1	no	—
91	C₂×#368	—	53	2	no	—
92	#1810	—	53	2	no	—
93	#1381	—	53	1	no	—
94	#1381	D₅₃	1	1	yes	characteristic, complemented
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#1808	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#368	1	2	yes	—
98	D₅₃	—	2	4	no	—
99	C₂³×C₈	—	53	1	no	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	—	53	4	no	—
102	C₂²×C₁₆	—	106	4	no	—
103	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
104	C₂×#108	—	53	2	no	—
105	C₂×#108	C₂×D₅₃	1	2	yes	—
106	#407	—	106	2	no	—
107	#368	—	106	4	no	—
108	#341	—	106	2	no	—
109	C₅₃	#12837	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	53	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	106	2	no	—
113	C₂²×C₈	—	53	1	no	—
114	C₂²×C₈	—	53	2	no	—
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	106	2	no	—
117	C₂²×C₈	—	106	2	no	—
118	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	4	no	—
122	C₂×C₁₆	—	106	8	no	—
123	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	106	2	no	—
127	#108	—	106	2	no	—
128	C₂²⋊C₈	—	212	4	no	—
129	C₂⁴	—	53	1	no	—
130	C₂²×C₄	—	53	1	no	—
131	C₂²×C₄	—	53	2	no	—
132	C₂²×C₄	—	53	2	no	—
133	C₂²×C₄	—	106	2	no	—
134	C₂²×C₄	—	106	2	no	—
135	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	212	2	no	—
139	M₁₆	—	212	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	212	2	no	—
147	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₈	#11719945	1	2	yes	—
149	C₂×C₈	D₄×D₅₃	1	2	yes	—
150	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
151	C₁₆	—	2	2	no	—
152	C₁₆	—	53	4	no	—
153	C₂³	—	53	2	no	—
154	C₂³	—	106	4	no	—
155	C₂³	—	106	2	no	—
156	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	4	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	#13133190	1	2	yes	—
164	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133238	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	106	2	no	—
168	C₈	—	424	2	no	—
169	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
170	C₈	#13133174	1	1	yes	characteristic
171	C₈	#13133195	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	53	4	no	—
174	C₂²	—	106	8	no	—
175	C₂²	—	106	2	no	—
176	C₂²	—	106	2	no	—
177	C₂²	#107×D₅₃	1	1	yes	characteristic
178	C₂²	#16810863	1	2	yes	—
179	C₄	—	106	2	no	—
180	C₄	#16810482	1	1	yes	characteristic
181	C₄	#16810658	1	1	yes	characteristic
182	C₄	#16810858	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	106	4	no	—
185	C₂	#20274743	1	1	yes	characteristic
186	C₂	#368×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#16810452

C₂

yes

maximal, complemented

#20274750

C₂

yes

maximal

#20272809

C₂

yes

maximal, characteristic, complemented

C₅₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₃

C₂²

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₁₆×D₅₃

—

C₂×C₁₆×D₅₃

C₄

yes

—

C₂×#13133153

C₂²

yes

—

#16810586

—

#16810452

—

#16810387

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

D₄

yes

characteristic

C₂×#11719917

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×C₈×D₅₃

Q₈

yes

—

C₂×#11719908

C₂×C₄

yes

—

C₂×#11719908

D₄

yes

—

C₁₆×D₅₃

—

#13133153

—

#13133153

—

#13133129

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

—

C₂³×D₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

D₈

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

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

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

#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

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#107

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#467

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂³×C₁₆

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

#1808

yes

characteristic

C₂×C₅₃

C₂×#368

yes

—

D₅₃

—

C₂³×C₈

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

106

—

103

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

104

C₂×#108

—

105

C₂×#108

C₂×D₅₃

yes

—

106

#407

—

106

—

107

#368

—

106

—

108

#341

—

106

—

109

C₅₃

#12837

yes

characteristic, complemented, Sylow

110

C₂³×C₄

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

106

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

106

—

117

C₂²×C₈

—

106

—

118

C₂²×C₈

C₂²×D₅₃

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

106

—

123

C₂×C₁₆

C₄×D₅₃

yes

—

124

#108

—

125

#108

—

126

#108

—

106

—

127

#108

—

106

—

128

C₂²⋊C₈

—

212

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

106

—

134

C₂²×C₄

—

106

—

135

C₂²×C₄

D₄×D₅₃

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

212

—

139

M₁₆

—

212

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

212

—

147

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

148

C₂×C₈

#11719945

yes

—

149

C₂×C₈

D₄×D₅₃

yes

—

150

C₂×C₈

Q₈×D₅₃

yes

characteristic

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

106

—

155

C₂³

—

106

—

156

C₂³

Q₁₆×D₅₃

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

163

C₂×C₄

#13133190

yes

—

164

C₂×C₄

D₈×D₅₃

yes

characteristic

165

C₂×C₄

#13133238

yes

—

166

C₈

—

167

C₈

—

106

—

168

C₈

—

424

—

169

C₈

C₂²⋊C₄×D₅₃

yes

characteristic

170

C₈

#13133174

yes

characteristic

171

C₈

#13133195

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

106

—

175

C₂²

—

106

—

176

C₂²

—

106

—

177

C₂²

#107×D₅₃

yes

characteristic

178

C₂²

#16810863

yes

—

179

C₄

—

106

—

180

C₄

#16810482

yes

characteristic

181

C₄

#16810658

yes

characteristic

182

C₄

#16810858

yes

—

183

C₂

—

184

C₂

—

106

—

185

C₂

#20274743

yes

characteristic

186

C₂

#368×D₅₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1381	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1381	D₅₃
hypocenter	C₅₃	#12837
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1381

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×D₅₃

Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#16810452	C₂	semidirect
4. #20272809	C₂	semidirect
5. C₅₃×#1381	C₂	semidirect
85. C₂×D₅₃	#368	semidirect
94. #1381	D₅₃	semidirect
109. C₅₃	#12837	semidirect

H ≅ G/N

type

2. C₂×#16810452

C₂

semidirect

4. #20272809

C₂

semidirect

5. C₅₃×#1381

C₂

semidirect

85. C₂×D₅₃

#368

semidirect

94. #1381

D₅₃

semidirect

109. C₅₃

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#16810452	2
#20274750	2
#20272809	2
C₅₃×#1381	2
#12837	53

subgroup

index

C₂²×C₁₆×D₅₃

C₅₃×#1381

Subgroups — #27300272

Conjugacy classes of subgroups

Complements

Maximal subgroups