Group #27300309 — subgroups

Subgroups — #27300309

Order 13568.

478 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20276326	C₂	1	2	yes	maximal
2	#20272799	C₂	1	4	yes	maximal, complemented
3	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₂²	1	4	yes	—
5	C₂²⋊#11719908	—	2	4	no	—
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₃	—	2	4	no	—
10	C₂×#97×C₅₃	C₄	1	2	yes	—
11	C₂×#97×C₅₃	C₂²	1	2	yes	—
12	#13133157	—	2	8	no	—
13	C₂×#11719908	—	2	4	no	—
14	C₂³×C₄×C₅₃	C₈	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₃	D₄	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₈	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	#97×C₅₃	—	2	4	no	—
28	#97×C₅₃	—	2	4	no	—
29	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	4	4	no	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	8	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	8	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	4	yes	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	#11719908	—	4	4	no	—
53	C₂³×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
56	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	#123	1	4	yes	—
73	#12846	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	8	no	—
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	#349	1	4	yes	—
79	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₃	#431	1	2	yes	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	—	4	4	no	—
83	C₄×C₅₃	#381	1	1	yes	characteristic
84	C₄×C₅₃	#407	1	1	yes	characteristic
85	C₄×C₅₃	#432	1	2	yes	—
86	C₂²×#97	D₅₃	1	1	yes	characteristic
87	#2112	—	53	2	no	—
88	#1377	—	53	4	no	—
89	C₂×C₅₃	—	2	4	no	—
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	#1817	1	1	yes	characteristic
92	C₂×C₅₃	C₂×#349	1	2	yes	—
93	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
94	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
96	C₂×M₃₂	—	53	4	no	—
97	C₂×#97	—	2	4	no	—
98	C₂×#97	Dic₅₃	1	2	yes	—
99	C₂×#97	C₂×D₅₃	1	2	yes	—
100	C₂²⋊C₁₆	—	106	4	no	—
101	C₅₃	#12846	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	#88322	1	2	yes	—
103	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
108	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
110	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	#88322	1	2	yes	—
114	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
115	M₃₂	—	106	8	no	—
116	C₂×C₁₆	—	106	4	no	—
117	#97	—	2	4	no	—
118	#97	—	2	4	no	—
119	C₂⁴	C₂×#88322	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
126	C₂²×C₄	C₂×#88322	1	4	yes	—
127	C₂²×C₄	#11719917	1	2	yes	—
128	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	#11719917	1	4	yes	—
136	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
137	C₂×C₈	#11719945	1	4	yes	—
138	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₁₆	—	212	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊#88322	1	2	yes	—
146	C₂³	C₂²×#88322	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂²⋊#88322	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
157	C₂×C₄	#13133176	1	2	yes	—
158	C₂×C₄	C₂×#11719917	1	2	yes	—
159	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
160	C₈	—	2	4	no	—
161	C₈	—	2	2	no	—
162	C₈	#13133200	1	4	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#16810405	1	4	yes	—
168	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
169	C₂²	#16810676	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#16810486	1	1	yes	characteristic
173	C₄	#16810587	1	1	yes	characteristic
174	C₄	#16810680	1	2	yes	—
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	#20274783	1	1	yes	characteristic
178	C₂	C₂×#16810405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276326

C₂

yes

maximal

#20272799

C₂

yes

maximal, complemented

C₂²×#97×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂²⋊#11719908

—

C₂³×C₈×C₅₃

C₄

yes

characteristic

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄²×C₅₃

C₄

yes

characteristic

C₂×#97×C₅₃

—

C₂×#97×C₅₃

C₄

yes

—

C₂×#97×C₅₃

C₂²

yes

—

#13133157

—

C₂×#11719908

—

C₂³×C₄×C₅₃

C₈

yes

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

#97×C₅₃

—

#97×C₅₃

—

C₂⁴×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#123

yes

—

#12846

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#349

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂²×#97

D₅₃

yes

characteristic

#2112

—

#1377

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1817

yes

characteristic

C₂×C₅₃

C₂×#349

yes

—

C₂³×C₈

Dic₅₃

yes

characteristic

C₂³×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄²

Dic₅₃

yes

characteristic

C₂×M₃₂

—

C₂×#97

—

C₂×#97

Dic₅₃

yes

—

C₂×#97

C₂×D₅₃

yes

—

100

C₂²⋊C₁₆

—

106

—

101

C₅₃

#12846

yes

characteristic, complemented, Sylow

102

C₂³×C₄

#88322

yes

—

103

C₂³×C₄

C₂×Dic₅₃

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×Dic₅₃

yes

—

108

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

109

C₂²×C₈

C₂²⋊D₅₃

yes

—

110

C₂²×C₈

C₂²×D₅₃

yes

characteristic

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

#88322

yes

—

114

C₂×C₄²

C₂×Dic₅₃

yes

—

115

M₃₂

—

106

—

116

C₂×C₁₆

—

106

—

117

#97

—

118

#97

—

119

C₂⁴

C₂×#88322

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×#88322

yes

characteristic

126

C₂²×C₄

C₂×#88322

yes

—

127

C₂²×C₄

#11719917

yes

—

128

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

129

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

#11719917

yes

—

136

C₂×C₈

C₂²⋊Dic₅₃

yes

—

137

C₂×C₈

#11719945

yes

—

138

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

212

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊#88322

yes

—

146

C₂³

C₂²×#88322

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊#88322

yes

—

156

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

157

C₂×C₄

#13133176

yes

—

158

C₂×C₄

C₂×#11719917

yes

—

159

C₂×C₄

C₂×#11719917

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

#13133200

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#16810405

yes

—

168

C₂²

C₂×C₂²⋊#88322

yes

characteristic

169

C₂²

#16810676

yes

—

170

C₄

—

171

C₄

—

172

C₄

#16810486

yes

characteristic

173

C₄

#16810587

yes

characteristic

174

C₄

#16810680

yes

—

175

C₂

—

176

C₂

—

177

C₂

#20274783

yes

characteristic

178

C₂

C₂×#16810405

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	#12846
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂²×#97×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₅₃

C₂×C₈

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂²×#97

D₅₃

hypocenter

C₅₃

#12846

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

N	H ≅ G/N	type
2. #20272799	C₂	semidirect
101. C₅₃	#12846	semidirect

H ≅ G/N

type

2. #20272799

C₂

semidirect

101. C₅₃

#12846

semidirect

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

subgroup	index
#20276326	2
#20272799	2
C₂²×#97×C₅₃	2
#12846	53

subgroup

index

#20276326

#20272799

C₂²×#97×C₅₃

#12846

Subgroups — #27300309

Conjugacy classes of subgroups

Complements

Maximal subgroups