Group #27217374 — subgroups

Subgroups — C₅₃×#9862 #27217374

Order 13568.

514 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₅₃	—	2	2	no	—
6	C₅₃×#532	C₂²	1	2	yes	—
7	C₅₃×#495	—	2	2	no	—
8	C₅₃×#494	C₄	1	1	yes	characteristic
9	C₅₃×#493	C₄	1	1	yes	characteristic
10	C₅₃×#492	C₄	1	2	yes	—
11	C₂×#108×C₅₃	—	2	2	no	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×D₄⋊C₈	—	2	8	no	—
16	C₅₃×#322	—	2	2	no	—
17	C₂×#41×C₅₃	—	2	2	no	—
18	#135×C₅₃	—	4	4	no	—
19	#135×C₅₃	—	4	4	no	—
20	C₂×Q₁₆×C₅₃	—	2	2	no	—
21	C₂×SD₁₆×C₅₃	—	2	4	no	—
22	C₂×D₈×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
27	#127×C₅₃	C₂×C₄	1	2	yes	—
28	#124×C₅₃	—	2	2	no	—
29	#121×C₅₃	—	2	4	no	—
30	C₄×D₄×C₅₃	—	2	4	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#108×C₅₃	—	4	4	no	—
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	4	yes	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	#41×C₅₃	—	4	4	no	—
40	#41×C₅₃	—	4	4	no	—
41	C₂×Q₈×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	—	2	2	no	—
43	C₂×D₄×C₅₃	—	2	4	no	—
44	C₂×D₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	2	yes	—
48	Q₁₆×C₅₃	—	4	4	no	—
49	SD₁₆×C₅₃	—	4	8	no	—
50	D₈×C₅₃	—	4	4	no	—
51	M₁₆×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	D₈	1	4	yes	—
58	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₂²⋊C₄×C₅₃	—	4	4	no	—
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₂³×C₅₃	—	4	2	no	—
65	C₂³×C₅₃	#124	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	4	no	—
67	D₄×C₅₃	—	4	4	no	—
68	D₄×C₅₃	—	4	8	no	—
69	D₄×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₈×C₅₃	—	4	4	no	—
81	C₈×C₅₃	—	8	2	no	—
82	C₈×C₅₃	#102	1	4	yes	—
83	C₈×C₅₃	#102	1	4	yes	—
84	#9862	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	8	2	no	—
87	C₂²×C₅₃	—	8	4	no	—
88	C₂²×C₅₃	#390	1	1	yes	characteristic
89	C₂²×C₅₃	#486	1	2	yes	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₅₃	C₂×#102	1	2	yes	—
93	C₄×C₅₃	#444	1	4	yes	—
94	C₄×C₅₃	#492	1	1	yes	characteristic
95	C₄×C₅₃	#493	1	1	yes	characteristic
96	#3147	C₂×C₅₃	1	1	yes	characteristic
97	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
98	#1492	C₂×C₅₃	1	4	yes	complemented
99	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	—	16	2	no	—
102	C₂×C₅₃	#1957	1	1	yes	characteristic
103	C₂×C₅₃	#1975	1	2	yes	—
104	C₂×#135	—	2	2	no	—
105	#532	C₂²×C₅₃	1	2	yes	—
106	#495	—	2	2	no	—
107	#494	C₄×C₅₃	1	1	yes	characteristic
108	#493	C₄×C₅₃	1	1	yes	characteristic
109	#492	C₄×C₅₃	1	2	yes	—
110	C₂×#108	—	2	2	no	—
111	#423	C₂²×C₅₃	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
114	D₄⋊C₈	—	2	8	no	—
115	#322	—	2	2	no	—
116	C₅₃	#9862	1	1	yes	characteristic, direct product factor, Sylow
117	C₂×#41	—	2	2	no	—
118	#135	—	4	4	no	—
119	#135	—	4	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	D₄×C₅₃	1	2	yes	—
126	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
127	#127	C₂×C₄×C₅₃	1	2	yes	—
128	#124	—	2	2	no	—
129	#121	—	2	4	no	—
130	C₄×D₄	—	2	4	no	—
131	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
137	C₄×C₈	D₄×C₅₃	1	4	yes	—
138	C₄×C₈	D₄×C₅₃	1	2	yes	—
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	#41×C₅₃	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
156	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
157	C₂×C₈	D₈×C₅₃	1	4	yes	—
158	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
162	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₅₃	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	#124×C₅₃	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
176	C₂×C₄	#121×C₅₃	1	2	yes	—
177	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
179	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	8	2	no	—
182	C₈	#102×C₅₃	1	4	yes	—
183	C₈	#102×C₅₃	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	C₅₃×#390	1	1	yes	characteristic
188	C₂²	C₅₃×#486	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#102×C₅₃	1	2	yes	—
192	C₄	C₅₃×#444	1	4	yes	—
193	C₄	C₅₃×#492	1	1	yes	characteristic
194	C₄	C₅₃×#493	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₅₃×#1957	1	1	yes	characteristic
198	C₂	C₅₃×#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3147

C₂

yes

maximal, characteristic

C₅₃×#1848

C₂

yes

maximal, characteristic, complemented

C₅₃×#1492

C₂

yes

maximal, complemented

C₂×C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₅₃

—

C₅₃×#532

C₂²

yes

—

C₅₃×#495

—

C₅₃×#494

C₄

yes

characteristic

C₅₃×#493

C₄

yes

characteristic

C₅₃×#492

C₄

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×D₄⋊C₈

—

C₅₃×#322

—

C₂×#41×C₅₃

—

#135×C₅₃

—

#135×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#127×C₅₃

C₂×C₄

yes

—

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#102

yes

—

#9862

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#486

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#444

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1848

C₂×C₅₃

yes

characteristic, complemented

#1492

C₂×C₅₃

yes

complemented

C₂×#322

C₂×C₅₃

yes

characteristic, complemented

100

C₂×C₅₃

—

101

C₂×C₅₃

—

102

C₂×C₅₃

#1957

yes

characteristic

103

C₂×C₅₃

#1975

yes

—

104

C₂×#135

—

105

#532

C₂²×C₅₃

yes

—

106

#495

—

107

#494

C₄×C₅₃

yes

characteristic

108

#493

C₄×C₅₃

yes

characteristic

109

#492

C₄×C₅₃

yes

—

110

C₂×#108

—

111

#423

C₂²×C₅₃

yes

—

112

C₂×C₄×C₈

C₂²×C₅₃

yes

—

113

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

114

D₄⋊C₈

—

115

#322

—

116

C₅₃

#9862

yes

characteristic, direct product factor, Sylow

117

C₂×#41

—

118

#135

—

119

#135

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

125

C₂²×C₈

D₄×C₅₃

yes

—

126

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

127

#127

C₂×C₄×C₅₃

yes

—

128

#124

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂³×C₅₃

yes

characteristic

132

#108

—

133

C₄⋊C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

136

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

137

C₄×C₈

D₄×C₅₃

yes

—

138

C₄×C₈

D₄×C₅₃

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

147

C₂²×C₄

#41×C₅₃

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

156

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

157

C₂×C₈

D₈×C₅₃

yes

—

158

C₂×C₈

SD₁₆×C₅₃

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×C₄×C₅₃

yes

characteristic

162

C₄²

C₂×D₄×C₅₃

yes

characteristic

163

C₄²

C₂×D₄×C₅₃

yes

—

164

C₂³

—

165

C₂³

#124×C₅₃

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

175

C₂×C₄

C₄×D₄×C₅₃

yes

—

176

C₂×C₄

#121×C₅₃

yes

—

177

C₂×C₄

#127×C₅₃

yes

characteristic

178

C₂×C₄

C₂×D₈×C₅₃

yes

—

179

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

180

C₈

—

181

C₈

—

182

C₈

#102×C₅₃

yes

—

183

C₈

#102×C₅₃

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₅₃×#390

yes

characteristic

188

C₂²

C₅₃×#486

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#102×C₅₃

yes

—

192

C₄

C₅₃×#444

yes

—

193

C₄

C₅₃×#492

yes

characteristic

194

C₄

C₅₃×#493

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₅₃×#1957

yes

characteristic

198

C₂

C₅₃×#1975

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#127

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#390

prime radical

C₅₃×#3147

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1848	C₂	semidirect
3. C₅₃×#1492	C₂	semidirect
4. C₂×C₅₃×#322	C₂	semidirect
84. #9862	C₅₃	direct (N × H)
97. #1848	C₂×C₅₃	semidirect
98. #1492	C₂×C₅₃	semidirect
99. C₂×#322	C₂×C₅₃	semidirect
116. C₅₃	#9862	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1848

C₂

semidirect

3. C₅₃×#1492

C₂

semidirect

4. C₂×C₅₃×#322

C₂

semidirect

84. #9862

C₅₃

direct (N × H)

97. #1848

C₂×C₅₃

semidirect

98. #1492

C₂×C₅₃

semidirect

99. C₂×#322

C₂×C₅₃

semidirect

116. C₅₃

#9862

direct (N × H)

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

subgroup	index
C₅₃×#3147	2
C₅₃×#1848	2
C₅₃×#1492	2
C₂×C₅₃×#322	2
#9862	53

subgroup

index

C₅₃×#3147

C₅₃×#1848

C₅₃×#1492

C₂×C₅₃×#322

#9862

Subgroups — C₅₃×#9862 #27217374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9862 #27217374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9862 #27217374