Group #27216150 — subgroups

Subgroups — C₅₃×#8638 #27216150

Order 13568.

404 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1560	C₂	1	2	yes	maximal
4	C₅₃×#1309	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₅₃	C₄	1	2	yes	—
6	C₅₃×#532	C₂²	1	1	yes	characteristic
7	C₅₃×#436	—	2	2	no	—
8	C₈×D₄×C₅₃	—	2	2	no	—
9	C₅₃×#423	C₂²	1	4	yes	—
10	C₅₃×#423	C₂²	1	1	yes	characteristic
11	C₅₃×#408	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#335	—	2	2	no	—
14	C₅₃×#335	—	2	2	no	—
15	C₅₃×#333	—	2	2	no	—
16	C₅₃×#331	—	2	2	no	—
17	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
18	#131×C₅₃	—	4	4	no	—
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂×M₁₆×C₅₃	—	2	4	no	—
21	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
26	#127×C₅₃	D₄	1	1	yes	characteristic
27	#124×C₅₃	Q₈	1	2	yes	—
28	#121×C₅₃	—	2	2	no	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄⋊C₈×C₅₃	—	2	4	no	—
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₂²⋊C₈×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	#41×C₅₃	—	4	2	no	—
38	#41×C₅₃	—	4	2	no	—
39	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	2	yes	—
46	M₁₆×C₅₃	—	4	4	no	—
47	M₁₆×C₅₃	—	4	2	no	—
48	M₁₆×C₅₃	—	8	4	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	4	no	—
53	C₂×C₈×C₅₃	—	4	4	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	2	4	no	—
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	#117	1	1	yes	characteristic
64	Q₈×C₅₃	—	4	2	no	—
65	D₄×C₅₃	—	4	2	no	—
66	D₄×C₅₃	—	4	4	no	—
67	D₄×C₅₃	—	4	2	no	—
68	D₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#121	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	4	2	no	—
83	C₈×C₅₃	—	8	4	no	—
84	C₈×C₅₃	#100	1	2	yes	—
85	#8638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	—	4	2	no	—
89	C₂²×C₅₃	—	4	4	no	—
90	C₂²×C₅₃	#380	1	1	yes	characteristic
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
95	C₄×C₅₃	#413	1	1	yes	characteristic
96	C₄×C₅₃	#472	1	2	yes	—
97	C₄×C₅₃	#473	1	2	yes	—
98	#2931	C₂×C₅₃	1	1	yes	characteristic
99	#1567	C₂×C₅₃	1	2	yes	complemented
100	#1560	C₂×C₅₃	1	2	yes	—
101	#1309	C₂×C₅₃	1	2	yes	complemented
102	C₂×C₅₃	—	4	1	no	—
103	C₂×C₅₃	—	8	2	no	—
104	C₂×C₅₃	#1800	1	1	yes	characteristic
105	C₂×C₅₃	#1816	1	2	yes	—
106	C₂×#131	C₄×C₅₃	1	2	yes	—
107	#532	C₂²×C₅₃	1	1	yes	characteristic
108	#436	—	2	2	no	—
109	C₈×D₄	—	2	2	no	—
110	#423	C₂²×C₅₃	1	4	yes	—
111	#423	C₂²×C₅₃	1	1	yes	characteristic
112	#408	C₄×C₅₃	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
114	#335	—	2	2	no	—
115	#335	—	2	2	no	—
116	#333	—	2	2	no	—
117	#331	—	2	2	no	—
118	C₅₃	#8638	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
120	#131	—	4	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
127	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
128	#127	D₄×C₅₃	1	1	yes	characteristic
129	#124	Q₈×C₅₃	1	2	yes	—
130	#121	—	2	2	no	—
131	C₄×D₄	—	2	2	no	—
132	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	—	2	4	no	—
136	C₂²⋊C₈	—	2	4	no	—
137	C₄×C₈	D₄×C₅₃	1	2	yes	—
138	C₄×C₈	D₄×C₅₃	1	2	yes	—
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	#41×C₅₃	1	2	yes	—
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	8	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
157	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂×D₄×C₅₃	1	2	yes	—
162	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
163	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
164	C₂³	—	2	2	no	—
165	C₂³	#117×C₅₃	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	2	no	—
170	D₄	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄	#121×C₅₃	1	2	yes	—
181	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
182	C₈	—	2	1	no	—
183	C₈	—	2	2	no	—
184	C₈	—	4	2	no	—
185	C₈	—	8	4	no	—
186	C₈	#100×C₅₃	1	2	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₅₃×#380	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
196	C₄	C₅₃×#413	1	1	yes	characteristic
197	C₄	C₅₃×#472	1	2	yes	—
198	C₄	C₅₃×#473	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	C₅₃×#1800	1	1	yes	characteristic
202	C₂	C₅₃×#1816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2931

C₂

yes

maximal, characteristic

C₅₃×#1567

C₂

yes

maximal, complemented

C₅₃×#1560

C₂

yes

maximal

C₅₃×#1309

C₂

yes

maximal, complemented

C₂×#131×C₅₃

C₄

yes

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#436

—

C₈×D₄×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#408

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#335

—

C₅₃×#333

—

C₅₃×#331

—

C₂×#41×C₅₃

C₂×C₄

yes

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#124×C₅₃

Q₈

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#100

yes

—

#8638

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#473

yes

—

#2931

C₂×C₅₃

yes

characteristic

#1567

C₂×C₅₃

yes

complemented

100

#1560

C₂×C₅₃

yes

—

101

#1309

C₂×C₅₃

yes

complemented

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1800

yes

characteristic

105

C₂×C₅₃

#1816

yes

—

106

C₂×#131

C₄×C₅₃

yes

—

107

#532

C₂²×C₅₃

yes

characteristic

108

#436

—

109

C₈×D₄

—

110

#423

C₂²×C₅₃

yes

—

111

#423

C₂²×C₅₃

yes

characteristic

112

#408

C₄×C₅₃

yes

—

113

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

114

#335

—

115

#335

—

116

#333

—

117

#331

—

118

C₅₃

#8638

yes

characteristic, direct product factor, Sylow

119

C₂×#41

C₂×C₄×C₅₃

yes

—

120

#131

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₂×C₄×C₅₃

yes

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

126

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

127

C₄⋊Q₈

D₄×C₅₃

yes

characteristic

128

#127

D₄×C₅₃

yes

characteristic

129

#124

Q₈×C₅₃

yes

—

130

#121

—

131

C₄×D₄

—

132

C₂×C₄²

C₂³×C₅₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₂²⋊C₈

—

137

C₄×C₈

D₄×C₅₃

yes

—

138

C₄×C₈

D₄×C₅₃

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

C₄⋊C₄×C₅₃

yes

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

C₄⋊C₄×C₅₃

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

147

C₂²×C₄

#41×C₅₃

yes

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

157

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

158

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂×D₄×C₅₃

yes

—

162

C₄²

C₂×D₄×C₅₃

yes

characteristic

163

C₄²

C₂×Q₈×C₅₃

yes

characteristic

164

C₂³

—

165

C₂³

#117×C₅₃

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

179

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

180

C₂×C₄

#121×C₅₃

yes

—

181

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

#100×C₅₃

yes

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₅₃×#380

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₂×#100×C₅₃

yes

characteristic

196

C₄

C₅₃×#413

yes

characteristic

197

C₄

C₅₃×#472

yes

—

198

C₄

C₅₃×#473

yes

—

199

C₂

—

200

C₂

—

201

C₂

C₅₃×#1800

yes

characteristic

202

C₂

C₅₃×#1816

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#380

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₅₃×#532

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1567	C₂	semidirect
4. C₅₃×#1309	C₂	semidirect
85. #8638	C₅₃	direct (N × H)
99. #1567	C₂×C₅₃	semidirect
101. #1309	C₂×C₅₃	semidirect
118. C₅₃	#8638	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1567

C₂

semidirect

4. C₅₃×#1309

C₂

semidirect

85. #8638

C₅₃

direct (N × H)

99. #1567

C₂×C₅₃

semidirect

101. #1309

C₂×C₅₃

semidirect

118. C₅₃

#8638

direct (N × H)

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

subgroup	index
C₅₃×#2931	2
C₅₃×#1567	2
C₅₃×#1560	2
C₅₃×#1309	2
#8638	53

subgroup

index

C₅₃×#2931

C₅₃×#1567

C₅₃×#1560

C₅₃×#1309

#8638

Subgroups — C₅₃×#8638 #27216150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8638 #27216150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8638 #27216150