Group #27216241 — subgroups

Subgroups — C₅₃×#8729 #27216241

Order 13568.

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 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×#2285	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1797	C₂	1	2	yes	maximal
3	C₅₃×#1296	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
7	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
8	C₅₃×#384	C₂²	1	2	yes	—
9	C₅₃×#382	C₄	1	2	yes	—
10	C₅₃×#380	—	2	2	no	—
11	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₅₃×#336	—	2	2	no	—
13	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₅₃	D₄	1	2	yes	—
15	C₂²×C₈×C₅₃	—	2	4	no	—
16	#117×C₅₃	—	4	4	no	—
17	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	—	2	4	no	—
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	8	no	—
27	C₂²⋊C₈×C₅₃	—	4	4	no	—
28	#95×C₅₃	—	2	4	no	—
29	C₂⁴×C₅₃	C₂²⋊C₄	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	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	—	4	4	no	—
44	C₂×C₈×C₅₃	—	4	4	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₂²⋊C₄×C₅₃	—	4	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	#41	1	4	yes	—
54	C₂³×C₅₃	—	2	1	no	—
55	C₂³×C₅₃	—	2	1	no	—
56	C₂³×C₅₃	—	2	2	no	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	#100	1	2	yes	—
59	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	8	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	#101	1	2	yes	—
74	C₂×C₄×C₅₃	#117	1	2	yes	—
75	C₂×C₄×C₅₃	#123	1	4	yes	—
76	C₂×C₄×C₅₃	#135	1	4	yes	—
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₅₃	#137	1	2	yes	—
79	C₈×C₅₃	—	8	4	no	—
80	#8729	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	—	2	4	no	—
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	4	1	no	—
84	C₂²×C₅₃	—	4	1	no	—
85	C₂²×C₅₃	—	4	2	no	—
86	C₂²×C₅₃	#381	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
89	C₂²×C₅₃	#416	1	2	yes	—
90	C₂²×C₅₃	#417	1	2	yes	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	—	8	4	no	—
95	C₄×C₅₃	#482	1	4	yes	—
96	C₄×C₅₃	#485	1	4	yes	—
97	#2285	C₂×C₅₃	1	1	yes	characteristic
98	#1797	C₂×C₅₃	1	2	yes	—
99	#1296	C₂×C₅₃	1	4	yes	complemented
100	C₂×C₅₃	—	4	2	no	—
101	C₂×C₅₃	#1544	1	1	yes	characteristic
102	C₂×C₅₃	#1551	1	2	yes	—
103	C₂×C₅₃	#1561	1	1	yes	characteristic
104	C₂×C₅₃	#1821	1	1	yes	characteristic
105	C₂×C₅₃	#1825	1	2	yes	—
106	C₂×#117	C₄×C₅₃	1	2	yes	—
107	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
108	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
109	C₂×C₂²⋊C₈	—	2	2	no	—
110	#384	C₂²×C₅₃	1	2	yes	—
111	#382	C₄×C₅₃	1	2	yes	—
112	#380	—	2	2	no	—
113	C₄×C₄⋊C₄	—	2	2	no	—
114	#336	—	2	2	no	—
115	C₅₃	#8729	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
117	C₂³×C₄	D₄×C₅₃	1	2	yes	—
118	C₂²×C₈	—	2	4	no	—
119	#117	—	4	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
123	C₂×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
127	C₂×C₄²	D₄×C₅₃	1	2	yes	—
128	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	8	no	—
130	C₂²⋊C₈	—	4	4	no	—
131	#95	—	2	4	no	—
132	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
142	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
145	C₂×C₈	—	4	4	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	#41×C₅₃	1	4	yes	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	#100×C₅₃	1	2	yes	—
162	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	#101×C₅₃	1	2	yes	—
177	C₂×C₄	#117×C₅₃	1	2	yes	—
178	C₂×C₄	#123×C₅₃	1	4	yes	—
179	C₂×C₄	#135×C₅₃	1	4	yes	—
180	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
181	C₂×C₄	#137×C₅₃	1	2	yes	—
182	C₈	—	8	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₅₃×#381	1	1	yes	characteristic
189	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
190	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
191	C₂²	C₅₃×#416	1	2	yes	—
192	C₂²	C₅₃×#417	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	4	no	—
197	C₄	C₅₃×#482	1	4	yes	—
198	C₄	C₅₃×#485	1	4	yes	—
199	C₂	—	4	2	no	—
200	C₂	C₅₃×#1544	1	1	yes	characteristic
201	C₂	C₅₃×#1551	1	2	yes	—
202	C₂	C₅₃×#1561	1	1	yes	characteristic
203	C₂	C₅₃×#1821	1	1	yes	characteristic
204	C₂	C₅₃×#1825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2285

C₂

yes

maximal, characteristic

C₅₃×#1797

C₂

yes

maximal

C₅₃×#1296

C₂

yes

maximal, complemented

C₂×#117×C₅₃

C₄

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₈×C₅₃

—

C₅₃×#384

C₂²

yes

—

C₅₃×#382

C₄

yes

—

C₅₃×#380

—

C₄×C₄⋊C₄×C₅₃

—

C₅₃×#336

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#95×C₅₃

—

C₂⁴×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#100

yes

—

C₂³×C₅₃

C₂×C₂²⋊C₄

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#101

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

#8729

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#417

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#482

yes

—

C₄×C₅₃

#485

yes

—

#2285

C₂×C₅₃

yes

characteristic

#1797

C₂×C₅₃

yes

—

#1296

C₂×C₅₃

yes

complemented

100

C₂×C₅₃

—

101

C₂×C₅₃

#1544

yes

characteristic

102

C₂×C₅₃

#1551

yes

—

103

C₂×C₅₃

#1561

yes

characteristic

104

C₂×C₅₃

#1821

yes

characteristic

105

C₂×C₅₃

#1825

yes

—

106

C₂×#117

C₄×C₅₃

yes

—

107

C₂²×C₄²

C₂²×C₅₃

yes

characteristic

108

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

109

C₂×C₂²⋊C₈

—

110

#384

C₂²×C₅₃

yes

—

111

#382

C₄×C₅₃

yes

—

112

#380

—

113

C₄×C₄⋊C₄

—

114

#336

—

115

C₅₃

#8729

yes

characteristic, direct product factor, Sylow

116

C₂³×C₄

C₂×C₄×C₅₃

yes

characteristic

117

C₂³×C₄

D₄×C₅₃

yes

—

118

C₂²×C₈

—

119

#117

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

123

C₂×C₂²⋊C₄

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

127

C₂×C₄²

D₄×C₅₃

yes

—

128

C₂×C₄²

C₂³×C₅₃

yes

characteristic

129

C₄⋊C₈

—

130

C₂²⋊C₈

—

131

#95

—

132

C₂⁴

C₂²⋊C₄×C₅₃

yes

characteristic

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

142

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

143

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

144

C₂²×C₄

C₂×D₄×C₅₃

yes

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

#41×C₅₃

yes

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

#100×C₅₃

yes

—

162

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#101×C₅₃

yes

—

177

C₂×C₄

#117×C₅₃

yes

—

178

C₂×C₄

#123×C₅₃

yes

—

179

C₂×C₄

#135×C₅₃

yes

—

180

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

181

C₂×C₄

#137×C₅₃

yes

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₅₃×#381

yes

characteristic

189

C₂²

C₂×#100×C₅₃

yes

characteristic

190

C₂²

C₂×#101×C₅₃

yes

characteristic

191

C₂²

C₅₃×#416

yes

—

192

C₂²

C₅₃×#417

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₅₃×#482

yes

—

198

C₄

C₅₃×#485

yes

—

199

C₂

—

200

C₂

C₅₃×#1544

yes

characteristic

201

C₂

C₅₃×#1551

yes

—

202

C₂

C₅₃×#1561

yes

characteristic

203

C₂

C₅₃×#1821

yes

characteristic

204

C₂

C₅₃×#1825

yes

—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₅₃

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#1296	C₂	semidirect
80. #8729	C₅₃	direct (N × H)
99. #1296	C₂×C₅₃	semidirect
115. C₅₃	#8729	direct (N × H)

H ≅ G/N

type

3. C₅₃×#1296

C₂

semidirect

80. #8729

C₅₃

direct (N × H)

99. #1296

C₂×C₅₃

semidirect

115. C₅₃

#8729

direct (N × H)

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

subgroup	index
C₅₃×#2285	2
C₅₃×#1797	2
C₅₃×#1296	2
#8729	53

subgroup

index

C₅₃×#2285

C₅₃×#1797

C₅₃×#1296

#8729

Subgroups — C₅₃×#8729 #27216241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8729 #27216241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8729 #27216241