Group #27218173 — subgroups

Subgroups — C₅₃×#10661 #27218173

Order 13568.

372 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1955	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1570	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1486	C₂	1	2	yes	maximal
5	C₅₃×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₂×#108×C₅₃	—	2	2	no	—
9	C₅₃×#423	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#334	—	2	4	no	—
14	C₅₃×D₄⋊C₈	—	2	4	no	—
15	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	—	2	1	no	—
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	D₄	1	2	yes	—
20	C₄×D₄×C₅₃	—	2	2	no	—
21	C₄×D₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#108×C₅₃	—	4	4	no	—
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	C₂²⋊C₈×C₅₃	—	4	4	no	—
32	#97×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	D₄	1	4	yes	—
34	C₂⁴×C₅₃	—	2	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	4	1	no	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	M₁₆×C₅₃	—	4	4	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂²⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	C₂×D₄	1	2	yes	—
56	C₄²×C₅₃	#41	1	2	yes	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	4	1	no	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	—	4	1	no	—
61	C₂³×C₅₃	—	4	2	no	—
62	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	D₄×C₅₃	—	2	4	no	—
64	D₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	—	4	1	no	—
67	C₂×C₄×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	8	1	no	—
71	C₂×C₄×C₅₃	—	8	1	no	—
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	#123	1	2	yes	—
74	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₅₃	—	2	4	no	—
79	C₈×C₅₃	—	4	4	no	—
80	C₈×C₅₃	—	8	1	no	—
81	C₈×C₅₃	—	8	2	no	—
82	#10661	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	—	8	2	no	—
88	C₂²×C₅₃	—	8	1	no	—
89	C₂²×C₅₃	—	8	1	no	—
90	C₂²×C₅₃	#399	1	1	yes	characteristic
91	C₂²×C₅₃	#476	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	8	1	no	—
95	C₄×C₅₃	#481	1	2	yes	—
96	C₄×C₅₃	#492	1	1	yes	characteristic
97	C₄×C₅₃	#496	1	1	yes	characteristic
98	#1955	C₂×C₅₃	1	1	yes	characteristic
99	#1849	C₂×C₅₃	1	1	yes	characteristic, complemented
100	#1570	C₂×C₅₃	1	2	yes	complemented
101	#1486	C₂×C₅₃	1	2	yes	—
102	#1275	C₂×C₅₃	1	1	yes	characteristic, complemented
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	8	2	no	—
105	C₂×C₅₃	#2091	1	1	yes	characteristic
106	C₂×C₅₃	#2094	1	1	yes	characteristic
107	C₂×C₅₃	#2098	1	1	yes	characteristic
108	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
109	#431	C₂²×C₅₃	1	2	yes	—
110	C₂×#108	—	2	2	no	—
111	#423	C₂²×C₅₃	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	—	2	2	no	—
114	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
115	#334	—	2	4	no	—
116	D₄⋊C₈	—	2	4	no	—
117	C₅₃	#10661	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
119	C₂³×C₄	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	D₄×C₅₃	1	2	yes	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
128	#108	—	4	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	4	4	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	D₄×C₅₃	1	4	yes	—
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	C₂×D₄×C₅₃	1	2	yes	—
159	C₄²	#41×C₅₃	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	1	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
176	C₂×C₄	#123×C₅₃	1	2	yes	—
177	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
178	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
180	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	C₅₃×#399	1	1	yes	characteristic
193	C₂²	C₅₃×#476	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	C₅₃×#481	1	2	yes	—
198	C₄	C₅₃×#492	1	1	yes	characteristic
199	C₄	C₅₃×#496	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	C₅₃×#2091	1	1	yes	characteristic
203	C₂	C₅₃×#2094	1	1	yes	characteristic
204	C₂	C₅₃×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1955

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic, complemented

C₅₃×#1570

C₂

yes

maximal, complemented

C₅₃×#1486

C₂

yes

maximal

C₅₃×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₅₃×D₄⋊C₈

—

C₂²×D₄×C₅₃

Q₈

yes

characteristic

C₂³×C₄×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#10661

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#476

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#481

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1849

C₂×C₅₃

yes

characteristic, complemented

100

#1570

C₂×C₅₃

yes

complemented

101

#1486

C₂×C₅₃

yes

—

102

#1275

C₂×C₅₃

yes

characteristic, complemented

103

C₂×C₅₃

—

104

C₂×C₅₃

—

105

C₂×C₅₃

#2091

yes

characteristic

106

C₂×C₅₃

#2094

yes

characteristic

107

C₂×C₅₃

#2098

yes

characteristic

108

C₂×C₄×D₄

C₂²×C₅₃

yes

characteristic

109

#431

C₂²×C₅₃

yes

—

110

C₂×#108

—

111

#423

C₂²×C₅₃

yes

—

112

C₂×C₄⋊C₈

C₂²×C₅₃

yes

characteristic

113

C₂×C₂²⋊C₈

—

114

C₂×#97

C₂²×C₅₃

yes

characteristic

115

#334

—

116

D₄⋊C₈

—

117

C₅₃

#10661

yes

characteristic, direct product factor, Sylow

118

C₂²×D₄

Q₈×C₅₃

yes

characteristic

119

C₂³×C₄

—

120

C₂×M₁₆

—

121

C₂²×C₈

—

122

C₂²×C₈

D₄×C₅₃

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

Q₈×C₅₃

yes

characteristic

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂³×C₅₃

yes

characteristic

128

#108

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

135

#97

—

136

C₄×C₈

D₄×C₅₃

yes

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

146

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

147

C₂²×C₄

#41×C₅₃

yes

characteristic

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

SD₁₆×C₅₃

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×D₄×C₅₃

yes

—

159

C₄²

#41×C₅₃

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

C₂²⋊Q₈×C₅₃

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

176

C₂×C₄

#123×C₅₃

yes

—

177

C₂×C₄

#123×C₅₃

yes

characteristic

178

C₂×C₄

#127×C₅₃

yes

characteristic

179

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

180

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₅₃×#399

yes

characteristic

193

C₂²

C₅₃×#476

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₅₃×#481

yes

—

198

C₄

C₅₃×#492

yes

characteristic

199

C₄

C₅₃×#496

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

C₅₃×#2091

yes

characteristic

203

C₂

C₅₃×#2094

yes

characteristic

204

C₂

C₅₃×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#399
prime radical	C₂²×D₄×C₅₃	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#399

prime radical

C₂²×D₄×C₅₃

Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1849	C₂	semidirect
3. C₅₃×#1570	C₂	semidirect
5. C₅₃×#1275	C₂	semidirect
82. #10661	C₅₃	direct (N × H)
99. #1849	C₂×C₅₃	semidirect
100. #1570	C₂×C₅₃	semidirect
102. #1275	C₂×C₅₃	semidirect
117. C₅₃	#10661	direct (N × H)

H ≅ G/N

type

2. C₅₃×#1849

C₂

semidirect

3. C₅₃×#1570

C₂

semidirect

5. C₅₃×#1275

C₂

semidirect

82. #10661

C₅₃

direct (N × H)

99. #1849

C₂×C₅₃

semidirect

100. #1570

C₂×C₅₃

semidirect

102. #1275

C₂×C₅₃

semidirect

117. C₅₃

#10661

direct (N × H)

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

subgroup	index
C₅₃×#1955	2
C₅₃×#1849	2
C₅₃×#1570	2
C₅₃×#1486	2
C₅₃×#1275	2
#10661	53

subgroup

index

C₅₃×#1955

C₅₃×#1849

C₅₃×#1570

C₅₃×#1486

C₅₃×#1275

#10661

Subgroups — C₅₃×#10661 #27218173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10661 #27218173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10661 #27218173