Group #27220384 — subgroups

Subgroups — C₅₃×#12872 #27220384

Order 13568.

398 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1810	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1387	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#117×C₅₃	—	2	1	no	—
7	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	—	2	2	no	—
10	C₅₃×#427	C₄	1	4	yes	—
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×#106×C₅₃	—	2	1	no	—
13	C₅₃×#407	—	2	1	no	—
14	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
15	C₅₃×#341	—	2	1	no	—
16	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	#117×C₅₃	—	2	4	no	—
24	#117×C₅₃	—	4	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
27	C₂×C₄²×C₅₃	—	4	1	no	—
28	M₃₂×C₅₃	—	2	4	no	—
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	#108×C₅₃	—	2	4	no	—
31	#107×C₅₃	—	2	2	no	—
32	#107×C₅₃	C₂×C₄	1	4	yes	—
33	#106×C₅₃	—	2	4	no	—
34	C₂²⋊C₈×C₅₃	—	4	2	no	—
35	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₅₃	—	4	2	no	—
44	M₁₆×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₂²⋊C₄×C₅₃	—	4	4	no	—
56	C₄²×C₅₃	—	4	4	no	—
57	C₁₆×C₅₃	—	4	2	no	—
58	C₂³×C₅₃	—	2	2	no	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	#102	1	2	yes	—
61	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	#102	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#123	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#137	1	2	yes	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	—	8	2	no	—
75	C₈×C₅₃	#117	1	2	yes	—
76	C₈×C₅₃	#123	1	2	yes	—
77	#12872	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	4	4	no	—
82	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
83	C₂²×C₅₃	#483	1	2	yes	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#381	1	1	yes	characteristic
87	C₄×C₅₃	#419	1	1	yes	characteristic
88	C₄×C₅₃	#480	1	2	yes	—
89	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
90	#2112	C₂×C₅₃	1	1	yes	characteristic
91	#1810	C₂×C₅₃	1	1	yes	characteristic
92	#1387	C₂×C₅₃	1	4	yes	complemented
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	—	4	2	no	—
95	C₂×C₅₃	#1823	1	1	yes	characteristic
96	C₂×C₅₃	#2148	1	2	yes	—
97	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
98	C₂×#117	—	2	1	no	—
99	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
100	C₂×#108	C₂²×C₅₃	1	2	yes	—
101	#427	—	2	2	no	—
102	#427	C₄×C₅₃	1	4	yes	—
103	C₂×#107	C₂²×C₅₃	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	#407	—	2	1	no	—
106	C₂²⋊C₁₆	—	2	2	no	—
107	#341	—	2	1	no	—
108	C₅₃	#12872	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
113	C₂²×C₈	D₄×C₅₃	1	2	yes	—
114	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
116	#117	—	2	4	no	—
117	#117	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₄²	—	4	1	no	—
121	M₃₂	—	2	4	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#108	—	2	4	no	—
124	#107	—	2	2	no	—
125	#107	C₂×C₄×C₅₃	1	4	yes	—
126	#106	—	2	4	no	—
127	C₂²⋊C₈	—	4	2	no	—
128	C₂⁴	SD₁₆×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
133	C₂²×C₄	D₈×C₅₃	1	2	yes	—
134	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
145	C₂×C₈	#41×C₅₃	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	4	no	—
149	C₄²	—	4	4	no	—
150	C₁₆	—	4	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	#102×C₅₃	1	2	yes	—
154	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	#102×C₅₃	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄	#123×C₅₃	1	2	yes	—
164	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
165	C₂×C₄	#137×C₅₃	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	#117×C₅₃	1	2	yes	—
169	C₈	#123×C₅₃	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
175	C₂²	C₅₃×#483	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	8	4	no	—
178	C₄	C₅₃×#381	1	1	yes	characteristic
179	C₄	C₅₃×#419	1	1	yes	characteristic
180	C₄	C₅₃×#480	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	C₅₃×#1823	1	1	yes	characteristic
184	C₂	C₅₃×#2148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#427

C₂

yes

maximal, characteristic

C₅₃×#2112

C₂

yes

maximal, characteristic

C₅₃×#1810

C₂

yes

maximal, characteristic

C₅₃×#1387

C₂

yes

maximal, complemented

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×#117×C₅₃

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#427

—

C₅₃×#427

C₄

yes

—

C₂×#107×C₅₃

C₂²

yes

—

C₂×#106×C₅₃

—

C₅₃×#407

—

C₅₃×C₂²⋊C₁₆

—

C₅₃×#341

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#108×C₅₃

—

#107×C₅₃

—

#107×C₅₃

C₂×C₄

yes

—

#106×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×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

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#123

yes

—

#12872

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#102

yes

characteristic

C₂²×C₅₃

#483

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#427

C₂×C₅₃

yes

characteristic

#2112

C₂×C₅₃

yes

characteristic

#1810

C₂×C₅₃

yes

characteristic

#1387

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1823

yes

characteristic

C₂×C₅₃

#2148

yes

—

C₂³×C₈

C₂²×C₅₃

yes

characteristic

C₂×#117

—

C₂×M₃₂

C₂²×C₅₃

yes

—

100

C₂×#108

C₂²×C₅₃

yes

—

101

#427

—

102

#427

C₄×C₅₃

yes

—

103

C₂×#107

C₂²×C₅₃

yes

—

104

C₂×#106

—

105

#407

—

106

C₂²⋊C₁₆

—

107

#341

—

108

C₅₃

#12872

yes

characteristic, direct product factor, Sylow

109

C₂³×C₄

D₄×C₅₃

yes

characteristic

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂×C₄×C₅₃

yes

—

113

C₂²×C₈

D₄×C₅₃

yes

—

114

C₂²×C₈

D₄×C₅₃

yes

characteristic

115

C₂²×C₈

C₂³×C₅₃

yes

characteristic

116

#117

—

117

#117

—

118

C₂×C₄⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₄²

—

121

M₃₂

—

122

C₂×C₁₆

—

123

#108

—

124

#107

—

125

#107

C₂×C₄×C₅₃

yes

—

126

#106

—

127

C₂²⋊C₈

—

128

C₂⁴

SD₁₆×C₅₃

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

133

C₂²×C₄

D₈×C₅₃

yes

—

134

C₂²×C₄

SD₁₆×C₅₃

yes

characteristic

135

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

143

C₂×C₈

C₂²×C₄×C₅₃

yes

characteristic

144

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

145

C₂×C₈

#41×C₅₃

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₁₆

—

151

C₂³

—

152

C₂³

—

153

C₂³

#102×C₅₃

yes

—

154

C₂³

C₂×SD₁₆×C₅₃

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#102×C₅₃

yes

—

162

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

163

C₂×C₄

#123×C₅₃

yes

—

164

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

165

C₂×C₄

#137×C₅₃

yes

—

166

C₈

—

167

C₈

—

168

C₈

#117×C₅₃

yes

—

169

C₈

#123×C₅₃

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#102×C₅₃

yes

characteristic

175

C₂²

C₅₃×#483

yes

—

176

C₄

—

177

C₄

—

178

C₄

C₅₃×#381

yes

characteristic

179

C₄

C₅₃×#419

yes

characteristic

180

C₄

C₅₃×#480

yes

—

181

C₂

—

182

C₂

—

183

C₂

C₅₃×#1823

yes

characteristic

184

C₂

C₅₃×#2148

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#1387	C₂	semidirect
77. #12872	C₅₃	direct (N × H)
92. #1387	C₂×C₅₃	semidirect
108. C₅₃	#12872	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1387

C₂

semidirect

77. #12872

C₅₃

direct (N × H)

92. #1387

C₂×C₅₃

semidirect

108. C₅₃

#12872

direct (N × H)

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

subgroup	index
C₂×C₅₃×#427	2
C₅₃×#2112	2
C₅₃×#1810	2
C₅₃×#1387	2
#12872	53

subgroup

index

C₂×C₅₃×#427

C₅₃×#2112

C₅₃×#1810

C₅₃×#1387

#12872

Subgroups — C₅₃×#12872 #27220384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12872 #27220384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12872 #27220384