Group #27215151 — subgroups

Subgroups — C₅₃×#7639 #27215151

Order 13568.

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

normal subgroups only

This group has 468 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	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1745	C₂	1	2	yes	maximal
3	C₅₃×#1272	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
6	C₂×#97×C₅₃	—	2	2	no	—
7	C₅₃×#391	C₂²	1	2	yes	—
8	C₅₃×#389	—	2	2	no	—
9	C₅₃×#384	C₄	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	2	no	—
13	C₅₃×#336	C₂²	1	4	yes	—
14	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₃	—	2	4	no	—
16	C₄⋊Q₈×C₅₃	—	2	4	no	—
17	C₄×Q₈×C₅₃	—	4	4	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#97×C₅₃	—	2	8	no	—
29	#95×C₅₃	—	2	4	no	—
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂×Q₈×C₅₃	—	4	2	no	—
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₅₃	—	2	8	no	—
42	C₂×C₈×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	2	8	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Q₈×C₅₃	—	4	4	no	—
56	Q₈×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	#100	1	2	yes	—
72	C₂×C₄×C₅₃	#104	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	4	yes	—
75	C₈×C₅₃	—	4	8	no	—
76	#7639	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	#407	1	1	yes	characteristic
78	C₂²×C₅₃	#410	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
80	C₂²×C₅₃	#419	1	2	yes	—
81	C₂²×C₅₃	C₂×#104	1	2	yes	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	—	4	1	no	—
88	C₄×C₅₃	—	8	2	no	—
89	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
90	#1745	C₂×C₅₃	1	2	yes	—
91	#1272	C₂×C₅₃	1	4	yes	—
92	C₂×C₅₃	#1466	1	1	yes	characteristic
93	C₂×C₅₃	#1491	1	2	yes	—
94	C₂×C₅₃	#1506	1	2	yes	—
95	C₂×C₅₃	#1525	1	1	yes	characteristic
96	C₂×C₅₃	#1558	1	1	yes	characteristic
97	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
99	C₂×#97	—	2	2	no	—
100	#391	C₂²×C₅₃	1	2	yes	—
101	#389	—	2	2	no	—
102	#384	C₄×C₅₃	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	2	no	—
104	C₄³	C₂²×C₅₃	1	1	yes	characteristic
105	#336	—	2	2	no	—
106	#336	C₂²×C₅₃	1	4	yes	—
107	C₅₃	#7639	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
109	C₂²×C₈	—	2	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
119	C₂×C₄²	D₄×C₅₃	1	2	yes	—
120	C₂×C₄²	D₄×C₅₃	1	2	yes	—
121	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
122	#97	—	2	8	no	—
123	#95	—	2	4	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
133	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	—	4	4	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	4	no	—
148	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
149	Q₈	—	4	4	no	—
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#100×C₅₃	1	2	yes	—
166	C₂×C₄	#104×C₅₃	1	4	yes	—
167	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
168	C₂×C₄	#137×C₅₃	1	4	yes	—
169	C₈	—	4	8	no	—
170	C₂²	C₅₃×#407	1	1	yes	characteristic
171	C₂²	C₅₃×#410	1	1	yes	characteristic
172	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
173	C₂²	C₅₃×#419	1	2	yes	—
174	C₂²	C₂×#104×C₅₃	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	4	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	—	4	1	no	—
181	C₄	—	8	2	no	—
182	C₂	C₅₃×#1466	1	1	yes	characteristic
183	C₂	C₅₃×#1491	1	2	yes	—
184	C₂	C₅₃×#1506	1	2	yes	—
185	C₂	C₅₃×#1525	1	1	yes	characteristic
186	C₂	C₅₃×#1558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#1745

C₂

yes

maximal

C₅₃×#1272

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×C₄×Q₈×C₅₃

C₄

yes

—

C₂×#97×C₅₃

—

C₅₃×#391

C₂²

yes

—

C₅₃×#389

—

C₅₃×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₅₃×#336

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

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

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

#7639

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₄⋊Q₈

C₂×C₅₃

yes

characteristic

#1745

C₂×C₅₃

yes

—

#1272

C₂×C₅₃

yes

—

C₂×C₅₃

#1466

yes

characteristic

C₂×C₅₃

#1491

yes

—

C₂×C₅₃

#1506

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

C₄×C₅₃

yes

characteristic

C₂×C₄×Q₈

C₄×C₅₃

yes

—

C₂×#97

—

100

#391

C₂²×C₅₃

yes

—

101

#389

—

102

#384

C₄×C₅₃

yes

characteristic

103

C₄×C₄⋊C₄

—

104

C₄³

C₂²×C₅₃

yes

characteristic

105

#336

—

106

#336

C₂²×C₅₃

yes

—

107

C₅₃

#7639

yes

characteristic, direct product factor, Sylow

108

C₂²×Q₈

C₂×C₄×C₅₃

yes

—

109

C₂²×C₈

—

110

C₄⋊Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

118

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

119

C₂×C₄²

D₄×C₅₃

yes

—

120

C₂×C₄²

D₄×C₅₃

yes

—

121

C₂×C₄²

C₂³×C₅₃

yes

characteristic

122

#97

—

123

#95

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

M₁₆×C₅₃

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

131

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

132

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

133

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

134

C₂²×C₄

C₂×D₄×C₅₃

yes

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

149

Q₈

—

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#100×C₅₃

yes

—

166

C₂×C₄

#104×C₅₃

yes

—

167

C₂×C₄

C₂×M₁₆×C₅₃

yes

—

168

C₂×C₄

#137×C₅₃

yes

—

169

C₈

—

170

C₂²

C₅₃×#407

yes

characteristic

171

C₂²

C₅₃×#410

yes

characteristic

172

C₂²

C₂×#100×C₅₃

yes

characteristic

173

C₂²

C₅₃×#419

yes

—

174

C₂²

C₂×#104×C₅₃

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

C₅₃×#1466

yes

characteristic

183

C₂

C₅₃×#1491

yes

—

184

C₂

C₅₃×#1506

yes

—

185

C₂

C₅₃×#1525

yes

characteristic

186

C₂

C₅₃×#1558

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

N	H ≅ G/N	type
76. #7639	C₅₃	direct (N × H)
107. C₅₃	#7639	direct (N × H)

H ≅ G/N

type

76. #7639

C₅₃

direct (N × H)

107. C₅₃

#7639

direct (N × H)

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1745	2
C₅₃×#1272	2
#7639	53

subgroup

index

C₄×C₄⋊Q₈×C₅₃

C₅₃×#1745

C₅₃×#1272

#7639

Subgroups — C₅₃×#7639 #27215151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7639 #27215151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7639 #27215151