Group #20279669 — subgroups

Subgroups — #20279669

Order 6784.

676 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 189 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×#13133193	C₂	1	4	yes	maximal, complemented
2	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810558	C₂	1	4	yes	maximal, complemented
4	#16810524	C₂	1	2	yes	maximal
5	#16810492	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
10	#13133193	—	2	16	no	—
11	C₄×D₂₁₂	—	2	4	no	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×#11719919	C₂²	1	8	yes	—
15	C₂×#11719921	C₂²	1	8	yes	—
16	C₂×#11719921	C₂²	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
18	#13133124	C₂²	1	2	yes	—
19	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₅₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂³×D₅₃	C₂³	1	2	yes	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	—	2	8	no	—
27	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂³	1	2	yes	—
32	C₂²×Dic₅₃	C₂³	1	4	yes	—
33	C₂×D₂₁₂	—	2	2	no	—
34	C₂×D₂₁₂	—	2	2	no	—
35	C₂×D₂₁₂	Q₈	1	4	yes	complemented
36	C₂×C₄×D₅₃	—	2	4	no	—
37	C₂×C₄×D₅₃	—	2	8	no	—
38	C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	D₄	1	4	yes	—
41	#11719921	—	2	4	no	—
42	#11719921	—	2	16	no	—
43	C₄⋊Dic₅₃	—	2	2	no	—
44	C₄⋊Dic₅₃	D₄	1	8	yes	complemented
45	#11719919	—	2	16	no	—
46	C₄×Dic₅₃	—	2	4	no	—
47	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂²×D₅₃	—	2	2	no	—
49	C₂²×D₅₃	—	2	4	no	—
50	C₂²×D₅₃	—	2	2	no	—
51	C₂²×D₅₃	—	2	4	no	—
52	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
53	Q₈×C₅₃	—	4	8	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₅₃	#41	1	2	yes	—
63	C₂×Dic₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	2	4	no	—
65	C₂×Dic₅₃	—	2	4	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	C₂×D₄	1	8	yes	—
68	D₂₁₂	—	2	8	no	—
69	C₄×D₅₃	—	4	8	no	—
70	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
71	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
75	C₂×D₅₃	—	2	4	no	—
76	C₂×D₅₃	—	2	8	no	—
77	C₂×D₅₃	—	4	2	no	—
78	C₂×D₅₃	—	4	2	no	—
79	C₂×D₅₃	—	4	4	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	4	4	no	—
82	C₄×C₅₃	C₂²⋊Q₈	1	4	yes	complemented
83	C₄×C₅₃	#127	1	4	yes	—
84	Dic₅₃	—	2	8	no	—
85	Dic₅₃	—	4	2	no	—
86	#2542	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
89	C₂×C₅₃	#536	1	1	yes	characteristic
90	C₂×C₅₃	#547	1	2	yes	—
91	C₂×C₅₃	D₄×Q₈	1	2	yes	—
92	D₅₃	—	4	4	no	—
93	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊Q₈	—	53	4	no	—
95	C₂×C₄×D₄	—	53	1	no	—
96	#399	—	53	4	no	—
97	#390	—	53	2	no	—
98	#384	—	53	2	no	—
99	C₄×C₄⋊C₄	—	53	1	no	—
100	C₅₃	#2542	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
102	C₂²×D₄	—	53	1	no	—
103	C₂³×C₄	—	53	2	no	—
104	C₄⋊Q₈	—	2	4	no	—
105	C₂²⋊Q₈	—	106	16	no	—
106	C₄×D₄	—	106	4	no	—
107	C₂×C₄⋊C₄	—	53	2	no	—
108	C₂×C₄⋊C₄	—	53	1	no	—
109	C₂×C₄⋊C₄	—	53	8	no	—
110	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
112	C₂×C₂²⋊C₄	—	53	2	no	—
113	C₂×C₂²⋊C₄	—	53	8	no	—
114	C₂×C₄²	—	53	2	no	—
115	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
116	#95	—	53	2	no	—
117	C₂⁴	—	53	2	no	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×D₄	—	53	4	no	—
121	C₂×D₄	—	106	2	no	—
122	C₂×D₄	—	106	2	no	—
123	C₂²×C₄	—	53	4	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	—	106	8	no	—
126	C₂²×C₄	—	106	4	no	—
127	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
128	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	53	8	no	—
134	C₄⋊C₄	—	106	16	no	—
135	C₄⋊C₄	—	106	2	no	—
136	C₂²⋊C₄	—	106	16	no	—
137	C₂²⋊C₄	—	106	4	no	—
138	C₄²	—	106	4	no	—
139	C₄²	C₂²⋊D₅₃	1	4	yes	—
140	C₂³	—	53	4	no	—
141	C₂³	—	106	4	no	—
142	C₂³	—	106	2	no	—
143	C₂³	—	106	2	no	—
144	C₂³	—	106	4	no	—
145	C₂³	C₂³×D₅₃	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	D₄	—	106	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	53	8	no	—
152	C₂×C₄	—	106	4	no	—
153	C₂×C₄	—	106	4	no	—
154	C₂×C₄	—	106	4	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	212	8	no	—
157	C₂×C₄	D₄×D₅₃	1	4	yes	—
158	C₂×C₄	Q₈×D₅₃	1	2	yes	—
159	C₂×C₄	#11719949	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
163	C₂²	—	106	8	no	—
164	C₂²	—	106	4	no	—
165	C₂²	—	212	4	no	—
166	C₂²	—	212	2	no	—
167	C₂²	—	212	2	no	—
168	C₂²	C₂×D₄×D₅₃	1	2	yes	—
169	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
171	C₂²	C₂×#11719949	1	1	yes	characteristic
172	C₂²	#13133265	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	4	4	no	—
175	C₄	—	106	8	no	—
176	C₄	—	212	2	no	—
177	C₄	#13133193	1	4	yes	complemented
178	C₄	#13133213	1	4	yes	—
179	C₂	—	212	4	no	—
180	C₂	C₂×#13133193	1	1	yes	characteristic
181	C₂	C₂×#13133213	1	1	yes	characteristic
182	C₂	#16811197	1	1	yes	characteristic
183	C₂	#16811276	1	2	yes	—
184	C₂	#16811288	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133193

C₂

yes

maximal, complemented

C₂×C₄×D₂₁₂

C₂

yes

maximal, characteristic

#16810558

C₂

yes

maximal, complemented

#16810524

C₂

yes

maximal

#16810492

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂²×C₄×D₅₃

C₂²

yes

—

#13133193

—

C₄×D₂₁₂

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

Q₈

yes

complemented

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

complemented

#11719919

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×Q₈

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

complemented

C₄×C₅₃

#127

yes

—

Dic₅₃

—

Dic₅₃

—

#2542

—

maximal, Sylow

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

D₄×Q₈

yes

—

D₅₃

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

—

#399

—

#390

—

#384

—

C₄×C₄⋊C₄

—

100

C₅₃

#2542

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

C₂×D₅₃

yes

—

102

C₂²×D₄

—

103

C₂³×C₄

—

104

C₄⋊Q₈

—

105

C₂²⋊Q₈

—

106

—

106

C₄×D₄

—

106

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

111

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₅₃

yes

characteristic

116

#95

—

117

C₂⁴

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

106

—

122

C₂×D₄

—

106

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

106

—

126

C₂²×C₄

—

106

—

127

C₂²×C₄

C₂²×D₅₃

yes

—

128

C₂²×C₄

C₂²×D₅₃

yes

characteristic

129

C₂²×C₄

C₂²×D₅₃

yes

characteristic

130

C₂²×C₄

C₂²×D₅₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₂²⋊C₄

—

106

—

137

C₂²⋊C₄

—

106

—

138

C₄²

—

106

—

139

C₄²

C₂²⋊D₅₃

yes

—

140

C₂³

—

141

C₂³

—

106

—

142

C₂³

—

106

—

143

C₂³

—

106

—

144

C₂³

—

106

—

145

C₂³

C₂³×D₅₃

yes

characteristic

146

Q₈

—

147

D₄

—

106

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

212

—

157

C₂×C₄

D₄×D₅₃

yes

—

158

C₂×C₄

Q₈×D₅₃

yes

—

159

C₂×C₄

#11719949

yes

—

160

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

161

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

162

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

163

C₂²

—

106

—

164

C₂²

—

106

—

165

C₂²

—

212

—

166

C₂²

—

212

—

167

C₂²

—

212

—

168

C₂²

C₂×D₄×D₅₃

yes

—

169

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

170

C₂²

C₂×Q₈×D₅₃

yes

characteristic

171

C₂²

C₂×#11719949

yes

characteristic

172

C₂²

#13133265

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

106

—

176

C₄

—

212

—

177

C₄

#13133193

yes

complemented

178

C₄

#13133213

yes

—

179

C₂

—

212

—

180

C₂

C₂×#13133193

yes

characteristic

181

C₂

C₂×#13133213

yes

characteristic

182

C₂

#16811197

yes

characteristic

183

C₂

#16811276

yes

—

184

C₂

#16811288

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2542
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#13133193	C₂	semidirect
3. #16810558	C₂	semidirect
5. #16810492	C₂	semidirect
6. C₄×C₄⋊Dic₅₃	C₂	semidirect
7. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
35. C₂×D₂₁₂	Q₈	semidirect
44. C₄⋊Dic₅₃	D₄	semidirect
82. C₄×C₅₃	C₂²⋊Q₈	semidirect
93. C₂×C₄⋊Q₈	D₅₃	semidirect
100. C₅₃	#2542	semidirect
177. C₄	#13133193	semidirect

H ≅ G/N

type

1. C₂×#13133193

C₂

semidirect

3. #16810558

C₂

semidirect

5. #16810492

C₂

semidirect

6. C₄×C₄⋊Dic₅₃

C₂

semidirect

7. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

35. C₂×D₂₁₂

Q₈

semidirect

44. C₄⋊Dic₅₃

D₄

semidirect

82. C₄×C₅₃

C₂²⋊Q₈

semidirect

93. C₂×C₄⋊Q₈

D₅₃

semidirect

100. C₅₃

#2542

semidirect

177. C₄

#13133193

semidirect

Subgroups — #20279669

Conjugacy classes of subgroups

Complements