Group #20287688 — subgroups

Subgroups — #20287688

Order 6784.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811293	C₂	1	2	yes	maximal, complemented
2	#16810998	C₂	1	2	yes	maximal, complemented
3	#16810816	C₂	1	4	yes	maximal, complemented
4	#16810815	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719947	—	2	2	no	—
10	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
11	C₂×#11719922	C₂²	1	4	yes	—
12	#13133213	C₂²	1	4	yes	—
13	#13133189	—	2	4	no	—
14	#13133188	—	2	2	no	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄×D₂₁₂	C₂²	1	2	yes	—
17	D₄×Dic₅₃	C₂²	1	4	yes	—
18	#13133151	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133139	C₂²	1	2	yes	—
22	#13133138	C₂²	1	4	yes	—
23	C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₂×D₈×C₅₃	C₂²	1	2	yes	—
25	C₂×D₈×C₅₃	C₂²	1	2	yes	—
26	#127×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×D₄×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	—	2	2	no	—
30	C₂×D₄×C₅₃	C₂³	1	4	yes	—
31	C₂×C₂²⋊D₅₃	—	2	4	no	—
32	C₂×C₂²⋊D₅₃	—	2	4	no	—
33	C₂²×Dic₅₃	—	2	4	no	—
34	#11719947	—	4	8	no	—
35	C₂×D₂₁₂	C₂³	1	2	yes	—
36	C₂×C₄×D₅₃	—	2	2	no	—
37	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
38	D₈×C₅₃	—	2	4	no	—
39	D₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₃	—	2	4	no	—
44	#11719922	—	4	4	no	—
45	#11719921	—	2	2	no	—
46	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
47	C₄⋊Dic₅₃	C₂³	1	2	yes	—
48	#11719919	D₄	1	2	yes	—
49	C₄×Dic₅₃	C₂³	1	2	yes	—
50	C₂×#88322	C₂³	1	2	yes	—
51	D₄₂₄	—	2	2	no	—
52	C₂³×C₅₃	—	2	4	no	—
53	C₂²×D₅₃	—	2	2	no	—
54	D₄×C₅₃	—	2	4	no	—
55	D₄×C₅₃	—	2	4	no	—
56	D₄×C₅₃	—	4	4	no	—
57	D₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₅₃	—	4	8	no	—
62	C₂²⋊D₅₃	—	8	4	no	—
63	C₂×Dic₅₃	—	2	2	no	—
64	C₂×Dic₅₃	—	4	8	no	—
65	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
66	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
67	D₂₁₂	—	2	2	no	—
68	D₂₁₂	—	2	2	no	—
69	C₄×D₅₃	—	4	4	no	—
70	Dic₁₀₆	D₈	1	4	yes	complemented
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	#41	1	2	yes	complemented
73	#88322	—	2	2	no	—
74	C₂²×C₅₃	—	4	4	no	—
75	C₂²×C₅₃	—	4	4	no	—
76	C₂²×C₅₃	—	4	4	no	—
77	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×D₅₃	—	4	2	no	—
79	C₂×D₅₃	—	4	2	no	—
80	C₂×D₅₃	—	4	2	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	C₂×D₈	1	2	yes	—
83	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
84	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
85	Dic₅₃	—	2	2	no	—
86	Dic₅₃	—	4	2	no	—
87	Dic₅₃	C₂×D₈	1	4	yes	—
88	#3391	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	—	8	4	no	—
90	C₂×C₅₃	#550	1	1	yes	characteristic
91	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
92	C₂×C₅₃	#576	1	1	yes	characteristic
93	D₅₃	—	8	2	no	—
94	#550	—	53	2	no	—
95	#493	—	53	2	no	—
96	#493	D₅₃	1	1	yes	characteristic, complemented
97	#459	—	53	4	no	—
98	#459	—	53	2	no	—
99	C₈×Q₈	—	53	1	no	—
100	C₄×D₈	—	53	2	no	—
101	C₄×D₈	—	53	1	no	—
102	C₅₃	#3391	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	106	2	no	—
104	C₂×D₈	—	53	4	no	—
105	C₂×D₈	—	53	1	no	—
106	C₂×D₈	C₂×D₅₃	1	2	yes	—
107	C₂×D₈	C₂×D₅₃	1	2	yes	—
108	#127	—	53	4	no	—
109	#127	C₂×D₅₃	1	2	yes	—
110	#121	—	106	2	no	—
111	#121	—	106	4	no	—
112	C₄×Q₈	—	53	1	no	—
113	C₄×D₄	—	53	4	no	—
114	C₄×D₄	—	53	2	no	—
115	#107	—	53	1	no	—
116	C₄⋊C₈	—	53	2	no	—
117	C₄⋊C₈	—	53	1	no	—
118	#102	—	53	4	no	—
119	#102	—	53	2	no	—
120	C₄×C₈	—	53	2	no	—
121	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	#41	—	212	8	no	—
123	C₂×Q₈	—	53	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	53	2	no	—
127	C₂×D₄	—	106	4	no	—
128	C₂×D₄	—	106	4	no	—
129	C₂×D₄	C₂²×D₅₃	1	4	yes	—
130	C₂²×C₄	—	106	4	no	—
131	C₂²×C₄	—	106	2	no	—
132	D₈	—	2	4	no	—
133	D₈	—	4	2	no	—
134	D₈	—	106	2	no	—
135	D₈	—	212	4	no	—
136	C₂×C₈	—	53	2	no	—
137	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
138	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
139	C₄⋊C₄	—	53	2	no	—
140	C₄⋊C₄	—	53	1	no	—
141	C₄⋊C₄	—	53	2	no	—
142	C₂²⋊C₄	—	106	4	no	—
143	C₂²⋊C₄	—	106	2	no	—
144	C₄²	—	53	2	no	—
145	C₄²	C₂²×D₅₃	1	1	yes	characteristic
146	C₂³	—	2	4	no	—
147	C₂³	—	106	2	no	—
148	Q₈	—	53	4	no	—
149	D₄	—	2	4	no	—
150	D₄	—	2	4	no	—
151	D₄	—	4	4	no	—
152	D₄	—	8	2	no	—
153	D₄	—	106	2	no	—
154	D₄	—	106	2	no	—
155	D₄	—	424	4	no	—
156	D₄	—	212	8	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	53	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	212	8	no	—
161	C₂×C₄	—	212	4	no	—
162	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	106	2	no	—
167	C₈	#11719947	1	2	yes	complemented
168	C₂²	—	4	4	no	—
169	C₂²	—	4	4	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	212	2	no	—
172	C₂²	—	212	2	no	—
173	C₂²	—	212	2	no	—
174	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	53	4	no	—
177	C₄	—	106	2	no	—
178	C₄	—	212	2	no	—
179	C₄	D₈×D₅₃	1	2	yes	—
180	C₄	C₂×#11719947	1	1	yes	characteristic
181	C₄	#13133262	1	1	yes	characteristic
182	C₂	—	8	4	no	—
183	C₂	—	424	2	no	—
184	C₂	#16811293	1	1	yes	characteristic
185	C₂	C₂×D₈×D₅₃	1	1	yes	characteristic
186	C₂	#16811424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811293

C₂

yes

maximal, complemented

#16810998

C₂

yes

maximal, complemented

#16810816

C₂

yes

maximal, complemented

#16810815

C₂

yes

maximal, complemented

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₃

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×#11719922

C₂²

yes

—

#13133213

C₂²

yes

—

#13133189

—

#13133188

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

D₄×Dic₅₃

C₂²

yes

—

#13133151

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133139

C₂²

yes

—

#13133138

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×D₈×C₅₃

C₂²

yes

—

C₂×D₈×C₅₃

C₂²

yes

—

#127×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719947

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719922

—

#11719921

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

D₄₂₄

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

D₈

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

#41

yes

complemented

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

—

#3391

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#550

yes

characteristic

C₂×C₅₃

C₂²×D₈

yes

characteristic

C₂×C₅₃

#576

yes

characteristic

D₅₃

—

#550

—

#493

—

#493

D₅₃

yes

characteristic, complemented

#459

—

#459

—

C₈×Q₈

—

100

C₄×D₈

—

101

C₄×D₈

—

102

C₅₃

#3391

yes

characteristic, complemented, Sylow

103

C₂×#41

—

106

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

C₂×D₅₃

yes

—

107

C₂×D₈

C₂×D₅₃

yes

—

108

#127

—

109

#127

C₂×D₅₃

yes

—

110

#121

—

106

—

111

#121

—

106

—

112

C₄×Q₈

—

113

C₄×D₄

—

114

C₄×D₄

—

115

#107

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#102

—

119

#102

—

120

C₄×C₈

—

121

C₄×C₈

C₂×D₅₃

yes

characteristic

122

#41

—

212

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

106

—

128

C₂×D₄

—

106

—

129

C₂×D₄

C₂²×D₅₃

yes

—

130

C₂²×C₄

—

106

—

131

C₂²×C₄

—

106

—

132

D₈

—

133

D₈

—

134

D₈

—

106

—

135

D₈

—

212

—

136

C₂×C₈

—

137

C₂×C₈

C₂²×D₅₃

yes

characteristic

138

C₂×C₈

C₂²×D₅₃

yes

characteristic

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

106

—

143

C₂²⋊C₄

—

106

—

144

C₄²

—

145

C₄²

C₂²×D₅₃

yes

characteristic

146

C₂³

—

147

C₂³

—

106

—

148

Q₈

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

106

—

154

D₄

—

106

—

155

D₄

—

424

—

156

D₄

—

212

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

212

—

161

C₂×C₄

—

212

—

162

C₂×C₄

D₄×D₅₃

yes

characteristic

163

C₂×C₄

D₄×D₅₃

yes

characteristic

164

C₂×C₄

C₂³×D₅₃

yes

characteristic

165

C₈

—

166

C₈

—

106

—

167

C₈

#11719947

yes

complemented

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

212

—

172

C₂²

—

212

—

173

C₂²

—

212

—

174

C₂²

C₂×D₄×D₅₃

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

106

—

178

C₄

—

212

—

179

C₄

D₈×D₅₃

yes

—

180

C₄

C₂×#11719947

yes

characteristic

181

C₄

#13133262

yes

characteristic

182

C₂

—

183

C₂

—

424

—

184

C₂

#16811293

yes

characteristic

185

C₂

C₂×D₈×D₅₃

yes

characteristic

186

C₂

#16811424

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#493	D₅₃
hypocenter	C₅₃	#3391
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#493

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16811293	C₂	semidirect
2. #16810998	C₂	semidirect
3. #16810816	C₂	semidirect
4. #16810815	C₂	semidirect
5. C₈×Dic₁₀₆	C₂	semidirect
6. C₄×D₄₂₄	C₂	semidirect
7. D₈×Dic₅₃	C₂	semidirect
8. C₅₃×#493	C₂	semidirect
70. Dic₁₀₆	D₈	semidirect
72. C₈×C₅₃	#41	semidirect
96. #493	D₅₃	semidirect
102. C₅₃	#3391	semidirect
167. C₈	#11719947	semidirect

H ≅ G/N

type

1. #16811293

C₂

semidirect

2. #16810998

C₂

semidirect

3. #16810816

C₂

semidirect

4. #16810815

C₂

semidirect

5. C₈×Dic₁₀₆

C₂

semidirect

6. C₄×D₄₂₄

C₂

semidirect

7. D₈×Dic₅₃

C₂

semidirect

8. C₅₃×#493

C₂

semidirect

70. Dic₁₀₆

D₈

semidirect

72. C₈×C₅₃

#41

semidirect

96. #493

D₅₃

semidirect

102. C₅₃

#3391

semidirect

167. C₈

#11719947

semidirect

Subgroups — #20287688

Conjugacy classes of subgroups

Complements