Group #20285868 — subgroups

Subgroups — #20285868

Order 6784.

360 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133217	C₂	1	1	yes	maximal, characteristic
3	#16811028	C₂	1	2	yes	maximal, complemented
4	#16810974	C₂	1	2	yes	maximal, complemented
5	#16810959	C₂	1	4	yes	maximal
6	#16810919	C₂	1	4	yes	maximal, complemented
7	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	—	2	1	no	—
9	#13133217	—	2	1	no	—
10	#13133217	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133217	C₂²	1	2	yes	—
13	#13133217	C₂²	1	2	yes	—
14	#13133217	C₂²	1	1	yes	characteristic
15	#13133209	—	2	1	no	—
16	#13133196	—	2	1	no	—
17	#13133196	C₂²	1	2	yes	—
18	C₄×Dic₁₀₆	—	2	1	no	—
19	C₄×Dic₁₀₆	C₂²	1	2	yes	—
20	#13133176	—	2	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	#13133151	C₂²	1	8	yes	—
24	#13133142	C₂²	1	8	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
28	C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	2	2	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	C₂×Dic₁₀₆	—	2	2	no	—
35	C₂×Dic₁₀₆	—	2	2	no	—
36	C₂×Dic₁₀₆	—	4	4	no	—
37	C₂×Dic₁₀₆	C₂³	1	2	yes	—
38	C₂×C₈×C₅₃	C₂³	1	4	yes	—
39	C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₃	—	2	2	no	—
44	C₄⋊Dic₅₃	—	2	2	no	—
45	C₄⋊Dic₅₃	—	2	2	no	—
46	C₄⋊Dic₅₃	—	2	2	no	—
47	C₄⋊Dic₅₃	—	2	1	no	—
48	C₄⋊Dic₅₃	—	2	1	no	—
49	C₄⋊Dic₅₃	—	2	1	no	—
50	C₄⋊Dic₅₃	C₂³	1	4	yes	—
51	C₄⋊Dic₅₃	C₂³	1	2	yes	—
52	#11719919	—	2	2	no	—
53	#11719919	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	Q₁₆	1	4	yes	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₅₃	—	2	4	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	4	4	no	—
69	Dic₁₀₆	—	2	4	no	—
70	Dic₁₀₆	—	4	4	no	—
71	Dic₁₀₆	—	4	2	no	—
72	Dic₁₀₆	—	4	4	no	—
73	C₈×C₅₃	—	2	4	no	—
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
77	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
81	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	—	4	4	no	—
86	Dic₅₃	—	4	2	no	—
87	#3230	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	#536	1	1	yes	characteristic
90	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
91	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
92	#533	—	53	1	no	—
93	C₂×C₄⋊Q₈	—	53	1	no	—
94	#500	—	53	2	no	—
95	#487	—	53	2	no	—
96	#484	—	53	4	no	—
97	#477	—	53	4	no	—
98	#431	D₅₃	1	1	yes	characteristic
99	C₅₃	#3230	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	—	106	1	no	—
101	C₄⋊Q₈	—	53	2	no	—
102	C₄⋊Q₈	—	53	2	no	—
103	C₄⋊Q₈	—	53	1	no	—
104	C₄⋊Q₈	—	53	1	no	—
105	C₄⋊Q₈	—	106	1	no	—
106	C₄⋊Q₈	—	106	1	no	—
107	#125	—	106	1	no	—
108	C₂²⋊Q₈	—	53	2	no	—
109	C₂²⋊Q₈	—	106	1	no	—
110	C₄×Q₈	—	53	2	no	—
111	C₄×Q₈	—	106	1	no	—
112	#117	—	106	1	no	—
113	C₂×C₄⋊C₄	—	53	2	no	—
114	C₂×C₄⋊C₄	—	106	1	no	—
115	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
116	#107	—	53	8	no	—
117	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
118	#103	—	53	8	no	—
119	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
120	C₄×C₈	C₂×D₅₃	1	2	yes	—
121	C₂×Q₈	—	53	2	no	—
122	C₂×Q₈	—	106	1	no	—
123	C₂×Q₈	—	106	2	no	—
124	C₂×Q₈	—	106	2	no	—
125	C₂×Q₈	—	212	4	no	—
126	C₂²×C₄	—	106	2	no	—
127	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
128	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₅₃	1	4	yes	—
131	C₄⋊C₄	—	53	4	no	—
132	C₄⋊C₄	—	53	2	no	—
133	C₄⋊C₄	—	106	2	no	—
134	C₄⋊C₄	—	106	2	no	—
135	C₄⋊C₄	—	106	2	no	—
136	C₄⋊C₄	—	106	1	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₄⋊C₄	—	106	2	no	—
139	C₄⋊C₄	—	106	2	no	—
140	C₄⋊C₄	—	106	1	no	—
141	C₂²⋊C₄	—	106	2	no	—
142	C₄²	—	106	2	no	—
143	C₄²	D₂₁₂	1	1	yes	characteristic
144	C₄²	D₂₁₂	1	1	yes	characteristic
145	C₄²	C₂²×D₅₃	1	1	yes	characteristic
146	C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
148	Q₈	—	106	4	no	—
149	Q₈	—	212	4	no	—
150	Q₈	—	212	2	no	—
151	Q₈	—	212	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	106	2	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	4	no	—
157	C₂×C₄	—	212	4	no	—
158	C₂×C₄	Dic₂₁₂	1	4	yes	—
159	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
161	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
162	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
163	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₂×Dic₂₁₂	1	2	yes	—
169	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	212	4	no	—
173	C₄	—	212	2	no	—
174	C₄	—	212	2	no	—
175	C₄	C₂×Dic₂₁₂	1	2	yes	—
176	C₄	C₂×Dic₂₁₂	1	2	yes	—
177	C₄	#13133245	1	2	yes	—
178	C₄	#13133264	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#16811199	1	1	yes	characteristic
181	C₂	C₂²×Dic₂₁₂	1	1	yes	characteristic
182	C₂	C₂×#13133245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

C₂×#13133217

C₂

yes

maximal, characteristic

#16811028

C₂

yes

maximal, complemented

#16810974

C₂

yes

maximal, complemented

#16810959

C₂

yes

maximal

#16810919

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

#13133217

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133151

C₂²

yes

—

#13133142

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#3230

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#536

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

C₂×C₅₃

yes

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#3230

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

—

106

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

106

—

106

C₄⋊Q₈

—

106

—

107

#125

—

106

—

108

C₂²⋊Q₈

—

109

C₂²⋊Q₈

—

106

—

110

C₄×Q₈

—

111

C₄×Q₈

—

106

—

112

#117

—

106

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

106

—

115

C₂×C₄²

C₂×D₅₃

yes

characteristic

116

#107

—

117

C₄⋊C₈

C₂×D₅₃

yes

—

118

#103

—

119

C₂²⋊C₈

C₂×D₅₃

yes

—

120

C₄×C₈

C₂×D₅₃

yes

—

121

C₂×Q₈

—

122

C₂×Q₈

—

106

—

123

C₂×Q₈

—

106

—

124

C₂×Q₈

—

106

—

125

C₂×Q₈

—

212

—

126

C₂²×C₄

—

106

—

127

C₂²×C₄

D₂₁₂

yes

characteristic

128

C₂²×C₄

D₂₁₂

yes

characteristic

129

C₂²×C₄

C₂²×D₅₃

yes

characteristic

130

C₂×C₈

C₂²×D₅₃

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

—

106

—

140

C₄⋊C₄

—

106

—

141

C₂²⋊C₄

—

106

—

142

C₄²

—

106

—

143

C₄²

D₂₁₂

yes

characteristic

144

C₄²

D₂₁₂

yes

characteristic

145

C₄²

C₂²×D₅₃

yes

characteristic

146

C₄²

C₂²×D₅₃

yes

characteristic

147

C₂³

C₂×D₂₁₂

yes

characteristic

148

Q₈

—

106

—

149

Q₈

—

212

—

150

Q₈

—

212

—

151

Q₈

—

212

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

212

—

158

C₂×C₄

Dic₂₁₂

yes

—

159

C₂×C₄

C₂×D₂₁₂

yes

characteristic

160

C₂×C₄

C₂×D₂₁₂

yes

characteristic

161

C₂×C₄

C₂×D₂₁₂

yes

characteristic

162

C₂×C₄

C₂×D₂₁₂

yes

characteristic

163

C₂×C₄

C₂×D₂₁₂

yes

characteristic

164

C₂×C₄

C₂³×D₅₃

yes

characteristic

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×Dic₂₁₂

yes

—

169

C₂²

C₂²×D₂₁₂

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

212

—

173

C₄

—

212

—

174

C₄

—

212

—

175

C₄

C₂×Dic₂₁₂

yes

—

176

C₄

C₂×Dic₂₁₂

yes

—

177

C₄

#13133245

yes

—

178

C₄

#13133264

yes

—

179

C₂

—

180

C₂

#16811199

yes

characteristic

181

C₂

C₂²×Dic₂₁₂

yes

characteristic

182

C₂

C₂×#13133245

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#3230
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16811028	C₂	semidirect
4. #16810974	C₂	semidirect
6. #16810919	C₂	semidirect
99. C₅₃	#3230	semidirect

H ≅ G/N

type

3. #16811028

C₂

semidirect

4. #16810974

C₂

semidirect

6. #16810919

C₂

semidirect

99. C₅₃

#3230

semidirect

Subgroups — #20285868

Conjugacy classes of subgroups

Complements