Group #20285043 — subgroups

Subgroups — C₂×#16810982 #20285043

Order 6784.

436 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133217	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810982	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133141	C₂	1	2	yes	maximal, complemented
4	C₂×#13133138	C₂	1	2	yes	maximal
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	—	2	1	no	—
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133217	—	2	2	no	—
10	#13133217	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₅₃	—	2	1	no	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	#13133141	C₂²	1	8	yes	complemented
14	#13133138	C₂²	1	8	yes	—
15	#13133128	C₂²	1	4	yes	complemented
16	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	#124×C₅₃	—	2	2	no	—
19	#124×C₅₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂⁴×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
27	C₂×D₄×C₅₃	—	2	1	no	—
28	C₂×D₄×C₅₃	—	2	4	no	—
29	C₂×D₄×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	2	2	no	—
35	C₂×Dic₁₀₆	—	2	4	no	—
36	C₂×Dic₁₀₆	—	4	4	no	—
37	C₂²⋊C₄×C₅₃	—	2	4	no	—
38	C₂²⋊C₄×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	D₄	1	2	yes	—
40	C₄²×C₅₃	C₂³	1	2	yes	—
41	C₄⋊Dic₅₃	—	2	4	no	—
42	C₄⋊Dic₅₃	—	2	2	no	—
43	C₄⋊Dic₅₃	C₂³	1	4	yes	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₂³	1	4	yes	—
46	C₂³×C₅₃	—	2	2	no	—
47	C₂³×C₅₃	—	4	2	no	—
48	C₂³×C₅₃	—	4	2	no	—
49	C₂³×C₅₃	—	4	2	no	—
50	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	4	2	no	—
53	D₄×C₅₃	—	2	4	no	—
54	D₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#41	1	4	yes	—
63	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₅₃	—	2	4	no	—
65	C₂×Dic₅₃	—	4	4	no	—
66	Dic₁₀₆	—	4	8	no	—
67	#88322	—	2	4	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	—	4	2	no	—
71	C₂²×C₅₃	—	8	4	no	—
72	C₂²×C₅₃	#124	1	2	yes	—
73	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂²×C₅₃	#137	1	2	yes	—
75	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#124	1	2	yes	—
80	C₄×C₅₃	C₂×#41	1	2	yes	—
81	Dic₅₃	—	4	4	no	—
82	C₂×#489	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	—	8	2	no	—
84	C₂×C₅₃	#489	1	4	yes	complemented
85	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
86	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
88	C₂×C₄⋊Q₈	—	53	1	no	—
89	C₂×#124	D₅₃	1	1	yes	characteristic
90	#489	—	53	8	no	—
91	C₂×#103	—	53	2	no	—
92	C₂×#102	—	53	2	no	—
93	C₂×#97	—	53	1	no	—
94	C₅₃	C₂×#489	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	106	1	no	—
96	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
97	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
98	C₂²×C₈	—	53	2	no	—
99	C₄⋊Q₈	—	53	4	no	—
100	C₄⋊Q₈	—	106	2	no	—
101	#124	—	2	2	no	—
102	#124	C₂×D₅₃	1	4	yes	—
103	C₂×C₄⋊C₄	—	53	2	no	—
104	C₂×C₄⋊C₄	—	106	1	no	—
105	C₂×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
108	#103	—	53	8	no	—
109	#102	—	53	8	no	—
110	#97	—	53	4	no	—
111	C₂⁴	—	2	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	—	106	4	no	—
115	C₂×Q₈	—	212	4	no	—
116	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	C₂²×D₅₃	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	106	2	no	—
122	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	106	4	no	—
127	C₄⋊C₄	—	53	4	no	—
128	C₄⋊C₄	—	106	2	no	—
129	C₄⋊C₄	—	106	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₄²	C₂²⋊D₅₃	1	2	yes	—
133	C₄²	C₂²×D₅₃	1	2	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	4	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	Q₈	—	2	4	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	212	8	no	—
142	D₄	—	2	4	no	—
143	D₄	—	4	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	106	4	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	#11719947	1	4	yes	—
151	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
155	C₈	—	106	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	4	no	—
160	C₂²	#13133204	1	2	yes	—
161	C₂²	#13133242	1	2	yes	—
162	C₂²	#13133247	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	212	4	no	—
168	C₄	#13133204	1	2	yes	—
169	C₄	C₂×#11719947	1	2	yes	—
170	C₂	—	8	2	no	—
171	C₂	#16810982	1	4	yes	direct product factor
172	C₂	C₂×#13133204	1	1	yes	characteristic
173	C₂	C₂×#13133242	1	1	yes	characteristic
174	C₂	C₂×#13133247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic, complemented

#16810982

C₂

yes

maximal, direct product factor

C₂×#13133141

C₂

yes

maximal, complemented

C₂×#13133138

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂²×#88322

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133141

C₂²

yes

complemented

#13133138

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×D₄×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#124×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×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

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

C₂×#489

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

—

C₂×#124

D₅₃

yes

—

—

—

C₂×#97

—

C₅₃

C₂×#489

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

106

—

101

#124

—

102

#124

C₂×D₅₃

yes

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

106

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

C₂×D₅₃

yes

characteristic

108

#103

—

109

#102

—

110

#97

—

111

C₂⁴

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

106

—

115

C₂×Q₈

—

212

—

116

C₂×Q₈

C₂²×D₅₃

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

C₂²×D₅₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

123

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

124

C₂²×C₄

C₂²×D₅₃

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

106

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

C₂²⋊D₅₃

yes

—

133

C₄²

C₂²×D₅₃

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

139

Q₈

—

140

Q₈

—

141

Q₈

—

212

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

212

—

150

C₂×C₄

#11719947

yes

—

151

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

152

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

153

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

154

C₂×C₄

C₂³×D₅₃

yes

characteristic

155

C₈

—

106

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#13133204

yes

—

161

C₂²

#13133242

yes

—

162

C₂²

#13133247

yes

—

163

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

212

—

168

C₄

#13133204

yes

—

169

C₄

C₂×#11719947

yes

—

170

C₂

—

171

C₂

#16810982

yes

direct product factor

172

C₂

C₂×#13133204

yes

characteristic

173

C₂

C₂×#13133242

yes

characteristic

174

C₂

C₂×#13133247

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#124×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#124	D₅₃
hypocenter	C₅₃	C₂×#489
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#124×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#13133217	C₂	semidirect
2. #16810982	C₂	direct (N × H)
3. C₂×#13133141	C₂	semidirect
5. C₂×#13133128	C₂	semidirect
10. #13133217	C₂²	semidirect
13. #13133141	C₂²	semidirect
15. #13133128	C₂²	semidirect
84. C₂×C₅₃	#489	semidirect
94. C₅₃	C₂×#489	semidirect
171. C₂	#16810982	direct (N × H)

H ≅ G/N

type

1. C₂×#13133217

C₂

semidirect

2. #16810982

C₂

direct (N × H)

3. C₂×#13133141

C₂

semidirect

5. C₂×#13133128

C₂

semidirect

10. #13133217

C₂²

semidirect

13. #13133141

C₂²

semidirect

15. #13133128

C₂²

semidirect

84. C₂×C₅₃

#489

semidirect

94. C₅₃

C₂×#489

semidirect

171. C₂

#16810982

direct (N × H)

Subgroups — C₂×#16810982 #20285043

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810982 #20285043

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810982 #20285043