Group #20284286 — subgroups

Subgroups — C₂×#16810861 #20284286

Order 6784.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133193	C₂	1	2	yes	maximal
3	#16810861	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133141	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₅₃	—	2	4	no	—
10	C₂×C₈×D₅₃	C₂²	1	4	yes	—
11	#13133193	—	2	4	no	—
12	#13133193	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×#11719919	—	2	2	no	—
15	C₂×#11719921	—	2	2	no	—
16	#13133151	C₂²	1	4	yes	complemented
17	#13133141	C₂²	1	8	yes	complemented
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₅₃	—	2	4	no	—
20	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₅₃	D₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	2	no	—
30	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₄×D₅₃	—	2	2	no	—
33	C₂×C₄×D₅₃	D₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂³	1	2	yes	—
35	Q₁₆×C₅₃	—	2	8	no	—
36	C₂×C₈×C₅₃	D₄	1	4	yes	—
37	C₂×C₈×C₅₃	C₂³	1	2	yes	—
38	#11719921	—	2	8	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	C₂³	1	4	yes	—
41	#11719919	—	2	8	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	C₂³	1	2	yes	—
44	C₈×D₅₃	—	2	8	no	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₅₃	—	2	2	no	—
47	C₂²×D₅₃	—	2	2	no	—
48	C₂²×D₅₃	Q₁₆	1	4	yes	complemented
49	C₂²×D₅₃	C₂×D₄	1	2	yes	—
50	Q₈×C₅₃	—	2	4	no	—
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	#41	1	2	yes	—
58	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	—	4	4	no	—
62	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
63	C₄×D₅₃	—	2	4	no	—
64	C₄×D₅₃	—	2	2	no	—
65	C₄×D₅₃	—	2	2	no	—
66	C₈×C₅₃	C₂×D₄	1	4	yes	—
67	#88322	—	2	2	no	—
68	C₂²×C₅₃	#121	1	2	yes	—
69	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
70	C₂²×C₅₃	#135	1	2	yes	—
71	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×D₅₃	—	2	2	no	—
73	C₂×D₅₃	—	2	2	no	—
74	C₂×D₅₃	—	2	8	no	—
75	C₂×D₅₃	C₂×Q₁₆	1	4	yes	complemented
76	C₄×C₅₃	—	4	4	no	—
77	C₄×C₅₃	#121	1	2	yes	—
78	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
80	Dic₅₃	—	2	2	no	—
81	Dic₅₃	—	4	4	no	—
82	C₂×#467	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	#467	1	4	yes	complemented
84	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
85	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
86	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
87	D₅₃	—	2	4	no	—
88	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
89	C₂³×C₈	—	53	1	no	—
90	C₂×C₂²⋊Q₈	—	53	2	no	—
91	#467	—	53	8	no	—
92	C₂×#107	—	53	1	no	—
93	C₂×#103	—	53	2	no	—
94	C₅₃	C₂×#467	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
96	C₂³×C₄	—	53	1	no	—
97	C₂×Q₁₆	—	2	4	no	—
98	C₂×Q₁₆	C₂×D₅₃	1	4	yes	complemented
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	4	no	—
101	C₂²×C₈	—	106	4	no	—
102	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂²⋊Q₈	—	53	8	no	—
104	C₂²⋊Q₈	—	106	4	no	—
105	C₂×C₄⋊C₄	—	53	2	no	—
106	C₂×C₄⋊C₄	—	106	2	no	—
107	C₂×C₂²⋊C₄	—	106	2	no	—
108	#107	—	53	4	no	—
109	#103	—	53	8	no	—
110	C₂⁴	—	53	1	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	2	4	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	2	no	—
118	C₂²×C₄	—	53	2	no	—
119	C₂²×C₄	—	106	2	no	—
120	C₂²×C₄	—	106	2	no	—
121	C₂²×C₄	—	106	2	no	—
122	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
123	Q₁₆	—	2	8	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	106	2	no	—
126	C₂×C₈	—	106	8	no	—
127	C₂×C₈	C₂²⋊D₅₃	1	4	yes	—
128	C₂×C₈	C₂²×D₅₃	1	2	yes	—
129	C₄⋊C₄	—	53	4	no	—
130	C₄⋊C₄	—	106	8	no	—
131	C₄⋊C₄	—	106	2	no	—
132	C₂²⋊C₄	—	106	8	no	—
133	C₂³	—	53	4	no	—
134	C₂³	—	53	2	no	—
135	C₂³	—	106	2	no	—
136	C₂³	—	106	2	no	—
137	C₂³	D₄×D₅₃	1	1	yes	characteristic
138	Q₈	—	2	4	no	—
139	Q₈	—	2	4	no	—
140	Q₈	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	106	4	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	4	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	#11719947	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
153	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
154	C₈	—	106	2	no	—
155	C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
156	C₂²	—	53	4	no	—
157	C₂²	—	106	2	no	—
158	C₂²	—	106	8	no	—
159	C₂²	—	106	2	no	—
160	C₂²	#13133188	1	2	yes	—
161	C₂²	Q₁₆×D₅₃	1	2	yes	—
162	C₂²	#13133240	1	2	yes	—
163	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
164	C₄	—	4	4	no	—
165	C₄	—	106	2	no	—
166	C₄	—	212	4	no	—
167	C₄	#13133188	1	2	yes	—
168	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄	C₂×#11719947	1	1	yes	characteristic
170	C₂	—	106	4	no	—
171	C₂	#16810861	1	4	yes	direct product factor
172	C₂	C₂×#13133188	1	1	yes	characteristic
173	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
174	C₂	C₂×#13133240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133193

C₂

yes

maximal

#16810861

C₂

yes

maximal, direct product factor

C₂×#13133151

C₂

yes

maximal, characteristic, complemented

C₂×#13133141

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133193

—

#13133193

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

C₂×#11719921

—

#13133151

C₂²

yes

complemented

#13133141

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

Q₁₆

yes

complemented

C₂²×D₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×Q₁₆

yes

complemented

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×#135

yes

characteristic

D₅₃

—

C₂²×Q₁₆

D₅₃

yes

characteristic, complemented

C₂³×C₈

—

C₂×C₂²⋊Q₈

—

#467

—

C₂×#107

—

C₂×#103

—

C₅₃

C₂×#467

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂³×C₄

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

complemented

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

106

—

102

C₂²×C₈

C₂×D₅₃

yes

characteristic

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

106

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

106

—

107

C₂×C₂²⋊C₄

—

106

—

108

#107

—

109

#103

—

110

C₂⁴

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

C₂²×D₅₃

yes

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

C₂²×D₅₃

yes

characteristic

123

Q₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₂×C₈

C₂²⋊D₅₃

yes

—

128

C₂×C₈

C₂²×D₅₃

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₂²⋊C₄

—

106

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

106

—

136

C₂³

—

106

—

137

C₂³

D₄×D₅₃

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

212

—

150

C₂×C₄

D₄×D₅₃

yes

characteristic

151

C₂×C₄

#11719947

yes

—

152

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

153

C₂×C₄

C₂³×D₅₃

yes

characteristic

154

C₈

—

106

—

155

C₈

C₂×C₂²⋊D₅₃

yes

—

156

C₂²

—

157

C₂²

—

106

—

158

C₂²

—

106

—

159

C₂²

—

106

—

160

C₂²

#13133188

yes

—

161

C₂²

Q₁₆×D₅₃

yes

—

162

C₂²

#13133240

yes

—

163

C₂²

C₂×D₄×D₅₃

yes

characteristic

164

C₄

—

165

C₄

—

106

—

166

C₄

—

212

—

167

C₄

#13133188

yes

—

168

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

169

C₄

C₂×#11719947

yes

characteristic

170

C₂

—

106

—

171

C₂

#16810861

yes

direct product factor

172

C₂

C₂×#13133188

yes

characteristic

173

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

174

C₂

C₂×#13133240

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×Q₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×#467
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂²×Q₁₆×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₃

hypocenter

C₅₃

C₂×#467

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #16810861	C₂	direct (N × H)
4. C₂×#13133151	C₂	semidirect
5. C₂×#13133141	C₂	semidirect
6. C₂²×Q₁₆×C₅₃	C₂	semidirect
16. #13133151	C₂²	semidirect
17. #13133141	C₂²	semidirect
20. C₂×Q₁₆×C₅₃	C₂²	semidirect
48. C₂²×D₅₃	Q₁₆	semidirect
75. C₂×D₅₃	C₂×Q₁₆	semidirect
83. C₂×C₅₃	#467	semidirect
88. C₂²×Q₁₆	D₅₃	semidirect
94. C₅₃	C₂×#467	semidirect
98. C₂×Q₁₆	C₂×D₅₃	semidirect
171. C₂	#16810861	direct (N × H)

H ≅ G/N

type

3. #16810861

C₂

direct (N × H)

4. C₂×#13133151

C₂

semidirect

5. C₂×#13133141

C₂

semidirect

6. C₂²×Q₁₆×C₅₃

C₂

semidirect

16. #13133151

C₂²

semidirect

17. #13133141

C₂²

semidirect

20. C₂×Q₁₆×C₅₃

C₂²

semidirect

48. C₂²×D₅₃

Q₁₆

semidirect

75. C₂×D₅₃

C₂×Q₁₆

semidirect

83. C₂×C₅₃

#467

semidirect

88. C₂²×Q₁₆

D₅₃

semidirect

94. C₅₃

C₂×#467

semidirect

98. C₂×Q₁₆

C₂×D₅₃

semidirect

171. C₂

#16810861

direct (N × H)

Subgroups — C₂×#16810861 #20284286

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810861 #20284286

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810861 #20284286