Group #20283785 — subgroups

Subgroups — #20283785

Order 6784.

394 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16811173	C₂	1	1	yes	maximal, characteristic
2	#16810696	C₂	1	4	yes	maximal, complemented
3	#16810693	C₂	1	4	yes	maximal, complemented
4	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810588	C₂	1	4	yes	maximal, complemented
6	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11719945	C₂²	1	2	yes	—
8	C₂×#11719912	C₂²	1	4	yes	—
9	C₂×C₈×D₅₃	C₂²	1	4	yes	—
10	#13133217	C₄	1	1	yes	characteristic
11	#13133214	C₄	1	1	yes	characteristic
12	#13133205	C₄	1	2	yes	—
13	#13133191	C₄	1	2	yes	—
14	#13133191	C₄	1	2	yes	—
15	C₄×D₂₁₂	C₂²	1	2	yes	—
16	C₄×D₂₁₂	C₂²	1	2	yes	—
17	C₄⋊#88322	C₂²	1	2	yes	—
18	C₂²⋊#88322	C₂²	1	2	yes	—
19	#13133129	C₂²	1	8	yes	—
20	#13133128	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
24	#97×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
29	#11719945	—	4	2	no	—
30	#11719945	—	4	2	no	—
31	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
32	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
33	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₂³	1	4	yes	—
35	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₅₃	C₂³	1	4	yes	—
37	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#11719921	C₂×C₄	1	4	yes	—
42	#11719921	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
45	C₂×#88322	C₂³	1	4	yes	—
46	#11719912	—	4	4	no	—
47	C₈×D₅₃	—	4	4	no	—
48	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₅₃	—	4	4	no	—
60	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
61	D₂₁₂	—	4	2	no	—
62	D₂₁₂	—	4	2	no	—
63	D₂₁₂	—	4	2	no	—
64	C₄×D₅₃	—	2	4	no	—
65	C₄×D₅₃	—	2	4	no	—
66	Dic₁₀₆	—	4	2	no	—
67	C₈×C₅₃	—	2	4	no	—
68	#88322	—	2	4	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₅₃	—	2	4	no	—
74	C₂×D₅₃	—	2	4	no	—
75	C₂×D₅₃	—	2	4	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
82	Dic₅₃	—	2	4	no	—
83	#2980	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	—	4	1	no	—
85	C₂×C₅₃	#518	1	1	yes	characteristic
86	C₂×C₅₃	#568	1	1	yes	characteristic
87	C₂×C₅₃	#568	1	1	yes	characteristic
88	D₅₃	—	4	4	no	—
89	#532	—	53	1	no	—
90	#435	—	53	4	no	—
91	#435	—	53	4	no	—
92	#432	—	53	1	no	—
93	#432	D₅₃	1	1	yes	characteristic, complemented
94	#408	—	53	4	no	—
95	C₅₃	#2980	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	53	2	no	—
97	C₂×M₁₆	—	53	4	no	—
98	C₂²×C₈	—	53	4	no	—
99	C₄⋊Q₈	—	53	1	no	—
100	#127	—	53	1	no	—
101	#124	—	53	2	no	—
102	#121	—	53	2	no	—
103	#121	—	53	2	no	—
104	C₄×D₄	—	53	2	no	—
105	C₄×D₄	—	53	2	no	—
106	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
107	C₄⋊C₈	—	53	2	no	—
108	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
109	C₂²⋊C₈	—	53	2	no	—
110	C₂²⋊C₈	—	53	8	no	—
111	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
112	#97	—	53	2	no	—
113	#97	C₂×D₅₃	1	2	yes	—
114	#41	—	212	2	no	—
115	#41	—	212	2	no	—
116	C₂×Q₈	—	53	2	no	—
117	C₂×D₄	—	53	4	no	—
118	C₂×D₄	—	53	2	no	—
119	C₂×D₄	—	53	2	no	—
120	C₂×D₄	—	53	2	no	—
121	C₂²×C₄	—	53	4	no	—
122	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	M₁₆	—	212	4	no	—
126	C₂×C₈	—	53	4	no	—
127	C₂×C₈	—	212	4	no	—
128	C₂×C₈	C₂²×D₅₃	1	4	yes	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	—	53	2	no	—
131	C₂²⋊C₄	—	53	4	no	—
132	C₂²⋊C₄	—	53	4	no	—
133	C₄²	C₄×D₅₃	1	1	yes	characteristic
134	C₄²	C₄×D₅₃	1	1	yes	characteristic
135	C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	53	4	no	—
138	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
139	Q₈	—	212	2	no	—
140	D₄	—	212	2	no	—
141	D₄	—	212	2	no	—
142	D₄	—	212	2	no	—
143	D₄	—	212	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	53	4	no	—
148	C₂×C₄	—	106	4	no	—
149	C₂×C₄	—	106	4	no	—
150	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	106	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	106	4	no	—
162	C₂²	—	106	4	no	—
163	C₂²	—	106	4	no	—
164	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	106	4	no	—
171	C₄	#13133263	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	212	4	no	—
174	C₂	#16811097	1	1	yes	characteristic
175	C₂	#16811389	1	1	yes	characteristic
176	C₂	#16811389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

#16810696

C₂

yes

maximal, complemented

#16810693

C₂

yes

maximal, complemented

#16810680

C₂

yes

maximal, characteristic, complemented

#16810588

C₂

yes

maximal, complemented

C₅₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

C₂×#11719912

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₄

yes

characteristic

#13133214

C₄

yes

characteristic

#13133205

C₄

yes

—

#13133191

C₄

yes

—

#13133191

C₄

yes

—

C₄×D₂₁₂

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133129

C₂²

yes

—

#13133128

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

#97×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

C₂×C₄

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

C₂×C₄

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#88322

C₂³

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

—

Dic₅₃

—

#2980

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

#408

—

C₅₃

#2980

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

100

#127

—

101

#124

—

102

#121

—

103

#121

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄²

C₂×D₅₃

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

C₂×D₅₃

yes

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

C₂×D₅₃

yes

—

112

#97

—

113

#97

C₂×D₅₃

yes

—

114

#41

—

212

—

115

#41

—

212

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×D₅₃

yes

characteristic

123

C₂²×C₄

C₄×D₅₃

yes

characteristic

124

C₂²×C₄

C₂²×D₅₃

yes

characteristic

125

M₁₆

—

212

—

126

C₂×C₈

—

127

C₂×C₈

—

212

—

128

C₂×C₈

C₂²×D₅₃

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

C₄×D₅₃

yes

characteristic

134

C₄²

C₄×D₅₃

yes

characteristic

135

C₄²

C₂²×D₅₃

yes

characteristic

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

C₂³

C₂×C₄×D₅₃

yes

characteristic

139

Q₈

—

212

—

140

D₄

—

212

—

141

D₄

—

212

—

142

D₄

—

212

—

143

D₄

—

212

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

151

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

152

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

153

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

154

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

155

C₂×C₄

C₂³×D₅₃

yes

characteristic

156

C₈

—

157

C₈

—

106

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

106

—

162

C₂²

—

106

—

163

C₂²

—

106

—

164

C₂²

C₂²×C₄×D₅₃

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

106

—

171

C₄

#13133263

yes

—

172

C₂

—

173

C₂

—

212

—

174

C₂

#16811097

yes

characteristic

175

C₂

#16811389

yes

characteristic

176

C₂

#16811389

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#432	C₂
hypercenter	#432	D₅₃
hypocenter	C₅₃	#2980
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#432

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16810696	C₂	semidirect
3. #16810693	C₂	semidirect
4. #16810680	C₂	semidirect
5. #16810588	C₂	semidirect
6. C₅₃×#432	C₂	semidirect
93. #432	D₅₃	semidirect
95. C₅₃	#2980	semidirect

H ≅ G/N

type

2. #16810696

C₂

semidirect

3. #16810693

C₂

semidirect

4. #16810680

C₂

semidirect

5. #16810588

C₂

semidirect

6. C₅₃×#432

C₂

semidirect

93. #432

D₅₃

semidirect

95. C₅₃

#2980

semidirect

Subgroups — #20283785

Conjugacy classes of subgroups

Complements