Group #20273451 — subgroups

Subgroups — #20273451

Order 6784.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16811178	C₂	1	1	yes	maximal, characteristic
2	#16810645	C₂	1	1	yes	maximal, characteristic
3	#16810645	C₂	1	1	yes	maximal, characteristic
4	#16810370	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	—	2	1	no	—
9	C₂×#11719912	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133208	C₄	1	2	yes	—
12	#13133197	—	2	1	no	—
13	#13133193	—	2	1	no	—
14	Q₈×Dic₅₃	—	2	1	no	—
15	#13133182	—	2	1	no	—
16	#13133175	C₄	1	2	yes	—
17	C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	—	2	1	no	—
21	#13133146	—	2	1	no	—
22	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719921	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	#11719919	—	2	2	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	D₄	1	1	yes	characteristic
43	C₄×Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	#11719912	—	4	2	no	—
47	#11719912	—	4	2	no	—
48	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₅₃	—	4	1	no	—
50	Q₈×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	2	2	no	—
64	Dic₁₀₆	—	4	1	no	—
65	Dic₁₀₆	—	4	1	no	—
66	C₈×C₅₃	—	4	1	no	—
67	C₈×C₅₃	—	4	1	no	—
68	#88322	—	4	1	no	—
69	#88322	—	4	1	no	—
70	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₅₃	—	2	1	no	—
72	C₂×D₅₃	—	2	1	no	—
73	C₂×D₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₅₃	#137	1	1	yes	characteristic
79	C₄×C₅₃	#137	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	4	2	no	—
83	Dic₅₃	#101	1	2	yes	—
84	#1561	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
86	C₂×C₅₃	#417	1	1	yes	characteristic
87	C₂×C₅₃	#417	1	1	yes	characteristic
88	D₅₃	—	4	1	no	—
89	#533	—	53	1	no	—
90	#423	—	53	1	no	—
91	#423	—	53	1	no	—
92	#332	—	53	1	no	—
93	#332	—	53	1	no	—
94	#332	—	53	1	no	—
95	#332	D₅₃	1	1	yes	characteristic, complemented
96	C₅₃	#1561	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	106	1	no	—
98	C₂×M₁₆	—	106	1	no	—
99	C₄⋊Q₈	—	53	1	no	—
100	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
101	#125	—	53	2	no	—
102	C₂²⋊Q₈	—	106	1	no	—
103	C₂²⋊Q₈	—	106	1	no	—
104	C₄×Q₈	—	106	1	no	—
105	C₄×Q₈	—	106	1	no	—
106	#117	—	53	2	no	—
107	C₂×C₄²	—	53	1	no	—
108	C₄⋊C₈	—	53	1	no	—
109	C₄⋊C₈	—	53	1	no	—
110	C₄⋊C₈	—	106	1	no	—
111	C₄⋊C₈	—	106	1	no	—
112	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	106	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	M₁₆	—	212	2	no	—
120	M₁₆	—	212	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₂×C₈	—	106	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	53	2	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄⋊C₄	—	106	2	no	—
129	C₄⋊C₄	—	106	1	no	—
130	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
131	C₂²⋊C₄	—	106	2	no	—
132	C₄²	—	53	1	no	—
133	C₄²	—	53	1	no	—
134	C₄²	—	53	1	no	—
135	C₄²	—	106	2	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	53	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	212	1	no	—
141	Q₈	—	212	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	1	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	212	1	no	—
158	C₈	—	212	1	no	—
159	C₂²	—	106	1	no	—
160	C₂²	—	106	1	no	—
161	C₂²	—	106	1	no	—
162	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	53	2	no	—
166	C₄	—	106	1	no	—
167	C₄	—	106	1	no	—
168	C₄	—	212	2	no	—
169	C₄	#13133241	1	1	yes	characteristic
170	C₄	#13133244	1	1	yes	characteristic
171	C₄	#13133246	1	1	yes	characteristic
172	C₄	#13133249	1	1	yes	characteristic
173	C₂	—	212	1	no	—
174	C₂	#101×D₅₃	1	1	yes	characteristic
175	C₂	#16810621	1	1	yes	characteristic
176	C₂	#16810623	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal, characteristic

#16810645

C₂

yes

maximal, characteristic

#16810645

C₂

yes

maximal, characteristic

#16810370

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

C₅₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

—

C₂×#11719912

—

#13133217

C₂²

yes

characteristic

#13133208

C₄

yes

—

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

—

#13133175

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

—

#13133146

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#101

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1561

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

101

#125

—

102

C₂²⋊Q₈

—

106

—

103

C₂²⋊Q₈

—

106

—

104

C₄×Q₈

—

106

—

105

C₄×Q₈

—

106

—

106

#117

—

107

C₂×C₄²

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

106

—

111

C₄⋊C₈

—

106

—

112

C₄⋊C₈

C₂×D₅₃

yes

characteristic

113

C₄⋊C₈

C₂×D₅₃

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

212

—

120

M₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

106

—

124

C₂×C₈

—

106

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

C₄×D₅₃

yes

—

131

C₂²⋊C₄

—

106

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

106

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

212

—

141

Q₈

—

212

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

153

C₂×C₄

D₄×D₅₃

yes

characteristic

154

C₂×C₄

D₄×D₅₃

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

212

—

158

C₈

—

212

—

159

C₂²

—

106

—

160

C₂²

—

106

—

161

C₂²

—

106

—

162

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

106

—

167

C₄

—

106

—

168

C₄

—

212

—

169

C₄

#13133241

yes

characteristic

170

C₄

#13133244

yes

characteristic

171

C₄

#13133246

yes

characteristic

172

C₄

#13133249

yes

characteristic

173

C₂

—

212

—

174

C₂

#101×D₅₃

yes

characteristic

175

C₂

#16810621

yes

characteristic

176

C₂

#16810623

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1561
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #16810370	C₂	semidirect
5. #16810369	C₂	semidirect
6. #16810368	C₂	semidirect
7. C₅₃×#332	C₂	semidirect
95. #332	D₅₃	semidirect
96. C₅₃	#1561	semidirect

H ≅ G/N

type

4. #16810370

C₂

semidirect

5. #16810369

C₂

semidirect

6. #16810368

C₂

semidirect

7. C₅₃×#332

C₂

semidirect

95. #332

D₅₃

semidirect

96. C₅₃

#1561

semidirect

Subgroups — #20273451

Conjugacy classes of subgroups

Complements