Group #20273962 — subgroups

Subgroups — #20273962

Order 6784.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16811026	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810900	C₂	1	1	yes	maximal, characteristic
4	#16810370	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810350	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	1	no	—
9	#13133217	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133197	—	2	1	no	—
12	#13133197	—	2	2	no	—
13	C₄×D₂₁₂	—	2	1	no	—
14	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	#13133150	—	2	2	no	—
16	#13133150	—	2	1	no	—
17	C₄⋊#88322	C₂²	1	1	yes	characteristic
18	#13133137	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₅₃	—	2	1	no	—
21	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#103×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	Q₈×D₅₃	—	4	2	no	—
28	Q₈×D₅₃	—	4	2	no	—
29	C₂×D₂₁₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	1	no	—
32	C₂×Dic₁₀₆	—	2	2	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	Q₁₆×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#11719921	—	2	2	no	—
43	#11719921	—	2	1	no	—
44	C₄⋊Dic₅₃	—	2	2	no	—
45	C₄⋊Dic₅₃	—	2	1	no	—
46	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
47	C₄⋊Dic₅₃	D₄	1	2	yes	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂²×D₅₃	—	2	1	no	—
51	Q₈×C₅₃	—	2	1	no	—
52	Q₈×C₅₃	—	4	1	no	—
53	Q₈×C₅₃	—	4	1	no	—
54	Q₈×C₅₃	D₈	1	2	yes	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	—	2	2	no	—
63	D₂₁₂	—	2	1	no	—
64	D₂₁₂	Q₁₆	1	2	yes	complemented
65	C₄×D₅₃	—	4	4	no	—
66	C₄×D₅₃	—	4	1	no	—
67	C₄×D₅₃	—	4	1	no	—
68	Dic₁₀₆	—	4	4	no	—
69	Dic₁₀₆	—	4	1	no	—
70	Dic₁₀₆	—	4	1	no	—
71	C₈×C₅₃	—	4	1	no	—
72	#88322	—	2	2	no	—
73	#88322	—	4	1	no	—
74	C₂²×C₅₃	#120	1	1	yes	characteristic
75	C₂×D₅₃	—	2	2	no	—
76	C₂×D₅₃	—	4	1	no	—
77	C₂×D₅₃	—	4	1	no	—
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	—	8	1	no	—
82	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₅₃	#137	1	1	yes	characteristic
86	Dic₅₃	—	4	1	no	—
87	Dic₅₃	—	4	2	no	—
88	#1649	—	53	1	no	maximal, Sylow
89	C₂×C₅₃	#447	1	1	yes	characteristic
90	C₂×C₅₃	#451	1	1	yes	characteristic
91	C₂×C₅₃	#456	1	1	yes	characteristic
92	D₅₃	—	4	2	no	—
93	D₄×Q₈	—	53	1	no	—
94	#500	—	53	1	no	—
95	#474	—	53	1	no	—
96	#462	D₅₃	1	1	yes	characteristic, complemented
97	#332	—	53	1	no	—
98	Q₈⋊C₈	—	53	1	no	—
99	D₄⋊C₈	—	53	1	no	—
100	C₅₃	#1649	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	106	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₄⋊Q₈	—	53	1	no	—
104	C₄⋊Q₈	—	106	1	no	—
105	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂²⋊Q₈	—	106	2	no	—
107	C₂²⋊Q₈	—	106	1	no	—
108	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄×D₄	—	53	1	no	—
110	C₄×D₄	—	106	1	no	—
111	#107	—	106	1	no	—
112	#107	—	106	2	no	—
113	C₄⋊C₈	—	53	1	no	—
114	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
115	#103	—	2	1	no	—
116	#102	—	106	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	106	2	no	—
120	C₂×Q₈	—	106	1	no	—
121	C₂×Q₈	—	212	2	no	—
122	C₂×Q₈	—	212	2	no	—
123	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂×D₄	—	53	1	no	—
125	C₂²×C₄	—	106	2	no	—
126	C₂²×C₄	—	106	1	no	—
127	Q₁₆	—	4	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	106	1	no	—
130	C₂×C₈	—	106	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	53	1	no	—
135	C₄⋊C₄	—	53	2	no	—
136	C₄⋊C₄	—	106	2	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²⋊C₄	—	106	2	no	—
140	C₂²⋊C₄	—	106	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	C₂³	—	106	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	212	4	no	—
148	Q₈	—	212	1	no	—
149	Q₈	—	212	1	no	—
150	Q₈	#11719922	1	2	yes	—
151	D₄	—	53	2	no	—
152	D₄	—	106	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	—	212	4	no	—
159	C₂×C₄	—	212	1	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
162	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	106	2	no	—
166	C₈	—	212	1	no	—
167	C₂²	—	106	2	no	—
168	C₂²	—	212	1	no	—
169	C₂²	—	212	1	no	—
170	C₂²	#13133185	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	212	2	no	—
176	C₄	—	212	1	no	—
177	C₄	C₂×#11719922	1	1	yes	characteristic
178	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
179	C₄	#13133242	1	1	yes	characteristic
180	C₄	#13133249	1	1	yes	characteristic
181	C₂	—	212	2	no	—
182	C₂	#16810756	1	1	yes	characteristic
183	C₂	#16810777	1	1	yes	characteristic
184	C₂	#16810804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₁₂

C₂

yes

maximal, characteristic

#16811026

C₂

yes

maximal, characteristic, complemented

#16810900

C₂

yes

maximal, characteristic

#16810370

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, characteristic, complemented

#16810350

C₂

yes

maximal, characteristic

C₅₃×#462

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133197

—

#13133197

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₂²

yes

—

—

C₂²

yes

characteristic

#13133137

—

C₄×#88322

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

Q₁₆

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₄×Q₈

—

#500

—

#474

—

#462

D₅₃

yes

characteristic, complemented

#332

—

Q₈⋊C₈

—

D₄⋊C₈

—

100

C₅₃

#1649

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

106

—

102

C₂×Q₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

106

—

105

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

106

C₂²⋊Q₈

—

106

—

107

C₂²⋊Q₈

—

106

—

108

C₄×Q₈

C₂×D₅₃

yes

characteristic

109

C₄×D₄

—

110

C₄×D₄

—

106

—

111

#107

—

106

—

112

#107

—

106

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₅₃

yes

characteristic

115

#103

—

116

#102

—

106

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

106

—

120

C₂×Q₈

—

106

—

121

C₂×Q₈

—

212

—

122

C₂×Q₈

—

212

—

123

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

124

C₂×D₄

—

125

C₂²×C₄

—

106

—

126

C₂²×C₄

—

106

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

106

—

130

C₂×C₈

—

106

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

139

C₂²⋊C₄

—

106

—

140

C₂²⋊C₄

—

106

—

141

C₄²

—

142

C₄²

C₂²×D₅₃

yes

characteristic

143

C₂³

—

106

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

212

—

148

Q₈

—

212

—

149

Q₈

—

212

—

150

Q₈

#11719922

yes

—

151

D₄

—

152

D₄

—

106

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

212

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

D₄×D₅₃

yes

characteristic

162

C₂×C₄

D₄×D₅₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₈

—

165

C₈

—

106

—

166

C₈

—

212

—

167

C₂²

—

106

—

168

C₂²

—

212

—

169

C₂²

—

212

—

170

C₂²

#13133185

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

212

—

176

C₄

—

212

—

177

C₄

C₂×#11719922

yes

characteristic

178

C₄

Q₁₆×D₅₃

yes

characteristic

179

C₄

#13133242

yes

characteristic

180

C₄

#13133249

yes

characteristic

181

C₂

—

212

—

182

C₂

#16810756

yes

characteristic

183

C₂

#16810777

yes

characteristic

184

C₂

#16810804

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#462	C₂
socle	C₂²×C₅₃	#120
hypercenter	#462	D₅₃
hypocenter	C₅₃	#1649
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#462

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
2. #16811026	C₂	semidirect
4. #16810370	C₂	semidirect
5. Q₈⋊#88322	C₂	semidirect
7. C₅₃×#462	C₂	semidirect
64. D₂₁₂	Q₁₆	semidirect
96. #462	D₅₃	semidirect
100. C₅₃	#1649	semidirect

H ≅ G/N

type

2. #16811026

C₂

semidirect

4. #16810370

C₂

semidirect

5. Q₈⋊#88322

C₂

semidirect

7. C₅₃×#462

C₂

semidirect

64. D₂₁₂

Q₁₆

semidirect

96. #462

D₅₃

semidirect

100. C₅₃

#1649

semidirect

Subgroups — #20273962

Conjugacy classes of subgroups

Complements