Group #20273882 — subgroups

Subgroups — #20273882

Order 6784.

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

normal subgroups only

This group has 228 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	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810918	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810914	C₂	1	1	yes	maximal, characteristic
5	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810352	C₂	1	1	yes	maximal, characteristic
7	C₅₃×Q₈⋊C₈	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	#13133217	C₂²	1	1	yes	characteristic
12	#13133197	—	2	1	no	—
13	#13133197	—	2	2	no	—
14	C₄×D₂₁₂	—	2	1	no	—
15	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	#13133150	—	2	1	no	—
17	#13133149	—	2	1	no	—
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	#13133142	—	2	2	no	—
20	#13133141	—	2	1	no	—
21	#13133139	—	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	D₄	1	1	yes	characteristic
26	Q₈×D₅₃	—	4	2	no	—
27	Q₈×D₅₃	—	4	2	no	—
28	C₂×D₂₁₂	D₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	—	2	2	no	—
30	C₂×C₄×D₅₃	—	2	1	no	—
31	C₂×Dic₁₀₆	—	2	2	no	—
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#11719921	—	2	2	no	—
41	#11719921	—	2	1	no	—
42	C₄⋊Dic₅₃	—	2	2	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	—	2	1	no	—
45	C₄⋊Dic₅₃	—	2	1	no	—
46	C₄⋊Dic₅₃	D₄	1	2	yes	—
47	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
48	C₂×#88322	—	2	1	no	—
49	C₂²×D₅₃	—	2	1	no	—
50	Q₈×C₅₃	—	2	1	no	—
51	Q₈×C₅₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	4	1	no	—
60	D₂₁₂	—	2	1	no	—
61	D₂₁₂	Q₁₆	1	2	yes	—
62	C₄×D₅₃	—	4	4	no	—
63	C₄×D₅₃	—	4	1	no	—
64	C₄×D₅₃	—	4	1	no	—
65	Dic₁₀₆	—	4	4	no	—
66	Dic₁₀₆	—	4	1	no	—
67	Dic₁₀₆	—	4	1	no	—
68	Dic₁₀₆	—	4	2	no	—
69	C₈×C₅₃	—	4	1	no	—
70	C₈×C₅₃	—	4	1	no	—
71	#88322	—	4	1	no	—
72	C₂²×C₅₃	#120	1	1	yes	characteristic
73	C₂×D₅₃	—	2	2	no	—
74	C₂×D₅₃	—	4	1	no	—
75	C₂×D₅₃	—	4	1	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	4	1	no	—
79	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₄×C₅₃	#137	1	1	yes	characteristic
83	Dic₅₃	—	4	1	no	—
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	—	8	1	no	—
86	#1637	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	#450	1	1	yes	characteristic
88	C₂×C₅₃	#451	1	1	yes	characteristic
89	C₂×C₅₃	#454	1	1	yes	characteristic
90	D₅₃	—	4	2	no	—
91	D₄×Q₈	—	53	1	no	—
92	#487	—	53	1	no	—
93	#477	—	53	1	no	—
94	#476	—	53	1	no	—
95	#332	—	53	1	no	—
96	Q₈⋊C₈	D₅₃	1	1	yes	characteristic, complemented
97	D₄⋊C₈	—	53	1	no	—
98	C₅₃	#1637	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	106	1	no	—
100	C₄⋊Q₈	—	53	1	no	—
101	C₄⋊Q₈	—	53	1	no	—
102	C₄⋊Q₈	—	106	1	no	—
103	C₂²⋊Q₈	—	106	2	no	—
104	C₂²⋊Q₈	—	106	1	no	—
105	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
106	C₄×D₄	—	53	1	no	—
107	C₄×D₄	—	106	1	no	—
108	#107	—	106	1	no	—
109	#106	—	106	1	no	—
110	C₄⋊C₈	—	53	1	no	—
111	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
112	#103	—	106	2	no	—
113	#103	—	106	1	no	—
114	#102	—	106	1	no	—
115	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₂×Q₈	—	106	2	no	—
117	C₂×Q₈	—	106	1	no	—
118	C₂×Q₈	—	106	1	no	—
119	C₂×Q₈	—	212	2	no	—
120	C₂×Q₈	—	212	2	no	—
121	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
122	C₂×D₄	—	53	1	no	—
123	C₂²×C₄	—	106	2	no	—
124	C₂²×C₄	—	106	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	106	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	—	53	1	no	—
131	C₄⋊C₄	—	106	2	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
136	C₂²⋊C₄	—	106	2	no	—
137	C₂²⋊C₄	—	106	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	C₂³	—	106	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	212	2	no	—
143	Q₈	—	212	4	no	—
144	Q₈	—	212	1	no	—
145	Q₈	—	212	1	no	—
146	Q₈	#11719913	1	2	yes	—
147	D₄	—	53	2	no	—
148	D₄	—	106	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	—	106	1	no	—
153	C₂×C₄	—	212	4	no	—
154	C₂×C₄	—	212	1	no	—
155	C₂×C₄	—	212	1	no	—
156	C₂×C₄	—	212	1	no	—
157	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
158	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	—	212	1	no	—
163	C₂²	—	106	2	no	—
164	C₂²	—	212	1	no	—
165	C₂²	—	212	1	no	—
166	C₂²	#13133187	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	424	1	no	—
171	C₄	—	212	2	no	—
172	C₄	—	212	1	no	—
173	C₄	C₂×#11719913	1	1	yes	characteristic
174	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
175	C₄	#13133245	1	1	yes	characteristic
176	C₄	#13133246	1	1	yes	characteristic
177	C₂	—	212	2	no	—
178	C₂	#16810776	1	1	yes	characteristic
179	C₂	#16810780	1	1	yes	characteristic
180	C₂	#16810797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₁₂

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic, complemented

#16810918

C₂

yes

maximal, characteristic, complemented

#16810914

C₂

yes

maximal, characteristic

#16810368

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, characteristic

C₅₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

#13133217

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133197

—

#13133197

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

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

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×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

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

—

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

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

Q₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₄×Q₈

—

—

—

—

—

Q₈⋊C₈

D₅₃

yes

characteristic, complemented

D₄⋊C₈

—

C₅₃

#1637

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

106

—

103

C₂²⋊Q₈

—

106

—

104

C₂²⋊Q₈

—

106

—

105

C₄×Q₈

C₂×D₅₃

yes

characteristic

106

C₄×D₄

—

107

C₄×D₄

—

106

—

108

#107

—

106

—

109

#106

—

106

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₅₃

yes

characteristic

112

#103

—

106

—

113

#103

—

106

—

114

#102

—

106

—

115

C₄×C₈

C₂×D₅₃

yes

characteristic

116

C₂×Q₈

—

106

—

117

C₂×Q₈

—

106

—

118

C₂×Q₈

—

106

—

119

C₂×Q₈

—

212

—

120

C₂×Q₈

—

212

—

121

C₂×Q₈

D₂₁₂

yes

characteristic

122

C₂×D₄

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

—

106

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

106

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

D₂₁₂

yes

characteristic

136

C₂²⋊C₄

—

106

—

137

C₂²⋊C₄

—

106

—

138

C₄²

—

139

C₄²

C₂²×D₅₃

yes

characteristic

140

C₂³

—

106

—

141

Q₈

—

142

Q₈

—

212

—

143

Q₈

—

212

—

144

Q₈

—

212

—

145

Q₈

—

212

—

146

Q₈

#11719913

yes

—

147

D₄

—

148

D₄

—

106

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

—

212

—

157

C₂×C₄

C₂×D₂₁₂

yes

characteristic

158

C₂×C₄

D₄×D₅₃

yes

characteristic

159

C₂×C₄

D₄×D₅₃

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

212

—

163

C₂²

—

106

—

164

C₂²

—

212

—

165

C₂²

—

212

—

166

C₂²

#13133187

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

424

—

171

C₄

—

212

—

172

C₄

—

212

—

173

C₄

C₂×#11719913

yes

characteristic

174

C₄

Q₁₆×D₅₃

yes

characteristic

175

C₄

#13133245

yes

characteristic

176

C₄

#13133246

yes

characteristic

177

C₂

—

212

—

178

C₂

#16810776

yes

characteristic

179

C₂

#16810780

yes

characteristic

180

C₂

#16810797

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	#120
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1637
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13133187

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×Q₈⋊C₈

C₂

socle

C₂²×C₅₃

#120

hypercenter

Q₈⋊C₈

D₅₃

hypocenter

C₅₃

#1637

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
2. #16810974	C₂	semidirect
3. #16810918	C₂	semidirect
5. #16810368	C₂	semidirect
7. C₅₃×Q₈⋊C₈	C₂	semidirect
96. Q₈⋊C₈	D₅₃	semidirect
98. C₅₃	#1637	semidirect

H ≅ G/N

type

2. #16810974

C₂

semidirect

3. #16810918

C₂

semidirect

5. #16810368

C₂

semidirect

7. C₅₃×Q₈⋊C₈

C₂

semidirect

96. Q₈⋊C₈

D₅₃

semidirect

98. C₅₃

#1637

semidirect

Subgroups — #20273882

Conjugacy classes of subgroups

Complements