Group #20290896 — subgroups

Subgroups — #20290896

Order 6784.

300 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₄×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#16811569	C₂	1	1	yes	maximal, characteristic
3	#16811496	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6157	—	2	1	no	—
7	C₂×C₄⋊#6157	—	2	1	no	—
8	C₂×C₄⋊#6157	C₂²	1	2	yes	—
9	C₄×#88324	—	2	1	no	—
10	C₄×#88324	—	2	1	no	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	#13133217	C₄	1	1	yes	characteristic, complemented
13	#13133216	—	2	1	no	—
14	#13133216	C₄	1	2	yes	complemented
15	#13133215	—	2	1	no	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6157	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	Q₈×D₅₃	—	4	2	no	—
23	Q₈×D₅₃	—	4	2	no	—
24	C₂×C₄×D₅₃	—	2	2	no	—
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×#88324	—	2	2	no	—
31	C₄⋊#6157	—	2	4	no	—
32	C₄⋊#6157	—	4	2	no	—
33	C₄⋊#6157	—	4	2	no	—
34	D₅₃⋊C₈	—	2	2	no	—
35	D₅₃⋊C₈	Q₈	1	4	yes	complemented
36	C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
41	#11719919	—	2	2	no	—
42	#11719919	—	2	2	no	—
43	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₄×Dic₅₃	D₄	1	1	yes	characteristic
46	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×#6157	—	4	2	no	—
48	C₂×#6157	—	4	2	no	—
49	C₂×#6157	—	4	2	no	—
50	Q₈×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
59	C₄×D₅₃	—	4	2	no	—
60	C₄×D₅₃	—	4	2	no	—
61	C₄×D₅₃	D₈	1	2	yes	—
62	C₄×D₅₃	SD₁₆	1	2	yes	—
63	C₄×D₅₃	SD₁₆	1	2	yes	—
64	C₄×D₅₃	Q₁₆	1	2	yes	—
65	C₄×D₅₃	C₂×Q₈	1	2	yes	—
66	C₄×D₅₃	#41	1	2	yes	—
67	Dic₁₀₆	—	4	2	no	—
68	Dic₁₀₆	—	4	4	no	—
69	#88324	—	2	4	no	—
70	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₅₃	#124	1	1	yes	characteristic
72	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	#6157	—	8	2	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#102	1	2	yes	—
80	C₄×C₅₃	#103	1	2	yes	—
81	C₄×C₅₃	C₄×Q₈	1	2	yes	—
82	Dic₅₃	—	4	2	no	—
83	Dic₅₃	#102	1	2	yes	—
84	Dic₅₃	#103	1	2	yes	—
85	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
86	#1990	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	#391	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
90	D₅₃	#486	1	1	yes	characteristic
91	D₅₃	#487	1	1	yes	characteristic
92	D₅₃	#498	1	1	yes	characteristic
93	D₅₃	#500	1	1	yes	characteristic
94	C₂×C₄⋊Q₈	—	53	1	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	#384	—	53	1	no	—
97	#340	—	53	4	no	—
98	C₅₃	#1990	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	106	1	no	—
100	C₂²×C₈	—	53	2	no	—
101	C₄⋊Q₈	—	53	2	no	—
102	C₄⋊Q₈	—	53	1	no	—
103	C₄⋊Q₈	—	106	1	no	—
104	C₄⋊Q₈	—	106	1	no	—
105	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
106	C₂×C₄⋊C₄	—	53	2	no	—
107	C₂×C₄⋊C₄	—	53	2	no	—
108	C₂×C₄⋊C₄	—	106	1	no	—
109	C₂×C₄⋊C₄	—	106	1	no	—
110	C₂×C₄⋊C₄	—	106	1	no	—
111	C₂×C₄²	—	53	1	no	—
112	C₄×C₈	—	106	1	no	—
113	C₄×C₈	—	106	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	106	2	no	—
116	C₂×Q₈	—	106	1	no	—
117	C₂×Q₈	—	212	2	no	—
118	C₂×Q₈	—	212	2	no	—
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	C₂²×C₄	—	106	2	no	—
123	C₂²×C₄	—	106	2	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	106	2	no	—
126	C₂×C₈	—	106	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	53	2	no	—
129	C₄⋊C₄	—	106	2	no	—
130	C₄⋊C₄	—	106	2	no	—
131	C₄⋊C₄	—	106	4	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	C₂×#6157	1	2	yes	—
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₄²	C₂×#6157	1	1	yes	characteristic
140	C₂³	—	53	1	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	212	4	no	—
143	Q₈	—	212	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	53	2	no	—
151	C₂×C₄	—	53	2	no	—
152	C₂×C₄	—	53	2	no	—
153	C₂×C₄	—	53	2	no	—
154	C₂×C₄	—	106	2	no	—
155	C₂×C₄	—	212	2	no	—
156	C₂×C₄	—	212	2	no	—
157	C₂×C₄	—	212	2	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	—	212	2	no	—
160	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
162	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
163	C₈	—	106	4	no	—
164	C₂²	—	53	1	no	—
165	C₂²	—	53	1	no	—
166	C₂²	—	53	1	no	—
167	C₂²	—	53	1	no	—
168	C₂²	—	53	1	no	—
169	C₂²	—	53	1	no	—
170	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	53	2	no	—
173	C₄	—	53	2	no	—
174	C₄	—	53	2	no	—
175	C₄	—	424	2	no	—
176	C₄	—	212	2	no	—
177	C₄	#13133279	1	2	yes	—
178	C₄	#13133280	1	2	yes	—
179	C₄	Q₈×#6157	1	2	yes	—
180	C₂	—	53	1	no	—
181	C₂	—	53	1	no	—
182	C₂	—	53	1	no	—
183	C₂	—	53	1	no	—
184	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
185	C₂	C₂×#13133279	1	1	yes	characteristic
186	C₂	C₂×#13133280	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#16811569

C₂

yes

maximal, characteristic

#16811496

C₂

yes

maximal

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

C₂²

yes

—

C₄×#88324

—

C₄×#88324

—

C₂×Q₈×D₅₃

—

#13133217

C₄

yes

characteristic, complemented

#13133216

—

#13133216

C₄

yes

complemented

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

—

—

—

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂²×D₅₃

C₂×D₄

yes

—

—

—

Q₈×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×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

#124

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

C₂×D₅₃

C₂×Q₁₆

yes

characteristic

#6157

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

#102

yes

—

Dic₅₃

#103

yes

—

Dic₅₃

C₂²⋊Q₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

D₅₃

yes

D₅₃

yes

D₅₃

yes

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

—

#384

—

#340

—

C₅₃

#1990

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

100

C₂²×C₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

106

—

104

C₄⋊Q₈

—

106

—

105

C₄⋊Q₈

#6157

yes

characteristic, complemented

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄⋊C₄

—

106

—

110

C₂×C₄⋊C₄

—

106

—

111

C₂×C₄²

—

112

C₄×C₈

—

106

—

113

C₄×C₈

—

106

—

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂×Q₈

—

106

—

117

C₂×Q₈

—

212

—

118

C₂×Q₈

—

212

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

—

106

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

212

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

C₂×#6157

yes

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×#6157

yes

characteristic

140

C₂³

—

141

Q₈

—

142

Q₈

—

212

—

143

Q₈

—

212

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

212

—

156

C₂×C₄

—

212

—

157

C₂×C₄

—

212

—

158

C₂×C₄

—

212

—

159

C₂×C₄

—

212

—

160

C₂×C₄

C₂²⋊#6157

yes

characteristic

161

C₂×C₄

C₂²⋊#6157

yes

characteristic

162

C₂×C₄

C₂²×#6157

yes

characteristic

163

C₈

—

106

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×C₂²⋊#6157

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

424

—

176

C₄

—

212

—

177

C₄

#13133279

yes

—

178

C₄

#13133280

yes

—

179

C₄

Q₈×#6157

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

185

C₂

C₂×#13133279

yes

characteristic

186

C₂

C₂×#13133280

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1990
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #13133217	C₄	semidirect
14. #13133216	C₄	semidirect
19. C₄⋊Q₈×C₅₃	C₄	semidirect
35. D₅₃⋊C₈	Q₈	semidirect
98. C₅₃	#1990	semidirect
105. C₄⋊Q₈	#6157	semidirect

H ≅ G/N

type

12. #13133217

C₄

semidirect

14. #13133216

C₄

semidirect

19. C₄⋊Q₈×C₅₃

C₄

semidirect

35. D₅₃⋊C₈

Q₈

semidirect

98. C₅₃

#1990

semidirect

105. C₄⋊Q₈

#6157

semidirect

Subgroups — #20290896

Conjugacy classes of subgroups

Complements