Group #20290063 — subgroups

Subgroups — #20290063

Order 6784.

232 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 72 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	#16811626	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88324	C₂	1	1	yes	maximal, characteristic
3	#16811473	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#88324	C₂	1	2	yes	maximal, complemented
5	#16811175	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719940	—	2	1	no	—
7	C₂²×#88324	—	2	1	no	—
8	#13133284	—	2	1	no	—
9	C₄⋊#88324	C₂²	1	2	yes	—
10	C₄×#88324	—	2	1	no	—
11	C₄×#88324	C₂²	1	2	yes	—
12	#13133216	C₄	1	1	yes	characteristic
13	#13133215	C₄	1	1	yes	characteristic
14	#13133207	—	2	1	no	—
15	#13133196	C₄	1	1	yes	characteristic
16	#13133194	—	2	1	no	—
17	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#13133182	—	2	1	no	—
19	#13133182	C₂²	1	1	yes	characteristic
20	#13133173	—	2	1	no	—
21	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
26	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
29	#11719940	—	4	2	no	—
30	C₂×#88324	—	2	4	no	—
31	C₂×#88324	—	2	2	no	—
32	C₂×#88324	—	2	2	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²⋊C₄×C₅₃	—	2	1	no	—
36	C₂²⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	#11719919	—	2	1	no	—
40	#11719919	—	2	1	no	—
41	#11719919	—	2	1	no	—
42	#11719919	—	2	1	no	—
43	#11719919	—	2	1	no	—
44	C₄×Dic₅₃	—	2	1	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₅₃	D₄	1	1	yes	characteristic
48	C₄×Dic₅₃	D₄	1	1	yes	characteristic
49	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₅₃	SD₁₆	1	2	yes	—
63	C₂×Dic₅₃	Q₁₆	1	2	yes	—
64	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
66	Dic₁₀₆	—	2	2	no	—
67	Dic₁₀₆	—	4	2	no	—
68	#88324	—	2	4	no	—
69	#88324	—	4	2	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	#103	1	2	yes	—
73	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	#131	1	2	yes	—
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	—	4	1	no	—
80	Dic₅₃	—	4	1	no	—
81	Dic₅₃	#103	1	2	yes	—
82	Dic₅₃	#104	1	2	yes	—
83	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
85	#1493	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#408	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
90	#533	—	53	1	no	—
91	#423	—	53	1	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	Q₈⋊C₈	—	53	2	no	—
94	Q₈⋊C₈	—	53	2	no	—
95	C₅₃	#1493	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	106	1	no	—
97	C₂²×C₈	—	106	1	no	—
98	C₄⋊Q₈	—	53	1	no	—
99	C₄⋊Q₈	—	53	1	no	—
100	#125	—	106	1	no	—
101	C₂²⋊Q₈	—	53	1	no	—
102	C₂²⋊Q₈	—	106	1	no	—
103	C₂²⋊Q₈	#6157	1	1	yes	characteristic
104	C₄×Q₈	—	53	1	no	—
105	C₄×Q₈	—	53	1	no	—
106	C₄×Q₈	—	106	1	no	—
107	#117	—	106	1	no	—
108	C₂×C₄²	—	53	1	no	—
109	C₄⋊C₈	—	53	2	no	—
110	C₄⋊C₈	—	106	1	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	—	106	1	no	—
113	C₂×Q₈	—	53	1	no	—
114	C₂×Q₈	—	106	1	no	—
115	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
119	M₁₆	—	212	2	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	—	106	4	no	—
122	C₂×C₈	—	106	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	53	1	no	—
125	C₄⋊C₄	—	106	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₄⋊C₄	—	106	1	no	—
128	C₄⋊C₄	—	106	1	no	—
129	C₄⋊C₄	—	106	1	no	—
130	C₄⋊C₄	—	106	1	no	—
131	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	106	1	no	—
134	C₄²	—	53	1	no	—
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	106	1	no	—
139	C₄²	—	106	1	no	—
140	C₂³	C₂²⋊#6157	1	1	yes	characteristic
141	Q₈	—	2	2	no	—
142	Q₈	—	106	2	no	—
143	Q₈	—	212	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	106	2	no	—
154	C₂×C₄	—	106	1	no	—
155	C₂×C₄	—	106	1	no	—
156	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
157	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
158	C₈	—	106	4	no	—
159	C₈	—	212	2	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	#13133280	1	2	yes	—
163	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	53	1	no	—
168	C₄	—	53	2	no	—
169	C₄	—	53	1	no	—
170	C₄	—	53	2	no	—
171	C₄	—	106	1	no	—
172	C₄	—	212	1	no	—
173	C₄	—	212	1	no	—
174	C₄	#13133304	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	#16811596	1	1	yes	characteristic
177	C₂	C₂×#13133280	1	1	yes	characteristic
178	C₂	D₅₃⋊#104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811626

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂

yes

maximal, characteristic

#16811473

C₂

yes

maximal, complemented

Q₈⋊#88324

C₂

yes

maximal, complemented

#16811175

C₂

yes

maximal, characteristic

C₂×#11719940

—

C₂²×#88324

—

#13133284

—

C₄⋊#88324

C₂²

yes

—

C₄×#88324

—

C₄×#88324

C₂²

yes

—

#13133216

C₄

yes

characteristic

#13133215

C₄

yes

characteristic

#13133207

—

#13133196

C₄

yes

characteristic

#13133194

—

Q₈×Dic₅₃

C₂²

yes

characteristic

#13133182

—

#13133182

C₂²

yes

characteristic

#13133173

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

—

—

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

—

C₂×Dic₅₃

Q₁₆

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#103

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#103

yes

—

Dic₅₃

#104

yes

—

Dic₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

characteristic

#1493

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1493

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#125

—

106

—

101

C₂²⋊Q₈

—

102

C₂²⋊Q₈

—

106

—

103

C₂²⋊Q₈

#6157

yes

characteristic

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₄×Q₈

—

106

—

107

#117

—

106

—

108

C₂×C₄²

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

106

—

111

C₄×C₈

—

112

C₄×C₈

—

106

—

113

C₂×Q₈

—

114

C₂×Q₈

—

106

—

115

C₂×Q₈

C₂×#6157

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂×#6157

yes

characteristic

119

M₁₆

—

212

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

C₂×#6157

yes

characteristic

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

106

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

106

—

139

C₄²

—

106

—

140

C₂³

C₂²⋊#6157

yes

characteristic

141

Q₈

—

142

Q₈

—

106

—

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

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

C₂²⋊#6157

yes

characteristic

157

C₂×C₄

C₂²×#6157

yes

characteristic

158

C₈

—

106

—

159

C₈

—

212

—

160

C₂²

—

161

C₂²

—

162

C₂²

#13133280

yes

—

163

C₂²

C₂×C₂²⋊#6157

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

106

—

172

C₄

—

212

—

173

C₄

—

212

—

174

C₄

#13133304

yes

—

175

C₂

—

176

C₂

#16811596

yes

characteristic

177

C₂

C₂×#13133280

yes

characteristic

178

C₂

D₅₃⋊#104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂²⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂²⋊Q₈	#6157
hypocenter	C₅₃	#1493
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂²⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₂²⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

N	H ≅ G/N	type
3. #16811473	C₂	semidirect
4. Q₈⋊#88324	C₂	semidirect
95. C₅₃	#1493	semidirect

H ≅ G/N

type

3. #16811473

C₂

semidirect

4. Q₈⋊#88324

C₂

semidirect

95. C₅₃

#1493

semidirect

Subgroups — #20290063

Conjugacy classes of subgroups

Complements