Group #20290018 — subgroups

Subgroups — #20290018

Order 6784.

248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811642	C₂	1	1	yes	maximal, characteristic
2	#16811641	C₂	1	1	yes	maximal, characteristic
3	#16811473	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#88324	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133284	—	2	1	no	—
7	C₄⋊#88324	C₂²	1	2	yes	—
8	#13133274	—	2	1	no	—
9	#13133274	—	2	1	no	—
10	C₄×#88324	—	2	1	no	—
11	C₄×#88324	C₂²	1	2	yes	—
12	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₀₆	—	2	1	no	—
15	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
16	#13133182	—	2	1	no	—
17	#13133182	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₅₃	—	2	1	no	—
19	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
20	C₄²×D₅₃	—	2	1	no	—
21	C₄²×D₅₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	Q₈×D₅₃	—	2	2	no	—
25	Q₈×D₅₃	C₈	1	4	yes	complemented
26	C₂×C₄×D₅₃	—	2	1	no	—
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₆	—	2	1	no	—
32	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
33	C₂×#88324	—	2	2	no	—
34	C₂×#88324	—	2	2	no	—
35	C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	—	2	1	no	—
38	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
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	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	1	no	—
52	Q₈×C₅₃	C₂×C₈	1	2	yes	—
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×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄×D₅₃	—	2	4	no	—
64	C₄×D₅₃	—	2	2	no	—
65	C₄×D₅₃	—	4	1	no	—
66	C₄×D₅₃	—	4	2	no	—
67	C₄×D₅₃	C₂×C₈	1	2	yes	—
68	C₄×D₅₃	M₁₆	1	2	yes	—
69	Dic₁₀₆	—	2	1	no	—
70	Dic₁₀₆	—	2	4	no	—
71	Dic₁₀₆	C₂×C₈	1	2	yes	—
72	#88324	—	4	2	no	—
73	#88324	—	4	2	no	—
74	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₅₃	—	2	1	no	—
76	C₂×D₅₃	—	2	1	no	—
77	C₂×D₅₃	C₂²⋊C₈	1	2	yes	—
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	4	1	no	—
82	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	Dic₅₃	—	2	2	no	—
85	Dic₅₃	—	2	1	no	—
86	Dic₅₃	—	4	1	no	—
87	Dic₅₃	C₂²⋊C₈	1	2	yes	—
88	Dic₅₃	#137	1	2	yes	—
89	#1479	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₅₃	#419	1	1	yes	characteristic
92	C₂×C₅₃	#421	1	1	yes	characteristic
93	D₅₃	—	2	2	no	—
94	C₂×C₄×Q₈	—	53	1	no	—
95	#431	—	53	1	no	—
96	#431	—	53	1	no	—
97	Q₈⋊C₈	—	53	2	no	—
98	Q₈⋊C₈	—	53	2	no	—
99	C₅₃	#1479	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	—	53	1	no	—
101	C₄×Q₈	—	53	1	no	—
102	C₄×Q₈	—	53	1	no	—
103	C₄×Q₈	—	53	1	no	—
104	C₄×Q₈	—	106	1	no	—
105	C₄×Q₈	—	106	1	no	—
106	C₄×Q₈	#6157	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	53	1	no	—
108	C₂×C₄⋊C₄	—	106	1	no	—
109	C₂×C₄²	—	53	1	no	—
110	C₂×C₄²	—	106	1	no	—
111	C₄⋊C₈	—	53	2	no	—
112	C₄⋊C₈	—	106	1	no	—
113	C₂²⋊C₈	—	106	1	no	—
114	C₂²⋊C₈	—	106	1	no	—
115	C₄×C₈	—	53	2	no	—
116	C₄×C₈	—	106	1	no	—
117	C₂×Q₈	—	53	4	no	—
118	C₂×Q₈	—	53	1	no	—
119	C₂×Q₈	—	106	1	no	—
120	C₂×Q₈	—	106	2	no	—
121	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	—	53	1	no	—
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	106	1	no	—
126	C₂²×C₄	—	106	1	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	53	1	no	—
131	C₄⋊C₄	—	106	1	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₄²	—	53	1	no	—
140	C₄²	—	106	1	no	—
141	C₄²	—	106	1	no	—
142	C₄²	—	106	1	no	—
143	C₄²	C₂×#6157	1	1	yes	characteristic
144	C₂³	—	53	1	no	—
145	Q₈	—	2	1	no	—
146	Q₈	—	53	2	no	—
147	Q₈	—	106	4	no	—
148	Q₈	—	106	1	no	—
149	Q₈	D₅₃⋊C₈	1	2	yes	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	2	no	—
155	C₂×C₄	—	53	2	no	—
156	C₂×C₄	—	53	1	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	4	no	—
159	C₂×C₄	—	106	1	no	—
160	C₂×C₄	—	106	1	no	—
161	C₂×C₄	—	212	1	no	—
162	C₂×C₄	—	212	2	no	—
163	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
164	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
165	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
166	C₈	—	212	2	no	—
167	C₈	—	212	2	no	—
168	C₂²	—	53	2	no	—
169	C₂²	—	106	1	no	—
170	C₂²	—	106	1	no	—
171	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	53	2	no	—
177	C₄	—	53	2	no	—
178	C₄	—	106	2	no	—
179	C₄	—	106	1	no	—
180	C₄	—	212	1	no	—
181	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
182	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
183	C₂	—	106	2	no	—
184	C₂	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
185	C₂	#16811616	1	1	yes	characteristic
186	C₂	#16811620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811642

C₂

yes

maximal, characteristic

#16811641

C₂

yes

maximal, characteristic

#16811473

C₂

yes

maximal, complemented

Q₈⋊#88324

C₂

yes

maximal, complemented

C₄×Q₈×D₅₃

C₂

yes

maximal, characteristic

#13133284

—

C₄⋊#88324

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Q₈×D₅₃

C₄

yes

characteristic

Q₈×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

characteristic

#13133182

—

#13133182

C₂²

yes

characteristic

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₄

yes

characteristic

C₄²×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

Q₈×D₅₃

—

Q₈×D₅₃

C₈

yes

complemented

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×#88324

—

C₂×#88324

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

C₂×C₈

yes

—

C₂×C₄×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

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

M₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×C₈

yes

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊C₈

yes

—

Dic₅₃

#137

yes

—

#1479

—

maximal, Sylow

C₂×C₅₃

C₂×C₂²⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×C₄×Q₈

—

#431

—

#431

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1479

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

—

101

C₄×Q₈

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₄×Q₈

—

106

—

105

C₄×Q₈

—

106

—

106

C₄×Q₈

#6157

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄²

—

110

C₂×C₄²

—

106

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

106

—

113

C₂²⋊C₈

—

106

—

114

C₂²⋊C₈

—

106

—

115

C₄×C₈

—

116

C₄×C₈

—

106

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

106

—

120

C₂×Q₈

—

106

—

121

C₂×Q₈

C₂×#6157

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

106

—

126

C₂²×C₄

—

106

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

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

C₂×#6157

yes

characteristic

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

106

—

141

C₄²

—

106

—

142

C₄²

—

106

—

143

C₄²

C₂×#6157

yes

characteristic

144

C₂³

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

106

—

148

Q₈

—

106

—

149

Q₈

D₅₃⋊C₈

yes

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

212

—

162

C₂×C₄

—

212

—

163

C₂×C₄

C₂²⋊#6157

yes

characteristic

164

C₂×C₄

C₂²⋊#6157

yes

characteristic

165

C₂×C₄

C₂²×#6157

yes

characteristic

166

C₈

—

212

—

167

C₈

—

212

—

168

C₂²

—

169

C₂²

—

106

—

170

C₂²

—

106

—

171

C₂²

C₂×C₂²⋊#6157

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

106

—

179

C₄

—

106

—

180

C₄

—

212

—

181

C₄

C₂×D₅₃⋊C₈

yes

characteristic

182

C₄

D₅₃⋊M₁₆

yes

characteristic

183

C₂

—

106

—

184

C₂

D₅₃⋊(C₂²⋊C₈)

yes

characteristic

185

C₂

#16811616

yes

characteristic

186

C₂

#16811620

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	#1479
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄×Q₈

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #16811473	C₂	semidirect
4. Q₈⋊#88324	C₂	semidirect
25. Q₈×D₅₃	C₈	semidirect
99. C₅₃	#1479	semidirect

H ≅ G/N

type

3. #16811473

C₂

semidirect

4. Q₈⋊#88324

C₂

semidirect

25. Q₈×D₅₃

C₈

semidirect

99. C₅₃

#1479

semidirect

Subgroups — #20290018

Conjugacy classes of subgroups

Complements