Group #22588333 — subgroups

Subgroups — #22588333

Order 9088.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	#17986023	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556046	C₂	1	2	yes	maximal
4	#17985767	C₂	1	2	yes	maximal
5	#423×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
7	C₂×#11765849	C₂²	1	2	yes	—
8	C₄×Dic₁₄₂	—	2	1	no	—
9	C₄×Dic₁₄₂	—	2	1	no	—
10	C₄×Dic₁₄₂	—	2	1	no	—
11	C₄×Dic₁₄₂	—	2	1	no	—
12	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₁⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₇₁	C₄	1	2	yes	—
15	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
17	#14556046	—	2	4	no	—
18	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₁	—	2	1	no	—
21	C₄⋊C₈×C₇₁	—	2	1	no	—
22	C₂×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	D₄	1	2	yes	—
27	C₂×Dic₁₄₂	Q₈	1	2	yes	—
28	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
29	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
30	#11765849	—	2	2	no	—
31	#11765849	—	2	2	no	—
32	C₂×C₇₁⋊C₈	—	2	2	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	—	2	1	no	—
35	Dic₇₁⋊C₄	—	2	2	no	—
36	Dic₇₁⋊C₄	—	2	2	no	—
37	C₄×Dic₇₁	—	2	2	no	—
38	C₄×Dic₇₁	—	2	2	no	—
39	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	M₁₆×C₇₁	—	2	2	no	—
43	M₁₆×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	2	2	no	—
45	C₄²×C₇₁	D₄	1	1	yes	characteristic
46	C₄²×C₇₁	D₄	1	1	yes	characteristic
47	C₄²×C₇₁	D₄	1	1	yes	characteristic
48	C₄²×C₇₁	D₄	1	1	yes	characteristic
49	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₂×Dic₇₁	—	2	2	no	—
63	C₂×Dic₇₁	—	2	2	no	—
64	C₂×Dic₇₁	—	2	2	no	—
65	C₂×Dic₇₁	—	4	2	no	—
66	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
67	Dic₁₄₂	—	2	2	no	—
68	Dic₁₄₂	—	2	2	no	—
69	Dic₁₄₂	—	2	2	no	—
70	Dic₁₄₂	—	2	2	no	—
71	C₈×C₇₁	—	4	2	no	—
72	C₇₁⋊C₈	—	4	2	no	—
73	C₂²×C₇₁	—	2	1	no	—
74	C₂²×C₇₁	—	2	1	no	—
75	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₁	#117	1	1	yes	characteristic
78	C₄×C₇₁	—	2	1	no	—
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	#101	1	2	yes	complemented
82	C₄×C₇₁	#121	1	1	yes	characteristic
83	C₄×C₇₁	#121	1	1	yes	characteristic
84	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
85	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₇₁	—	2	4	no	—
87	Dic₇₁	—	4	2	no	—
88	C₂×C₇₁	—	2	1	no	—
89	C₂×C₇₁	—	2	1	no	—
90	C₂×C₇₁	#380	1	1	yes	characteristic
91	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
92	C₂×C₇₁	#413	1	1	yes	characteristic
93	#1801	—	71	1	no	maximal, Sylow
94	C₇₁	#1801	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄×Q₈	—	71	1	no	—
96	#423	—	71	1	no	—
97	#423	D₇₁	1	1	yes	characteristic
98	C₂×#101	—	71	2	no	—
99	#343	—	71	2	no	—
100	C₂²×Q₈	—	71	1	no	—
101	C₂×M₁₆	—	71	2	no	—
102	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
103	C₄×Q₈	—	142	1	no	—
104	C₄×Q₈	—	142	1	no	—
105	C₄×Q₈	—	142	1	no	—
106	C₄×Q₈	—	142	1	no	—
107	C₂×C₄⋊C₄	—	71	2	no	—
108	C₂×C₄⋊C₄	—	71	1	no	—
109	C₂×C₄²	—	71	2	no	—
110	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	142	1	no	—
114	C₄⋊C₈	—	142	1	no	—
115	#101	—	142	4	no	—
116	C₂×Q₈	—	71	2	no	—
117	C₂×Q₈	—	71	2	no	—
118	C₂×Q₈	—	142	1	no	—
119	C₂×Q₈	—	142	1	no	—
120	C₂×Q₈	—	142	1	no	—
121	C₂×Q₈	—	142	1	no	—
122	C₂²×C₄	—	71	2	no	—
123	C₂²×C₄	—	71	2	no	—
124	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
125	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	2	no	—
129	M₁₆	—	142	2	no	—
130	M₁₆	—	142	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	142	2	no	—
133	C₄⋊C₄	—	142	2	no	—
134	C₄⋊C₄	—	142	2	no	—
135	C₄⋊C₄	—	142	1	no	—
136	C₄⋊C₄	—	142	1	no	—
137	C₄²	—	142	2	no	—
138	C₄²	—	142	2	no	—
139	C₄²	D₂₈₄	1	1	yes	characteristic
140	C₄²	D₂₈₄	1	1	yes	characteristic
141	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
143	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
144	Q₈	—	142	2	no	—
145	Q₈	—	142	2	no	—
146	Q₈	—	142	2	no	—
147	Q₈	—	142	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	284	2	no	—
151	C₂×C₄	—	71	4	no	—
152	C₂×C₄	—	142	2	no	—
153	C₂×C₄	—	142	2	no	—
154	C₂×C₄	—	142	2	no	—
155	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
157	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
158	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
161	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
163	C₂×C₄	#11765864	1	1	yes	characteristic
164	C₂×C₄	#11765866	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₈	—	284	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
170	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
171	C₂²	#14556086	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	284	2	no	—
176	C₄	—	142	4	no	—
177	C₄	#14556046	1	2	yes	complemented
178	C₄	#14556099	1	1	yes	characteristic
179	C₄	#14556103	1	1	yes	characteristic
180	C₄	#14556104	1	1	yes	characteristic
181	C₄	#14556108	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#17985856	1	1	yes	characteristic
185	C₂	C₂×#14556046	1	1	yes	characteristic
186	C₂	#17985981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₂

C₂

yes

maximal, characteristic

#17986023

C₂

yes

maximal, characteristic

C₂×#14556046

C₂

yes

maximal

#17985767

C₂

yes

maximal

#423×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₇₁

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556046

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

—

C₂×Dic₁₄₂

Q₈

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#117

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#423

—

#423

D₇₁

yes

characteristic

C₂×#101

—

#343

—

100

C₂²×Q₈

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₇₁

yes

—

103

C₄×Q₈

—

142

—

104

C₄×Q₈

—

142

—

105

C₄×Q₈

—

142

—

106

C₄×Q₈

—

142

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×D₇₁

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

142

—

114

C₄⋊C₈

—

142

—

115

#101

—

142

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

142

—

119

C₂×Q₈

—

142

—

120

C₂×Q₈

—

142

—

121

C₂×Q₈

—

142

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₄×D₇₁

yes

characteristic

125

C₂²×C₄

C₄×D₇₁

yes

characteristic

126

C₂²×C₄

C₂²×D₇₁

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

M₁₆

—

142

—

130

M₁₆

—

142

—

131

C₂×C₈

—

132

C₂×C₈

—

142

—

133

C₄⋊C₄

—

142

—

134

C₄⋊C₄

—

142

—

135

C₄⋊C₄

—

142

—

136

C₄⋊C₄

—

142

—

137

C₄²

—

142

—

138

C₄²

—

142

—

139

C₄²

D₂₈₄

yes

characteristic

140

C₄²

D₂₈₄

yes

characteristic

141

C₄²

C₂²⋊D₇₁

yes

characteristic

142

C₄²

C₂²⋊D₇₁

yes

characteristic

143

C₂³

C₂×C₄×D₇₁

yes

characteristic

144

Q₈

—

142

—

145

Q₈

—

142

—

146

Q₈

—

142

—

147

Q₈

—

142

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

284

—

151

C₂×C₄

—

152

C₂×C₄

—

142

—

153

C₂×C₄

—

142

—

154

C₂×C₄

—

142

—

155

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

156

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

157

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

158

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

159

C₂×C₄

D₄×D₇₁

yes

characteristic

160

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

161

C₂×C₄

C₂×D₂₈₄

yes

characteristic

162

C₂×C₄

Q₈×D₇₁

yes

characteristic

163

C₂×C₄

#11765864

yes

characteristic

164

C₂×C₄

#11765866

yes

characteristic

165

C₈

—

166

C₈

—

284

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

170

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

171

C₂²

#14556086

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

284

—

176

C₄

—

142

—

177

C₄

#14556046

yes

complemented

178

C₄

#14556099

yes

characteristic

179

C₄

#14556103

yes

characteristic

180

C₄

#14556104

yes

characteristic

181

C₄

#14556108

yes

characteristic

182

C₂

—

183

C₂

—

184

C₂

#17985856

yes

characteristic

185

C₂

C₂×#14556046

yes

characteristic

186

C₂

#17985981

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂³×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#423×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#423	D₇₁
hypocenter	C₇₁	#1801
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂³×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#423×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
81. C₄×C₇₁	#101	semidirect
94. C₇₁	#1801	semidirect
177. C₄	#14556046	semidirect

H ≅ G/N

type

81. C₄×C₇₁

#101

semidirect

94. C₇₁

#1801

semidirect

177. C₄

#14556046

semidirect

Subgroups — #22588333

Conjugacy classes of subgroups

Complements