Group #22589607 — subgroups

Subgroups — #22589607

Order 9088.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×#11765855	C₂	1	1	yes	maximal, characteristic
2	#17986026	C₂	1	1	yes	maximal, characteristic
3	#17986007	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556047	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556046	C₂	1	1	yes	maximal, characteristic
6	#17985763	C₂	1	1	yes	maximal, characteristic
7	#419×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	C₂×#11765855	—	2	2	no	—
10	C₂×#11765855	—	2	2	no	—
11	C₂×#11765855	—	2	2	no	—
12	C₂×#11765849	C₂²	1	1	yes	characteristic
13	C₂×#11765849	C₂²	1	1	yes	characteristic
14	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
16	#14556051	—	2	2	no	—
17	#14556047	—	2	2	no	—
18	#14556046	—	2	2	no	—
19	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	#117×C₇₁	C₂²	1	1	yes	characteristic
22	#103×C₇₁	—	2	2	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	C₂×Dic₁₄₂	—	2	1	no	—
28	C₂×Dic₁₄₂	D₄	1	2	yes	—
29	C₂²×Dic₇₁	—	2	1	no	—
30	#11765855	—	2	8	no	—
31	#11765849	—	2	1	no	—
32	#11765849	—	2	1	no	—
33	#11765849	—	2	1	no	—
34	#11765849	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₂×C₇₁⋊C₈	—	2	1	no	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×Q₈×C₇₁	—	2	1	no	—
42	C₂×Q₈×C₇₁	—	2	1	no	—
43	C₂×Q₈×C₇₁	—	2	1	no	—
44	C₂×Q₈×C₇₁	—	2	1	no	—
45	C₂×Q₈×C₇₁	D₄	1	2	yes	—
46	C₂²×C₄×C₇₁	—	2	1	no	—
47	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
48	M₁₆×C₇₁	—	2	1	no	—
49	M₁₆×C₇₁	—	2	1	no	—
50	C₂×C₈×C₇₁	—	2	1	no	—
51	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
52	C₂²⋊C₄×C₇₁	—	2	1	no	—
53	C₄²×C₇₁	—	2	1	no	—
54	C₂³×C₇₁	#41	1	1	yes	characteristic
55	Q₈×C₇₁	—	2	2	no	—
56	Q₈×C₇₁	—	2	2	no	—
57	Q₈×C₇₁	—	4	1	no	—
58	Q₈×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	4	1	no	—
62	C₂×C₄×C₇₁	—	4	1	no	—
63	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	—	4	1	no	—
71	C₂×Dic₇₁	—	4	1	no	—
72	Dic₁₄₂	—	2	2	no	—
73	Dic₁₄₂	—	2	2	no	—
74	Dic₁₄₂	—	4	1	no	—
75	Dic₁₄₂	—	4	1	no	—
76	C₈×C₇₁	—	4	1	no	—
77	C₇₁⋊C₈	—	2	2	no	—
78	C₇₁⋊C₈	—	4	1	no	—
79	C₇₁⋊C₈	—	4	1	no	—
80	C₂²×C₇₁	—	2	2	no	—
81	C₂²×C₇₁	#121	1	1	yes	characteristic
82	C₂²×C₇₁	#123	1	1	yes	characteristic
83	C₂²×C₇₁	#123	1	1	yes	characteristic
84	C₄×C₇₁	—	4	2	no	—
85	C₄×C₇₁	—	4	2	no	—
86	C₄×C₇₁	#120	1	1	yes	characteristic
87	C₄×C₇₁	#121	1	1	yes	characteristic
88	C₄×C₇₁	#121	1	1	yes	characteristic
89	C₄×C₇₁	#124	1	1	yes	characteristic
90	Dic₇₁	—	4	2	no	—
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#394	1	1	yes	characteristic
94	C₂×C₇₁	#473	1	2	yes	—
95	#2047	—	71	1	no	maximal, Sylow
96	C₇₁	#2047	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₁₆	—	71	1	no	—
98	#424	—	71	1	no	—
99	#419	—	71	1	no	—
100	#419	D₇₁	1	1	yes	characteristic
101	C₂×#101	—	71	1	no	—
102	C₂×#101	—	71	1	no	—
103	#341	—	71	1	no	—
104	C₂²×Q₈	—	71	1	no	—
105	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂×Q₁₆	—	142	2	no	—
107	C₂×Q₁₆	—	142	2	no	—
108	C₂×Q₁₆	—	142	2	no	—
109	C₂×M₁₆	—	71	1	no	—
110	C₂×M₁₆	—	71	1	no	—
111	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
112	C₂²×C₈	—	71	1	no	—
113	#117	C₂×D₇₁	1	1	yes	characteristic
114	C₄⋊C₈	—	142	2	no	—
115	#103	—	2	2	no	—
116	#103	—	142	2	no	—
117	#101	—	142	2	no	—
118	#101	—	142	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	1	no	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	—	71	2	no	—
125	C₂×Q₈	—	142	1	no	—
126	C₂×Q₈	—	142	1	no	—
127	C₂×Q₈	—	142	1	no	—
128	C₂×Q₈	—	142	1	no	—
129	C₂×Q₈	—	142	1	no	—
130	C₂×Q₈	D₂₈₄	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	142	1	no	—
133	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
134	Q₁₆	—	142	8	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	142	1	no	—
138	M₁₆	—	142	1	no	—
139	M₁₆	—	142	1	no	—
140	M₁₆	—	142	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	71	2	no	—
143	C₂×C₈	—	142	1	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	—	142	1	no	—
146	C₂×C₈	—	142	1	no	—
147	C₄⋊C₄	C₂²⋊D₇₁	1	2	yes	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₂³	#11765864	1	1	yes	characteristic
151	Q₈	—	2	2	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	4	1	no	—
154	Q₈	—	4	1	no	—
155	Q₈	—	284	1	no	—
156	Q₈	—	284	1	no	—
157	Q₈	—	142	2	no	—
158	Q₈	—	142	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	284	1	no	—
164	C₂×C₄	—	284	1	no	—
165	C₂×C₄	—	142	2	no	—
166	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
167	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
168	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
169	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
170	C₂×C₄	#11765864	1	1	yes	characteristic
171	C₂×C₄	#11765865	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	284	1	no	—
174	C₈	—	284	1	no	—
175	C₈	—	142	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#14556100	1	1	yes	characteristic
178	C₂²	#14556110	1	1	yes	characteristic
179	C₂²	#14556112	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	284	2	no	—
183	C₄	#14556098	1	1	yes	characteristic
184	C₄	#14556099	1	1	yes	characteristic
185	C₄	#14556102	1	1	yes	characteristic
186	C₄	#14556114	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#17985915	1	1	yes	characteristic
190	C₂	#17986272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11765855

C₂

yes

maximal, characteristic

#17986026

C₂

yes

maximal, characteristic

#17986007

C₂

yes

maximal, characteristic

C₂×#14556047

C₂

yes

maximal, characteristic

C₂×#14556046

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal, characteristic

#419×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

#14556051

—

#14556047

—

#14556046

—

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

—

C₂²×Dic₇₁

—

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

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×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#394

yes

characteristic

C₂×C₇₁

#473

yes

—

#2047

—

maximal, Sylow

C₇₁

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

#424

—

#419

—

100

#419

D₇₁

yes

characteristic

101

C₂×#101

—

102

C₂×#101

—

103

#341

—

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂×D₇₁

yes

characteristic

106

C₂×Q₁₆

—

142

—

107

C₂×Q₁₆

—

142

—

108

C₂×Q₁₆

—

142

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

C₂×D₇₁

yes

characteristic

112

C₂²×C₈

—

113

#117

C₂×D₇₁

yes

characteristic

114

C₄⋊C₈

—

142

—

115

#103

—

116

#103

—

142

—

117

#101

—

142

—

118

#101

—

142

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

142

—

126

C₂×Q₈

—

142

—

127

C₂×Q₈

—

142

—

128

C₂×Q₈

—

142

—

129

C₂×Q₈

—

142

—

130

C₂×Q₈

D₂₈₄

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

142

—

133

C₂²×C₄

C₂²×D₇₁

yes

characteristic

134

Q₁₆

—

142

—

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

142

—

138

M₁₆

—

142

—

139

M₁₆

—

142

—

140

M₁₆

—

142

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

142

—

144

C₂×C₈

—

142

—

145

C₂×C₈

—

142

—

146

C₂×C₈

—

142

—

147

C₄⋊C₄

C₂²⋊D₇₁

yes

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₂³

#11765864

yes

characteristic

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

Q₈

—

284

—

156

Q₈

—

284

—

157

Q₈

—

142

—

158

Q₈

—

142

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

284

—

164

C₂×C₄

—

284

—

165

C₂×C₄

—

142

—

166

C₂×C₄

D₄×D₇₁

yes

characteristic

167

C₂×C₄

D₄×D₇₁

yes

characteristic

168

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

169

C₂×C₄

C₂×D₂₈₄

yes

characteristic

170

C₂×C₄

#11765864

yes

characteristic

171

C₂×C₄

#11765865

yes

characteristic

172

C₈

—

173

C₈

—

284

—

174

C₈

—

284

—

175

C₈

—

142

—

176

C₂²

—

177

C₂²

#14556100

yes

characteristic

178

C₂²

#14556110

yes

characteristic

179

C₂²

#14556112

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

284

—

183

C₄

#14556098

yes

characteristic

184

C₄

#14556099

yes

characteristic

185

C₄

#14556102

yes

characteristic

186

C₄

#14556114

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#17985915

yes

characteristic

190

C₂

#17986272

yes

—

	subgroup	G/H
center	C₂²	#14556100
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#419×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#419	D₇₁
hypocenter	C₇₁	#2047
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14556100

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#419×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
96. C₇₁	#2047	semidirect

H ≅ G/N

type

96. C₇₁

#2047

semidirect

Subgroups — #22589607

Conjugacy classes of subgroups

Complements