Group #22598882 — subgroups

Subgroups — #22598882

Order 9088.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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	#17986556	C₂	1	1	yes	maximal, characteristic
2	#17986405	C₂	1	4	yes	maximal, complemented
3	#17986402	C₂	1	2	yes	maximal, complemented
4	#17986395	C₂	1	2	yes	maximal, complemented
5	#17986039	C₂	1	4	yes	maximal
6	C₄×#11765849	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	2	yes	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556120	C₂²	1	2	yes	—
12	#14556107	—	2	1	no	—
13	#14556107	—	2	1	no	—
14	C₄×Dic₁₄₂	—	2	1	no	—
15	C₄×Dic₁₄₂	—	2	1	no	—
16	#14556087	C₂²	1	2	yes	—
17	#14556061	C₂²	1	8	yes	—
18	#14556059	C₂²	1	8	yes	—
19	#14556039	C₂²	1	2	yes	—
20	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
21	C₂²×Q₈×C₇₁	—	2	1	no	—
22	C₄⋊Q₈×C₇₁	—	2	1	no	—
23	C₄⋊Q₈×C₇₁	—	2	1	no	—
24	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
25	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	2	1	no	—
30	C₂×Dic₁₄₂	—	2	1	no	—
31	#11765849	Q₈	1	8	yes	complemented
32	C₂×C₇₁⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	—	2	1	no	—
35	C₄⋊Dic₇₁	C₂³	1	2	yes	—
36	C₄⋊Dic₇₁	C₂³	1	2	yes	—
37	Dic₇₁⋊C₄	—	2	2	no	—
38	Dic₇₁⋊C₄	—	2	2	no	—
39	C₂²⋊Dic₇₁	—	2	2	no	—
40	C₄×Dic₇₁	—	2	2	no	—
41	C₂×Q₈×C₇₁	—	2	2	no	—
42	C₂×Q₈×C₇₁	—	2	2	no	—
43	C₂×Q₈×C₇₁	—	4	4	no	—
44	C₂²×C₄×C₇₁	—	2	2	no	—
45	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₇₁	—	2	2	no	—
49	C₄⋊C₄×C₇₁	—	2	2	no	—
50	C₄⋊C₄×C₇₁	—	2	2	no	—
51	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
52	C₄²×C₇₁	D₄	1	1	yes	characteristic
53	C₄²×C₇₁	D₄	1	1	yes	characteristic
54	C₄²×C₇₁	C₂³	1	1	yes	characteristic
55	C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₇₁	—	4	4	no	—
58	Q₈×C₇₁	—	4	4	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	4	no	—
62	C₂×C₄×C₇₁	—	4	4	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	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₇₁	—	2	2	no	—
71	C₂×Dic₇₁	—	2	2	no	—
72	Dic₁₄₂	—	4	2	no	—
73	Dic₁₄₂	—	4	2	no	—
74	C₇₁⋊C₈	C₂×Q₈	1	8	yes	complemented
75	C₂²×C₇₁	—	2	1	no	—
76	C₂²×C₇₁	—	2	1	no	—
77	C₂²×C₇₁	C₄⋊Q₈	1	2	yes	—
78	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	—	4	4	no	—
82	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
83	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
84	C₄×C₇₁	#137	1	2	yes	—
85	C₄×C₇₁	C₂²×Q₈	1	2	yes	—
86	Dic₇₁	—	4	2	no	—
87	Dic₇₁	—	4	2	no	—
88	C₂×C₇₁	—	2	2	no	—
89	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
92	#3165	—	71	1	no	maximal, Sylow
93	C₇₁	#3165	1	1	yes	characteristic, complemented, Sylow
94	#533	—	71	1	no	—
95	C₂×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
96	#501	—	71	4	no	—
97	#500	—	71	2	no	—
98	#498	—	71	2	no	—
99	#428	—	71	4	no	—
100	C₄×M₁₆	—	71	1	no	—
101	C₂²×Q₈	—	2	1	no	—
102	C₂×M₁₆	—	71	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	2	1	no	—
105	C₄⋊Q₈	—	71	1	no	—
106	C₄⋊Q₈	—	71	1	no	—
107	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
108	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
109	#125	—	71	2	no	—
110	C₂²⋊Q₈	—	142	1	no	—
111	C₂²⋊Q₈	—	142	1	no	—
112	C₄×Q₈	—	142	1	no	—
113	C₄×Q₈	—	142	1	no	—
114	#117	—	71	2	no	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
117	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
118	#107	—	71	8	no	—
119	#106	—	71	8	no	—
120	#97	—	71	2	no	—
121	C₄×C₈	—	71	2	no	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂×Q₈	—	142	1	no	—
126	C₂×Q₈	—	142	1	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
131	M₁₆	—	71	8	no	—
132	C₂×C₈	—	71	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	71	2	no	—
137	C₄⋊C₄	—	71	2	no	—
138	C₄⋊C₄	—	142	2	no	—
139	C₄⋊C₄	—	142	2	no	—
140	C₄⋊C₄	—	142	1	no	—
141	C₄⋊C₄	—	142	1	no	—
142	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
143	C₂²⋊C₄	—	142	2	no	—
144	C₄²	—	142	2	no	—
145	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₄²	C₂²×D₇₁	1	1	yes	characteristic
148	C₄²	C₂²×D₇₁	1	1	yes	characteristic
149	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	284	2	no	—
153	Q₈	—	284	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	142	2	no	—
159	C₂×C₄	—	142	2	no	—
160	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₇₁	1	4	yes	—
166	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
167	C₈	—	71	8	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#14556126	1	2	yes	—
171	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	284	2	no	—
176	C₄	—	284	2	no	—
177	C₄	#14556126	1	2	yes	—
178	C₄	#14556154	1	2	yes	—
179	C₄	#14556159	1	2	yes	—
180	C₄	C₂×Q₈×D₇₁	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	C₂×#14556126	1	1	yes	characteristic
183	C₂	C₂×#14556154	1	1	yes	characteristic
184	C₂	C₂×#14556159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

#17986405

C₂

yes

maximal, complemented

#17986402

C₂

yes

maximal, complemented

#17986395

C₂

yes

maximal, complemented

#17986039

C₂

yes

maximal

C₄×#11765849

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556120

C₂²

yes

—

#14556107

—

#14556107

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

#14556087

C₂²

yes

—

#14556061

C₂²

yes

—

#14556059

C₂²

yes

—

#14556039

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

#11765849

Q₈

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂×D₄

yes

characteristic

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

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

—

C₄×C₇₁

C₂²×Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

C₂×C₄⋊Q₈

D₇₁

yes

—

—

—

—

100

C₄×M₁₆

—

101

C₂²×Q₈

—

102

C₂×M₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

106

C₄⋊Q₈

—

107

C₄⋊Q₈

C₂×D₇₁

yes

—

108

C₄⋊Q₈

C₂×D₇₁

yes

—

109

#125

—

110

C₂²⋊Q₈

—

142

—

111

C₂²⋊Q₈

—

142

—

112

C₄×Q₈

—

142

—

113

C₄×Q₈

—

142

—

114

#117

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

117

C₂×C₄²

C₂×D₇₁

yes

characteristic

118

#107

—

119

#106

—

120

#97

—

121

C₄×C₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

142

—

126

C₂×Q₈

—

142

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

129

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

130

C₂²×C₄

C₂²×D₇₁

yes

characteristic

131

M₁₆

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

142

—

139

C₄⋊C₄

—

142

—

140

C₄⋊C₄

—

142

—

141

C₄⋊C₄

—

142

—

142

C₄⋊C₄

C₂²×D₇₁

yes

—

143

C₂²⋊C₄

—

142

—

144

C₄²

—

142

—

145

C₄²

C₂²⋊D₇₁

yes

characteristic

146

C₄²

C₂²⋊D₇₁

yes

characteristic

147

C₄²

C₂²×D₇₁

yes

characteristic

148

C₄²

C₂²×D₇₁

yes

characteristic

149

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

150

Q₈

—

151

Q₈

—

152

Q₈

—

284

—

153

Q₈

—

284

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

142

—

159

C₂×C₄

—

142

—

160

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

161

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

162

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

165

C₂×C₄

Q₈×D₇₁

yes

—

166

C₂×C₄

C₂³×D₇₁

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

#14556126

yes

—

171

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

284

—

176

C₄

—

284

—

177

C₄

#14556126

yes

—

178

C₄

#14556154

yes

—

179

C₄

#14556159

yes

—

180

C₄

C₂×Q₈×D₇₁

yes

—

181

C₂

—

182

C₂

C₂×#14556126

yes

characteristic

183

C₂

C₂×#14556154

yes

characteristic

184

C₂

C₂×#14556159

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₂×C₄⋊Q₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#3165
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂×C₄⋊Q₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₁

hypocenter

C₇₁

#3165

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #17986405	C₂	semidirect
3. #17986402	C₂	semidirect
4. #17986395	C₂	semidirect
31. #11765849	Q₈	semidirect
74. C₇₁⋊C₈	C₂×Q₈	semidirect
93. C₇₁	#3165	semidirect

H ≅ G/N

type

2. #17986405

C₂

semidirect

3. #17986402

C₂

semidirect

4. #17986395

C₂

semidirect

31. #11765849

Q₈

semidirect

74. C₇₁⋊C₈

C₂×Q₈

semidirect

93. C₇₁

#3165

semidirect

Subgroups — #22598882

Conjugacy classes of subgroups

Complements