Group #22587540 — subgroups

Subgroups — #22587540

Order 9088.

250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986345	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986284	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986204	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985743	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#550×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556059	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
13	#14556052	—	2	1	no	—
14	#14556051	—	2	1	no	—
15	#14556049	—	2	2	no	—
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#41×C₇₁	—	2	1	no	—
18	#127×C₇₁	—	2	1	no	—
19	#127×C₇₁	C₂²	1	1	yes	characteristic
20	#121×C₇₁	—	2	2	no	—
21	#121×C₇₁	—	2	1	no	—
22	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₁	—	2	1	no	—
24	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₂	—	2	1	no	—
26	#11765853	—	4	2	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	#41×C₇₁	—	4	2	no	—
33	#41×C₇₁	—	4	2	no	—
34	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂×D₄×C₇₁	—	2	2	no	—
36	C₂×D₄×C₇₁	—	2	2	no	—
37	C₂×D₄×C₇₁	—	2	1	no	—
38	C₂×D₄×C₇₁	—	2	1	no	—
39	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂×D₄×C₇₁	D₄	1	2	yes	—
41	C₂²×C₄×C₇₁	—	2	2	no	—
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₇₁	—	2	2	no	—
47	C₂²⋊C₄×C₇₁	—	2	1	no	—
48	C₄²×C₇₁	—	2	1	no	—
49	C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂³×C₇₁	—	2	1	no	—
51	C₂³×C₇₁	—	2	2	no	—
52	Q₈×C₇₁	—	2	1	no	—
53	Q₈×C₇₁	SD₁₆	1	2	yes	complemented
54	D₄×C₇₁	—	2	1	no	—
55	D₄×C₇₁	—	2	1	no	—
56	D₄×C₇₁	—	4	2	no	—
57	D₄×C₇₁	—	4	4	no	—
58	D₄×C₇₁	—	4	1	no	—
59	D₄×C₇₁	—	4	1	no	—
60	D₄×C₇₁	—	4	1	no	—
61	D₄×C₇₁	—	4	1	no	—
62	D₄×C₇₁	—	8	2	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	2	1	no	—
65	C₂×C₄×C₇₁	—	2	1	no	—
66	C₂×C₄×C₇₁	—	4	4	no	—
67	C₂×C₄×C₇₁	—	4	1	no	—
68	C₂×C₄×C₇₁	—	4	1	no	—
69	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₇₁	—	4	1	no	—
73	Dic₁₄₂	—	4	1	no	—
74	Dic₁₄₂	—	4	1	no	—
75	C₇₁⋊C₈	—	4	1	no	—
76	C₇₁⋊C₈	—	4	1	no	—
77	C₇₁⋊C₈	—	4	1	no	—
78	C₂²×C₇₁	—	4	1	no	—
79	C₂²×C₇₁	—	4	2	no	—
80	C₂²×C₇₁	—	4	2	no	—
81	C₂²×C₇₁	—	4	1	no	—
82	C₂²×C₇₁	—	4	2	no	—
83	C₂²×C₇₁	—	4	1	no	—
84	C₂²×C₇₁	#120	1	1	yes	characteristic
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	—	2	1	no	—
87	C₄×C₇₁	—	4	1	no	—
88	C₄×C₇₁	—	4	1	no	—
89	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₄×C₇₁	#135	1	1	yes	characteristic
91	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
93	Dic₇₁	—	8	1	no	—
94	C₂×C₇₁	—	8	1	no	—
95	C₂×C₇₁	—	8	2	no	—
96	C₂×C₇₁	#449	1	1	yes	characteristic
97	C₂×C₇₁	#450	1	1	yes	characteristic
98	C₂×C₇₁	#455	1	1	yes	characteristic
99	#1642	—	71	1	no	maximal, Sylow
100	C₇₁	#1642	1	1	yes	characteristic, complemented, Sylow
101	#550	D₇₁	1	1	yes	characteristic
102	#486	—	71	1	no	—
103	#475	—	71	1	no	—
104	#461	—	71	1	no	—
105	#331	—	71	1	no	—
106	Q₈⋊C₈	—	71	1	no	—
107	D₄⋊C₈	—	71	1	no	—
108	C₂×#41	—	2	1	no	—
109	C₂×SD₁₆	—	142	1	no	—
110	C₄⋊Q₈	—	71	1	no	—
111	#127	—	2	1	no	—
112	#127	C₂×D₇₁	1	1	yes	characteristic
113	#121	—	2	2	no	—
114	#121	—	2	1	no	—
115	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
116	C₄×D₄	—	2	1	no	—
117	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
118	#106	—	142	1	no	—
119	C₄⋊C₈	—	71	1	no	—
120	C₄⋊C₈	—	71	1	no	—
121	#103	—	142	1	no	—
122	#103	—	142	1	no	—
123	#102	—	142	2	no	—
124	C₄×C₈	—	71	1	no	—
125	#41	—	4	2	no	—
126	#41	—	4	2	no	—
127	C₂×Q₈	—	142	1	no	—
128	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
134	C₂×D₄	C₂²⋊D₇₁	1	2	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	SD₁₆	—	284	2	no	—
138	C₂×C₈	—	142	1	no	—
139	C₂×C₈	—	142	1	no	—
140	C₂×C₈	—	142	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	142	1	no	—
143	C₄⋊C₄	—	142	1	no	—
144	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
145	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	C₂²×D₇₁	1	1	yes	characteristic
150	C₂³	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	Q₈	—	2	1	no	—
153	Q₈	—	284	1	no	—
154	Q₈	—	284	1	no	—
155	Q₈	#11765853	1	2	yes	complemented
156	D₄	—	2	1	no	—
157	D₄	—	2	1	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	4	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	4	1	no	—
164	D₄	—	8	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	284	1	no	—
172	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
175	C₈	—	284	1	no	—
176	C₈	—	284	1	no	—
177	C₈	—	284	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	#14556097	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	568	1	no	—
190	C₄	C₂×#11765853	1	1	yes	characteristic
191	C₄	#14556148	1	1	yes	characteristic
192	C₄	#14556153	1	1	yes	characteristic
193	C₄	#14556154	1	1	yes	characteristic
194	C₂	—	8	1	no	—
195	C₂	—	8	2	no	—
196	C₂	#17986144	1	1	yes	characteristic
197	C₂	#17986150	1	1	yes	characteristic
198	C₂	#17986177	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986345

C₂

yes

maximal, characteristic, complemented

#17986284

C₂

yes

maximal, characteristic, complemented

#17986204

C₂

yes

maximal, characteristic, complemented

#17985743

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

#550×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

#14556128

C₂²

yes

characteristic

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

—

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×#41×C₇₁

—

#127×C₇₁

—

#127×C₇₁

C₂²

yes

characteristic

#121×C₇₁

—

#121×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

—

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

SD₁₆

yes

complemented

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×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×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

100

C₇₁

#1642

yes

characteristic, complemented, Sylow

101

#550

D₇₁

yes

characteristic

102

#486

—

103

#475

—

104

#461

—

105

#331

—

106

Q₈⋊C₈

—

107

D₄⋊C₈

—

108

C₂×#41

—

109

C₂×SD₁₆

—

142

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

C₂×D₇₁

yes

characteristic

113

#121

—

114

#121

—

115

C₄×Q₈

C₂×D₇₁

yes

characteristic

116

C₄×D₄

—

117

C₄×D₄

C₂×D₇₁

yes

characteristic

118

#106

—

142

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

142

—

122

#103

—

142

—

123

#102

—

142

—

124

C₄×C₈

—

125

#41

—

126

#41

—

127

C₂×Q₈

—

142

—

128

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

134

C₂×D₄

C₂²⋊D₇₁

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

SD₁₆

—

284

—

138

C₂×C₈

—

142

—

139

C₂×C₈

—

142

—

140

C₂×C₈

—

142

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

142

—

143

C₄⋊C₄

—

142

—

144

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

145

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

C₂²×D₇₁

yes

characteristic

150

C₂³

—

151

C₂³

—

152

Q₈

—

153

Q₈

—

284

—

154

Q₈

—

284

—

155

Q₈

#11765853

yes

complemented

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

284

—

172

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

173

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

175

C₈

—

284

—

176

C₈

—

284

—

177

C₈

—

284

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#14556097

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

568

—

190

C₄

C₂×#11765853

yes

characteristic

191

C₄

#14556148

yes

characteristic

192

C₄

#14556153

yes

characteristic

193

C₄

#14556154

yes

characteristic

194

C₂

—

195

C₂

—

196

C₂

#17986144

yes

characteristic

197

C₂

#17986150

yes

characteristic

198

C₂

#17986177

yes

characteristic

	subgroup	G/H
center	C₂²	#14556097
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#550×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#550	D₇₁
hypocenter	C₇₁	#1642
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #550

subgroup

G/H

center

C₂²

#14556097

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#550×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #550

N	H ≅ G/N	type
1. #17986345	C₂	semidirect
2. #17986284	C₂	semidirect
3. #17986204	C₂	semidirect
4. #17985743	C₂	semidirect
5. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
6. D₄⋊(C₇₁⋊C₈)	C₂	semidirect
53. Q₈×C₇₁	SD₁₆	semidirect
100. C₇₁	#1642	semidirect
155. Q₈	#11765853	semidirect

H ≅ G/N

type

1. #17986345

C₂

semidirect

2. #17986284

C₂

semidirect

3. #17986204

C₂

semidirect

4. #17985743

C₂

semidirect

5. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

6. D₄⋊(C₇₁⋊C₈)

C₂

semidirect

53. Q₈×C₇₁

SD₁₆

semidirect

100. C₇₁

#1642

semidirect

155. Q₈

#11765853

semidirect

Subgroups — #22587540

Conjugacy classes of subgroups

Complements