Group #22587601 — subgroups

Subgroups — #22587601

Order 9088.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	D₄×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	#17986395	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986281	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985744	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985729	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#461×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₂	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556127	—	2	1	no	—
11	#14556107	—	2	1	no	—
12	#14556105	—	2	2	no	—
13	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
14	D₄×Dic₇₁	—	2	1	no	—
15	#14556059	—	2	2	no	—
16	#14556059	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	#14556051	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₇₁	—	2	1	no	—
21	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#103×C₇₁	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	2	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	C₂×Dic₁₄₂	—	4	2	no	—
28	C₂×Dic₁₄₂	—	4	2	no	—
29	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
30	C₂²×Dic₇₁	—	2	2	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₁	D₄	1	2	yes	—
36	Dic₇₁⋊C₄	—	2	2	no	—
37	Dic₇₁⋊C₄	—	2	1	no	—
38	C₂²⋊Dic₇₁	—	2	2	no	—
39	C₄×Dic₇₁	—	2	1	no	—
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	SD₁₆×C₇₁	—	4	2	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₁	—	2	1	no	—
49	C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂³×C₇₁	—	2	1	no	—
51	Q₈×C₇₁	—	4	1	no	—
52	Q₈×C₇₁	—	4	1	no	—
53	D₄×C₇₁	—	2	1	no	—
54	D₄×C₇₁	SD₁₆	1	2	yes	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₇₁	—	2	1	no	—
63	C₂×Dic₇₁	—	2	1	no	—
64	C₂×Dic₇₁	—	2	2	no	—
65	C₂×Dic₇₁	—	4	4	no	—
66	Dic₁₄₂	—	2	1	no	—
67	Dic₁₄₂	—	4	4	no	—
68	Dic₁₄₂	—	4	1	no	—
69	Dic₁₄₂	—	4	1	no	—
70	Dic₁₄₂	SD₁₆	1	2	yes	complemented
71	C₈×C₇₁	—	4	1	no	—
72	C₇₁⋊C₈	—	2	2	no	—
73	C₇₁⋊C₈	—	4	1	no	—
74	C₂²×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	—	4	1	no	—
76	C₂²×C₇₁	—	4	1	no	—
77	C₂²×C₇₁	#120	1	1	yes	characteristic
78	C₄×C₇₁	—	2	1	no	—
79	C₄×C₇₁	—	4	1	no	—
80	C₄×C₇₁	—	8	1	no	—
81	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₇₁	#137	1	1	yes	characteristic
85	Dic₇₁	—	2	2	no	—
86	Dic₇₁	—	4	2	no	—
87	Dic₇₁	—	4	1	no	—
88	C₂×C₇₁	—	4	2	no	—
89	C₂×C₇₁	#448	1	1	yes	characteristic
90	C₂×C₇₁	#450	1	1	yes	characteristic
91	C₂×C₇₁	#456	1	1	yes	characteristic
92	#1651	—	71	1	no	maximal, Sylow
93	C₇₁	#1651	1	1	yes	characteristic, complemented, Sylow
94	D₄×Q₈	—	71	1	no	—
95	#498	—	71	1	no	—
96	#475	—	71	1	no	—
97	#461	D₇₁	1	1	yes	characteristic
98	#332	—	71	1	no	—
99	Q₈⋊C₈	—	71	1	no	—
100	D₄⋊C₈	—	71	1	no	—
101	C₂²×Q₈	—	142	1	no	—
102	C₂×SD₁₆	—	2	1	no	—
103	C₄⋊Q₈	—	71	1	no	—
104	C₄⋊Q₈	—	142	1	no	—
105	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂²⋊Q₈	—	142	2	no	—
107	C₂²⋊Q₈	—	142	1	no	—
108	C₄×Q₈	—	71	1	no	—
109	C₄×D₄	—	142	1	no	—
110	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
111	#106	—	142	1	no	—
112	#106	—	142	2	no	—
113	C₄⋊C₈	—	71	1	no	—
114	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
115	#103	—	2	1	no	—
116	#103	—	142	1	no	—
117	C₄×C₈	—	71	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	284	2	no	—
120	C₂×Q₈	—	284	2	no	—
121	C₂×Q₈	—	71	1	no	—
122	C₂×Q₈	—	142	2	no	—
123	C₂×Q₈	—	142	1	no	—
124	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	142	2	no	—
127	SD₁₆	—	4	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	142	1	no	—
130	C₂×C₈	—	142	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	71	1	no	—
134	C₄⋊C₄	—	71	2	no	—
135	C₄⋊C₄	—	142	2	no	—
136	C₄⋊C₄	—	142	1	no	—
137	C₄⋊C₄	—	142	1	no	—
138	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	142	2	no	—
141	C₄²	—	142	1	no	—
142	C₄²	C₂²×D₇₁	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	284	4	no	—
147	Q₈	—	284	1	no	—
148	Q₈	—	284	1	no	—
149	Q₈	—	71	2	no	—
150	Q₈	—	142	1	no	—
151	D₄	—	2	1	no	—
152	D₄	#11765853	1	2	yes	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	284	4	no	—
158	C₂×C₄	—	142	1	no	—
159	C₂×C₄	—	142	2	no	—
160	C₂×C₄	—	142	1	no	—
161	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
162	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	284	1	no	—
166	C₈	—	142	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	#14556096	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	8	1	no	—
174	C₄	—	284	2	no	—
175	C₄	—	284	1	no	—
176	C₄	—	142	2	no	—
177	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
178	C₄	C₂×#11765853	1	1	yes	characteristic
179	C₄	#14556155	1	1	yes	characteristic
180	C₄	#14556158	1	1	yes	characteristic
181	C₂	—	4	2	no	—
182	C₂	#17986137	1	1	yes	characteristic
183	C₂	#17986148	1	1	yes	characteristic
184	C₂	#17986180	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Dic₁₄₂

C₂

yes

maximal, characteristic

#17986395

C₂

yes

maximal, characteristic, complemented

#17986281

C₂

yes

maximal, characteristic, complemented

#17985744

C₂

yes

maximal, characteristic, complemented

#17985729

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#461×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

#14556128

C₂²

yes

—

—

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

D₄×Dic₇₁

—

#14556059

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556051

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

SD₁₆

yes

—

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

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

SD₁₆

yes

complemented

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#498

—

#475

—

#461

D₇₁

yes

characteristic

#332

—

Q₈⋊C₈

—

100

D₄⋊C₈

—

101

C₂²×Q₈

—

142

—

102

C₂×SD₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

142

—

105

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

106

C₂²⋊Q₈

—

142

—

107

C₂²⋊Q₈

—

142

—

108

C₄×Q₈

—

109

C₄×D₄

—

142

—

110

C₄×D₄

C₂×D₇₁

yes

characteristic

111

#106

—

142

—

112

#106

—

142

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₇₁

yes

characteristic

115

#103

—

116

#103

—

142

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

284

—

120

C₂×Q₈

—

284

—

121

C₂×Q₈

—

122

C₂×Q₈

—

142

—

123

C₂×Q₈

—

142

—

124

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

142

—

127

SD₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

142

—

130

C₂×C₈

—

142

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

142

—

136

C₄⋊C₄

—

142

—

137

C₄⋊C₄

—

142

—

138

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

142

—

141

C₄²

—

142

—

142

C₄²

C₂²×D₇₁

yes

characteristic

143

C₂³

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

284

—

147

Q₈

—

284

—

148

Q₈

—

284

—

149

Q₈

—

150

Q₈

—

142

—

151

D₄

—

152

D₄

#11765853

yes

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

142

—

159

C₂×C₄

—

142

—

160

C₂×C₄

—

142

—

161

C₂×C₄

D₄×D₇₁

yes

characteristic

162

C₂×C₄

D₄×D₇₁

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₈

—

165

C₈

—

284

—

166

C₈

—

142

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#14556096

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

284

—

175

C₄

—

284

—

176

C₄

—

142

—

177

C₄

SD₁₆×D₇₁

yes

characteristic

178

C₄

C₂×#11765853

yes

characteristic

179

C₄

#14556155

yes

characteristic

180

C₄

#14556158

yes

characteristic

181

C₂

—

182

C₂

#17986137

yes

characteristic

183

C₂

#17986148

yes

characteristic

184

C₂

#17986180

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#461×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#461	D₇₁
hypocenter	C₇₁	#1651
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#461×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
2. #17986395	C₂	semidirect
3. #17986281	C₂	semidirect
4. #17985744	C₂	semidirect
5. #17985729	C₂	semidirect
70. Dic₁₄₂	SD₁₆	semidirect
93. C₇₁	#1651	semidirect

H ≅ G/N

type

2. #17986395

C₂

semidirect

3. #17986281

C₂

semidirect

4. #17985744

C₂

semidirect

5. #17985729

C₂

semidirect

70. Dic₁₄₂

SD₁₆

semidirect

93. C₇₁

#1651

semidirect

Subgroups — #22587601

Conjugacy classes of subgroups

Complements