Group #22587023 — subgroups

Subgroups — #22587023

Order 9088.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄⋊Q₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#17986051	C₂	1	1	yes	maximal, characteristic
3	#17986021	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985744	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₁	—	2	1	no	—
9	C₂×#11765848	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556127	—	2	1	no	—
12	#14556127	C₄	1	2	yes	—
13	#14556126	—	2	1	no	—
14	C₄⋊C₄×D₇₁	—	2	1	no	—
15	C₄⋊C₄×D₇₁	C₄	1	2	yes	—
16	C₄²×D₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₇₁⋊C₈	—	2	1	no	—
20	(C₂×D₇₁)⋊C₈	—	2	1	no	—
21	C₈×Dic₇₁	—	2	1	no	—
22	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	2	no	—
27	Q₈×D₇₁	—	4	2	no	—
28	Q₈×D₇₁	—	4	2	no	—
29	C₂×C₄×D₇₁	—	2	2	no	—
30	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
33	#11765848	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₄⋊Dic₇₁	—	2	1	no	—
37	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
38	Dic₇₁⋊C₄	—	2	2	no	—
39	Dic₇₁⋊C₄	—	2	2	no	—
40	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₇₁	D₄	1	1	yes	characteristic
42	C₄×Dic₇₁	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
48	C₄²×C₇₁	C₂³	1	1	yes	characteristic
49	C₂²×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	Q₈×C₇₁	—	4	1	no	—
51	Q₈×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₇₁	—	2	2	no	—
57	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×D₇₁	—	2	1	no	—
61	C₄×D₇₁	—	2	1	no	—
62	C₄×D₇₁	—	2	2	no	—
63	C₄×D₇₁	—	4	4	no	—
64	C₄×D₇₁	SD₁₆	1	2	yes	—
65	C₄×D₇₁	Q₁₆	1	2	yes	—
66	Dic₁₄₂	—	4	4	no	—
67	Dic₁₄₂	—	4	1	no	—
68	Dic₁₄₂	—	4	1	no	—
69	C₈×C₇₁	—	4	1	no	—
70	C₈×C₇₁	—	4	1	no	—
71	C₇₁⋊C₈	—	4	1	no	—
72	C₇₁⋊C₈	—	4	1	no	—
73	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×D₇₁	—	2	1	no	—
75	C₂×D₇₁	—	2	1	no	—
76	C₂×D₇₁	#103	1	2	yes	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	4	2	no	—
79	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
80	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
81	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
83	Dic₇₁	—	2	1	no	—
84	Dic₇₁	—	4	2	no	—
85	Dic₇₁	#101	1	2	yes	—
86	Dic₇₁	#103	1	2	yes	—
87	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
88	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
89	C₂×C₇₁	#418	1	1	yes	characteristic
90	D₇₁	—	2	2	no	—
91	#1549	—	71	1	no	maximal, Sylow
92	C₇₁	#1549	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊Q₈	—	71	1	no	—
94	#431	—	71	1	no	—
95	#423	—	71	1	no	—
96	#332	—	71	1	no	—
97	#332	—	71	1	no	—
98	#332	—	71	1	no	—
99	#332	D₇₁	1	1	yes	characteristic, complemented
100	C₂²×Q₈	—	142	1	no	—
101	C₂×M₁₆	—	142	1	no	—
102	C₄⋊Q₈	—	71	2	no	—
103	C₄⋊Q₈	—	71	1	no	—
104	C₄⋊Q₈	—	142	1	no	—
105	C₄⋊Q₈	—	142	1	no	—
106	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	71	2	no	—
108	C₂×C₄⋊C₄	—	142	1	no	—
109	C₂×C₄²	—	71	1	no	—
110	C₄⋊C₈	—	71	1	no	—
111	C₄⋊C₈	—	71	1	no	—
112	C₄⋊C₈	—	142	1	no	—
113	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₂²⋊C₈	—	142	1	no	—
116	C₄×C₈	—	142	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	284	2	no	—
119	C₂×Q₈	—	284	2	no	—
120	C₂×Q₈	—	142	2	no	—
121	C₂×Q₈	—	142	1	no	—
122	C₂²×C₄	—	71	1	no	—
123	C₂²×C₄	—	71	1	no	—
124	C₂²×C₄	—	71	1	no	—
125	C₂²×C₄	—	142	2	no	—
126	M₁₆	—	284	2	no	—
127	C₂×C₈	—	2	1	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	—	71	2	no	—
133	C₄⋊C₄	—	142	2	no	—
134	C₄⋊C₄	—	142	2	no	—
135	C₄⋊C₄	—	142	1	no	—
136	C₄⋊C₄	C₄×D₇₁	1	2	yes	—
137	C₄²	—	71	1	no	—
138	C₄²	—	71	1	no	—
139	C₄²	—	71	1	no	—
140	C₄²	C₂²×D₇₁	1	1	yes	characteristic
141	C₂³	—	71	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	284	4	no	—
145	Q₈	—	284	1	no	—
146	Q₈	—	284	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	284	4	no	—
149	C₂×C₄	—	71	1	no	—
150	C₂×C₄	—	71	1	no	—
151	C₂×C₄	—	71	2	no	—
152	C₂×C₄	—	71	2	no	—
153	C₂×C₄	—	71	1	no	—
154	C₂×C₄	—	142	2	no	—
155	C₂×C₄	—	142	2	no	—
156	C₂×C₄	—	142	1	no	—
157	C₂×C₄	—	142	1	no	—
158	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
159	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
160	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
161	C₈	—	4	1	no	—
162	C₈	—	4	1	no	—
163	C₈	—	284	1	no	—
164	C₈	—	284	1	no	—
165	C₂²	—	71	2	no	—
166	C₂²	—	142	1	no	—
167	C₂²	—	142	1	no	—
168	C₂²	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	4	2	no	—
171	C₄	—	284	2	no	—
172	C₄	—	71	2	no	—
173	C₄	—	71	2	no	—
174	C₄	—	142	1	no	—
175	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
176	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
177	C₄	#14556152	1	1	yes	characteristic
178	C₄	#14556155	1	1	yes	characteristic
179	C₂	—	142	2	no	—
180	C₂	#101×D₇₁	1	1	yes	characteristic
181	C₂	#103×D₇₁	1	1	yes	characteristic
182	C₂	#17986002	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₇₁

C₂

yes

maximal, characteristic

#17986051

C₂

yes

maximal, characteristic

#17986021

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#17985744

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

—

C₂×#11765848

—

#14556128

C₂²

yes

characteristic

#14556127

—

#14556127

C₄

yes

—

#14556126

—

C₄⋊C₄×D₇₁

—

C₄⋊C₄×D₇₁

C₄

yes

—

C₄²×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

(C₂×D₇₁)⋊C₈

—

C₈×Dic₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Q₈×D₇₁

—

Q₈×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

SD₁₆

yes

—

C₄×D₇₁

Q₁₆

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#103

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

#101

yes

—

Dic₇₁

#103

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1549

—

maximal, Sylow

C₇₁

#1549

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

100

C₂²×Q₈

—

142

—

101

C₂×M₁₆

—

142

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

142

—

105

C₄⋊Q₈

—

142

—

106

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

142

—

109

C₂×C₄²

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

142

—

113

C₄⋊C₈

C₂×D₇₁

yes

characteristic

114

C₄⋊C₈

C₂×D₇₁

yes

characteristic

115

C₂²⋊C₈

—

142

—

116

C₄×C₈

—

142

—

117

C₂×Q₈

—

118

C₂×Q₈

—

284

—

119

C₂×Q₈

—

284

—

120

C₂×Q₈

—

142

—

121

C₂×Q₈

—

142

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

142

—

126

M₁₆

—

284

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

142

—

130

C₂×C₈

—

142

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

142

—

134

C₄⋊C₄

—

142

—

135

C₄⋊C₄

—

142

—

136

C₄⋊C₄

C₄×D₇₁

yes

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

C₂²×D₇₁

yes

characteristic

141

C₂³

—

142

Q₈

—

143

Q₈

—

144

Q₈

—

284

—

145

Q₈

—

284

—

146

Q₈

—

284

—

147

C₂×C₄

—

148

C₂×C₄

—

284

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

142

—

155

C₂×C₄

—

142

—

156

C₂×C₄

—

142

—

157

C₂×C₄

—

142

—

158

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

159

C₂×C₄

D₄×D₇₁

yes

characteristic

160

C₂×C₄

D₄×D₇₁

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

—

284

—

164

C₈

—

284

—

165

C₂²

—

166

C₂²

—

142

—

167

C₂²

—

142

—

168

C₂²

C₂²⋊C₄×D₇₁

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

284

—

172

C₄

—

173

C₄

—

174

C₄

—

142

—

175

C₄

SD₁₆×D₇₁

yes

characteristic

176

C₄

Q₁₆×D₇₁

yes

characteristic

177

C₄

#14556152

yes

characteristic

178

C₄

#14556155

yes

characteristic

179

C₂

—

142

—

180

C₂

#101×D₇₁

yes

characteristic

181

C₂

#103×D₇₁

yes

characteristic

182

C₂

#17986002

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1549
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #17985746	C₂	semidirect
5. #17985745	C₂	semidirect
6. #17985744	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
92. C₇₁	#1549	semidirect
99. #332	D₇₁	semidirect

H ≅ G/N

type

4. #17985746

C₂

semidirect

5. #17985745

C₂

semidirect

6. #17985744

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

92. C₇₁

#1549

semidirect

99. #332

D₇₁

semidirect

Subgroups — #22587023

Conjugacy classes of subgroups

Complements