Group #22587029 — subgroups

Subgroups — #22587029

Order 9088.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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	#17986554	C₂	1	1	yes	maximal, characteristic
2	#17986021	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765848	—	2	1	no	—
9	C₂×C₈×D₇₁	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556119	—	2	1	no	—
12	#14556108	C₄	1	2	yes	—
13	#14556104	—	2	1	no	—
14	Q₈×Dic₇₁	—	2	1	no	—
15	#14556093	C₄	1	2	yes	—
16	#14556086	—	2	1	no	—
17	C₄²×D₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₁⋊C₈	—	2	1	no	—
21	Dic₇₁⋊C₈	—	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	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
29	#11765848	—	4	2	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₈×D₇₁	—	4	2	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	—	2	1	no	—
35	C₄⋊Dic₇₁	—	2	2	no	—
36	Dic₇₁⋊C₄	—	2	2	no	—
37	(C₂×D₇₁)⋊C₄	—	2	2	no	—
38	C₄×Dic₇₁	—	2	2	no	—
39	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₁	D₄	1	1	yes	characteristic
41	C₄×Dic₇₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₇₁	—	4	1	no	—
50	Q₈×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	2	no	—
56	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₇₁	—	2	1	no	—
60	C₄×D₇₁	—	2	1	no	—
61	C₄×D₇₁	—	4	1	no	—
62	C₄×D₇₁	SD₁₆	1	2	yes	—
63	C₄×D₇₁	Q₁₆	1	2	yes	—
64	Dic₁₄₂	—	4	2	no	—
65	C₈×C₇₁	—	4	1	no	—
66	C₈×C₇₁	—	4	1	no	—
67	C₇₁⋊C₈	—	4	1	no	—
68	C₇₁⋊C₈	—	4	1	no	—
69	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₇₁	—	2	1	no	—
71	C₂×D₇₁	—	2	1	no	—
72	C₂×D₇₁	#103	1	2	yes	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₇₁	#137	1	1	yes	characteristic
79	Dic₇₁	—	2	1	no	—
80	Dic₇₁	—	2	1	no	—
81	Dic₇₁	—	4	2	no	—
82	Dic₇₁	#103	1	2	yes	—
83	C₂×C₇₁	#413	1	1	yes	characteristic
84	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
85	C₂×C₇₁	#417	1	1	yes	characteristic
86	D₇₁	—	2	2	no	—
87	#1550	—	71	1	no	maximal, Sylow
88	C₇₁	#1550	1	1	yes	characteristic, complemented, Sylow
89	#533	—	71	1	no	—
90	#423	—	71	1	no	—
91	C₂×C₄⋊C₈	—	71	1	no	—
92	#333	—	71	1	no	—
93	#333	—	71	1	no	—
94	#332	—	71	1	no	—
95	#332	D₇₁	1	1	yes	characteristic, complemented
96	C₂×M₁₆	—	142	1	no	—
97	C₂²×C₈	—	142	1	no	—
98	C₄⋊Q₈	—	71	1	no	—
99	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
100	#125	—	142	1	no	—
101	C₂²⋊Q₈	—	71	2	no	—
102	C₂²⋊Q₈	—	142	1	no	—
103	C₄×Q₈	—	71	2	no	—
104	C₄×Q₈	—	142	1	no	—
105	#117	—	142	1	no	—
106	C₂×C₄²	—	71	1	no	—
107	C₄⋊C₈	—	71	1	no	—
108	C₄⋊C₈	—	71	1	no	—
109	C₄⋊C₈	—	142	1	no	—
110	C₄⋊C₈	—	142	1	no	—
111	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
112	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	—	71	2	no	—
115	C₂²×C₄	—	71	1	no	—
116	C₂²×C₄	—	71	1	no	—
117	C₂²×C₄	—	71	1	no	—
118	M₁₆	—	284	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	284	2	no	—
122	C₂×C₈	—	142	1	no	—
123	C₂×C₈	—	142	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	142	2	no	—
126	C₄⋊C₄	—	142	2	no	—
127	C₄⋊C₄	—	142	1	no	—
128	C₄⋊C₄	—	142	1	no	—
129	C₄⋊C₄	C₄×D₇₁	1	2	yes	—
130	C₂²⋊C₄	—	142	2	no	—
131	C₄²	—	71	1	no	—
132	C₄²	—	71	1	no	—
133	C₄²	—	71	1	no	—
134	C₄²	—	142	2	no	—
135	C₄²	C₂²×D₇₁	1	1	yes	characteristic
136	C₂³	—	71	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	284	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	284	1	no	—
142	C₂×C₄	—	71	1	no	—
143	C₂×C₄	—	71	2	no	—
144	C₂×C₄	—	71	2	no	—
145	C₂×C₄	—	71	1	no	—
146	C₂×C₄	—	71	1	no	—
147	C₂×C₄	—	142	2	no	—
148	C₂×C₄	—	142	1	no	—
149	C₂×C₄	—	142	1	no	—
150	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
151	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
152	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
153	C₈	—	4	1	no	—
154	C₈	—	4	1	no	—
155	C₈	—	284	1	no	—
156	C₈	—	284	1	no	—
157	C₂²	—	71	2	no	—
158	C₂²	—	142	1	no	—
159	C₂²	—	142	1	no	—
160	C₂²	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	284	2	no	—
164	C₄	—	71	2	no	—
165	C₄	—	142	1	no	—
166	C₄	—	142	1	no	—
167	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
168	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
169	C₄	#14556152	1	1	yes	characteristic
170	C₄	#14556157	1	1	yes	characteristic
171	C₂	—	142	2	no	—
172	C₂	#17985982	1	1	yes	characteristic
173	C₂	#103×D₇₁	1	1	yes	characteristic
174	C₂	#17985997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986554

C₂

yes

maximal, characteristic

#17986021

C₂

yes

maximal, characteristic

C₄⋊C₈×D₇₁

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, characteristic, complemented

#17985747

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765848

—

C₂×C₈×D₇₁

—

#14556128

C₂²

yes

characteristic

#14556119

—

#14556108

C₄

yes

—

#14556104

—

Q₈×Dic₇₁

—

#14556093

C₄

yes

—

#14556086

—

C₄²×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

Dic₇₁⋊C₈

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₂×C₄

yes

—

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

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

—

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

SD₁₆

yes

—

C₄×D₇₁

Q₁₆

yes

—

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

—

Dic₇₁

#103

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1550

—

maximal, Sylow

C₇₁

#1550

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄⋊C₈

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×M₁₆

—

142

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

100

#125

—

142

—

101

C₂²⋊Q₈

—

102

C₂²⋊Q₈

—

142

—

103

C₄×Q₈

—

104

C₄×Q₈

—

142

—

105

#117

—

142

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

142

—

110

C₄⋊C₈

—

142

—

111

C₄⋊C₈

C₂×D₇₁

yes

characteristic

112

C₄⋊C₈

C₂×D₇₁

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

284

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

284

—

122

C₂×C₈

—

142

—

123

C₂×C₈

—

142

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

142

—

126

C₄⋊C₄

—

142

—

127

C₄⋊C₄

—

142

—

128

C₄⋊C₄

—

142

—

129

C₄⋊C₄

C₄×D₇₁

yes

—

130

C₂²⋊C₄

—

142

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

142

—

135

C₄²

C₂²×D₇₁

yes

characteristic

136

C₂³

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

284

—

140

C₂×C₄

—

141

C₂×C₄

—

284

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

142

—

148

C₂×C₄

—

142

—

149

C₂×C₄

—

142

—

150

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

151

C₂×C₄

D₄×D₇₁

yes

characteristic

152

C₂×C₄

D₄×D₇₁

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

284

—

156

C₈

—

284

—

157

C₂²

—

158

C₂²

—

142

—

159

C₂²

—

142

—

160

C₂²

C₂²⋊C₄×D₇₁

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

284

—

164

C₄

—

165

C₄

—

142

—

166

C₄

—

142

—

167

C₄

SD₁₆×D₇₁

yes

characteristic

168

C₄

Q₁₆×D₇₁

yes

characteristic

169

C₄

#14556152

yes

characteristic

170

C₄

#14556157

yes

characteristic

171

C₂

—

142

—

172

C₂

#17985982

yes

characteristic

173

C₂

#103×D₇₁

yes

characteristic

174

C₂

#17985997

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	#1550
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. #17985747	C₂	semidirect
5. #17985747	C₂	semidirect
6. #17985745	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
88. C₇₁	#1550	semidirect
95. #332	D₇₁	semidirect

H ≅ G/N

type

4. #17985747

C₂

semidirect

5. #17985747

C₂

semidirect

6. #17985745

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

88. C₇₁

#1550

semidirect

95. #332

D₇₁

semidirect

Subgroups — #22587029

Conjugacy classes of subgroups

Complements