Group #31918246 — subgroups

Subgroups — #31918246

Order 18176.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×Dic₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985839	C₂	1	1	yes	maximal, characteristic
3	#22590591	C₂	1	1	yes	maximal, characteristic
4	#22590198	C₂	1	1	yes	maximal, characteristic
5	Dic₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#22586520	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
10	C₂×#14556074	—	2	1	no	—
11	#17986377	—	2	1	no	—
12	#17986093	—	2	1	no	—
13	Dic₁₁₃₆	—	2	2	no	—
14	#17985839	—	2	2	no	—
15	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
16	#17985802	—	2	1	no	—
17	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
18	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
20	#366×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₈₄	—	2	1	no	—
22	C₂×Dic₂₈₄	—	2	1	no	—
23	C₂×#11765855	—	4	1	no	—
24	#14556126	—	4	1	no	—
25	#14556093	—	4	1	no	—
26	Dic₅₆₈	—	2	1	no	—
27	Dic₅₆₈	—	2	1	no	—
28	Dic₅₆₈	—	2	1	no	—
29	Dic₅₆₈	D₄	1	2	yes	—
30	#14556074	—	4	2	no	—
31	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
32	#14556051	—	4	1	no	—
33	C₈×Dic₇₁	D₄	1	1	yes	characteristic
34	C₇₁⋊C₃₂	—	2	1	no	—
35	C₂×Q₁₆×C₇₁	—	2	1	no	—
36	Q₃₂×C₇₁	—	2	1	no	—
37	Q₃₂×C₇₁	—	2	1	no	—
38	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
39	#107×C₇₁	—	2	1	no	—
40	C₃₂×C₇₁	—	2	1	no	—
41	C₂×Dic₁₄₂	—	4	1	no	—
42	C₂×Dic₁₄₂	—	4	1	no	—
43	Dic₂₈₄	—	2	2	no	—
44	Dic₂₈₄	—	4	1	no	—
45	Dic₂₈₄	—	4	1	no	—
46	Dic₂₈₄	—	4	1	no	—
47	#11765855	—	8	2	no	—
48	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
49	Dic₇₁⋊C₄	—	8	1	no	—
50	Dic₇₁⋊C₄	—	8	1	no	—
51	Dic₇₁⋊C₄	—	8	1	no	—
52	C₄×Dic₇₁	—	8	1	no	—
53	C₄×Dic₇₁	D₈	1	1	yes	characteristic
54	C₇₁⋊C₁₆	—	2	1	no	—
55	C₂×Q₈×C₇₁	—	4	1	no	—
56	Q₁₆×C₇₁	—	4	1	no	—
57	Q₁₆×C₇₁	—	4	1	no	—
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₇₁	—	4	1	no	—
60	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₁₆×C₇₁	#41	1	1	yes	characteristic
62	Q₈×C₇₁	—	8	1	no	—
63	Q₈×C₇₁	—	8	1	no	—
64	C₂×C₄×C₇₁	—	8	1	no	—
65	C₂×C₄×C₇₁	—	8	1	no	—
66	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
67	C₂×Dic₇₁	—	8	1	no	—
68	C₂×Dic₇₁	—	8	1	no	—
69	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
70	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
71	Dic₁₄₂	—	4	2	no	—
72	Dic₁₄₂	—	8	1	no	—
73	Dic₁₄₂	—	8	1	no	—
74	Dic₁₄₂	—	8	1	no	—
75	C₈×C₇₁	#121	1	1	yes	characteristic
76	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₇₁⋊C₈	—	2	1	no	—
78	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
79	C₄×C₇₁	—	16	1	no	—
80	C₄×C₇₁	—	16	1	no	—
81	C₄×C₇₁	#459	1	1	yes	characteristic
82	C₄×C₇₁	#509	1	1	yes	characteristic
83	Dic₇₁	—	2	1	no	—
84	Dic₇₁	—	8	2	no	—
85	Dic₇₁	—	16	1	no	—
86	Dic₇₁	Q₆₄	1	2	yes	complemented
87	#13688	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	#2206	1	1	yes	characteristic
89	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
90	C₂×C₇₁	#2264	1	1	yes	characteristic
91	C₂×Q₆₄	—	71	1	no	—
92	C₂×Q₆₄	—	71	1	no	—
93	#2247	—	71	1	no	—
94	C₄×Q₃₂	—	71	1	no	—
95	C₄⋊C₃₂	—	71	1	no	—
96	#1416	—	71	1	no	—
97	#1416	D₇₁	1	1	yes	characteristic
98	C₇₁	#13688	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₃₂	—	71	1	no	—
100	C₂×Q₃₂	—	71	1	no	—
101	C₂×Q₃₂	—	142	1	no	—
102	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
103	#494	—	142	1	no	—
104	C₄×Q₁₆	—	142	1	no	—
105	Q₆₄	—	142	2	no	—
106	Q₆₄	—	142	2	no	—
107	C₂×C₃₂	—	71	1	no	—
108	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
109	#366	C₂×D₇₁	1	1	yes	characteristic
110	#358	—	142	1	no	—
111	C₄×C₁₆	—	71	1	no	—
112	C₂×Q₁₆	—	2	1	no	—
113	C₂×Q₁₆	—	284	1	no	—
114	C₂×Q₁₆	—	142	1	no	—
115	C₂×Q₁₆	—	142	1	no	—
116	C₄⋊Q₈	—	284	1	no	—
117	C₄×Q₈	—	284	1	no	—
118	Q₃₂	—	2	1	no	—
119	Q₃₂	—	2	1	no	—
120	Q₃₂	—	284	2	no	—
121	Q₃₂	—	71	2	no	—
122	Q₃₂	—	142	1	no	—
123	Q₃₂	—	142	1	no	—
124	Q₃₂	—	142	1	no	—
125	C₂×C₁₆	—	71	1	no	—
126	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
127	#107	—	2	1	no	—
128	#103	—	284	1	no	—
129	C₄×C₈	—	71	1	no	—
130	C₃₂	—	2	1	no	—
131	C₃₂	—	142	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	284	1	no	—
134	C₂×Q₈	—	284	1	no	—
135	Q₁₆	—	4	1	no	—
136	Q₁₆	—	4	1	no	—
137	Q₁₆	—	284	1	no	—
138	Q₁₆	—	284	1	no	—
139	Q₁₆	—	284	1	no	—
140	Q₁₆	—	568	2	no	—
141	Q₁₆	—	142	2	no	—
142	C₂×C₈	—	71	1	no	—
143	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	568	1	no	—
146	C₄⋊C₄	—	568	1	no	—
147	C₄⋊C₄	—	568	1	no	—
148	C₄²	—	568	1	no	—
149	C₄²	—	71	1	no	—
150	C₁₆	—	142	1	no	—
151	C₁₆	D₄×D₇₁	1	1	yes	characteristic
152	C₁₆	#11765865	1	1	yes	characteristic
153	Q₈	—	8	1	no	—
154	Q₈	—	8	1	no	—
155	Q₈	—	284	2	no	—
156	Q₈	—	568	1	no	—
157	Q₈	—	568	1	no	—
158	Q₈	—	568	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	568	1	no	—
162	C₂×C₄	—	568	1	no	—
163	C₂×C₄	—	71	1	no	—
164	C₂×C₄	—	71	1	no	—
165	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
166	C₈	—	142	1	no	—
167	C₈	#14556101	1	1	yes	characteristic
168	C₈	#14556155	1	1	yes	characteristic
169	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
170	C₄	—	16	1	no	—
171	C₄	—	16	1	no	—
172	C₄	—	568	2	no	—
173	C₄	—	71	2	no	—
174	C₄	—	1136	1	no	—
175	C₄	—	142	1	no	—
176	C₄	#17986193	1	1	yes	characteristic
177	C₄	#17986436	1	1	yes	characteristic
178	C₂	#22590374	1	1	yes	characteristic
179	C₂	Q₆₄×D₇₁	1	1	yes	characteristic
180	C₂	#22590656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₃₆

C₂

yes

maximal, characteristic

C₂×#17985839

C₂

yes

maximal, characteristic

#22590591

C₂

yes

maximal, characteristic

#22590198

C₂

yes

maximal, characteristic

Dic₇₁⋊C₃₂

C₂

yes

maximal, characteristic

#22586520

C₂

yes

maximal, characteristic

C₇₁×#1416

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₂²

yes

characteristic

C₂×Dic₅₆₈

C₂²

yes

—

—

—

Dic₁₁₃₆

—

#17985839

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985802

—

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#366×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×#11765855

—

#14556126

—

#14556093

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

D₄

yes

—

#14556074

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

#14556051

—

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂×Q₁₆×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

C₃₂×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765855

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

D₁₆

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

Q₆₄

yes

—

C₂×C₇₁

yes

C₂×C₇₁

C₂×Q₆₄

yes

characteristic

C₂×C₇₁

#2264

yes

characteristic

C₂×Q₆₄

—

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

—

C₄⋊C₃₂

—

#1416

—

#1416

D₇₁

yes

characteristic

C₇₁

#13688

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

100

C₂×Q₃₂

—

101

C₂×Q₃₂

—

142

—

102

C₂×Q₃₂

C₂×D₇₁

yes

characteristic

103

#494

—

142

—

104

C₄×Q₁₆

—

142

—

105

Q₆₄

—

142

—

106

Q₆₄

—

142

—

107

C₂×C₃₂

—

108

C₂×C₃₂

C₂×D₇₁

yes

characteristic

109

#366

C₂×D₇₁

yes

characteristic

110

#358

—

142

—

111

C₄×C₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

284

—

114

C₂×Q₁₆

—

142

—

115

C₂×Q₁₆

—

142

—

116

C₄⋊Q₈

—

284

—

117

C₄×Q₈

—

284

—

118

Q₃₂

—

119

Q₃₂

—

120

Q₃₂

—

284

—

121

Q₃₂

—

122

Q₃₂

—

142

—

123

Q₃₂

—

142

—

124

Q₃₂

—

142

—

125

C₂×C₁₆

—

126

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

127

#107

—

128

#103

—

284

—

129

C₄×C₈

—

130

C₃₂

—

131

C₃₂

—

142

—

132

C₂×Q₈

—

133

C₂×Q₈

—

284

—

134

C₂×Q₈

—

284

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

—

284

—

138

Q₁₆

—

284

—

139

Q₁₆

—

284

—

140

Q₁₆

—

568

—

141

Q₁₆

—

142

—

142

C₂×C₈

—

143

C₂×C₈

D₄×D₇₁

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

568

—

146

C₄⋊C₄

—

568

—

147

C₄⋊C₄

—

568

—

148

C₄²

—

568

—

149

C₄²

—

150

C₁₆

—

142

—

151

C₁₆

D₄×D₇₁

yes

characteristic

152

C₁₆

#11765865

yes

characteristic

153

Q₈

—

154

Q₈

—

155

Q₈

—

284

—

156

Q₈

—

568

—

157

Q₈

—

568

—

158

Q₈

—

568

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

568

—

162

C₂×C₄

—

568

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

D₈×D₇₁

yes

characteristic

166

C₈

—

142

—

167

C₈

#14556101

yes

characteristic

168

C₈

#14556155

yes

characteristic

169

C₂²

D₁₆×D₇₁

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

568

—

173

C₄

—

174

C₄

—

1136

—

175

C₄

—

142

—

176

C₄

#17986193

yes

characteristic

177

C₄

#17986436

yes

characteristic

178

C₂

#22590374

yes

characteristic

179

C₂

Q₆₄×D₇₁

yes

characteristic

180

C₂

#22590656

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1416	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1416	D₇₁
hypocenter	C₇₁	#13688
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1416

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

D₇₁

C₇₁

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
86. Dic₇₁	Q₆₄	semidirect
98. C₇₁	#13688	semidirect

H ≅ G/N

type

86. Dic₇₁

Q₆₄

semidirect

98. C₇₁

#13688

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₁₁₃₆	2
C₂×#17985839	2
#22590591	2
#22590198	2
Dic₇₁⋊C₃₂	2
#22586520	2
C₇₁×#1416	2
#13688	71

subgroup

index

C₂×Dic₁₁₃₆

Dic₇₁⋊C₃₂

C₇₁×#1416

Subgroups — #31918246

Conjugacy classes of subgroups

Complements

Maximal subgroups