Group #31917101 — subgroups

Subgroups — #31917101

Order 18176.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17986397	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985822	C₂	1	2	yes	maximal
3	#22589088	C₂	1	1	yes	maximal, characteristic
4	#22586446	C₂	1	2	yes	maximal
5	C₇₁×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	—	2	1	no	—
7	#17986397	—	2	4	no	—
8	C₂×#14556064	C₂²	1	1	yes	characteristic
9	#17986037	C₂²	1	1	yes	characteristic
10	C₂×#14556060	—	2	1	no	—
11	C₂×#14556060	C₂²	1	2	yes	—
12	#17986025	—	2	1	no	—
13	#17985822	—	2	4	no	—
14	#17985759	—	2	1	no	—
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₂	—	4	1	no	—
18	C₂×#11765849	—	2	1	no	—
19	#14556128	—	2	4	no	—
20	#14556128	—	4	2	no	—
21	#14556087	—	2	1	no	—
22	C₂×C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×C₄⋊Dic₇₁	—	4	1	no	—
24	#14556064	—	2	2	no	—
25	#14556062	Q₈	1	2	yes	complemented
26	#14556060	—	2	4	no	—
27	#14556060	—	2	2	no	—
28	#14556060	—	2	1	no	—
29	#14556060	D₄	1	4	yes	—
30	Dic₇₁⋊C₈	—	4	2	no	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	M₃₂×C₇₁	—	2	2	no	—
35	M₃₂×C₇₁	Q₈	1	4	yes	complemented
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	C₂×Dic₁₄₂	—	4	4	no	—
39	C₂×Dic₁₄₂	—	8	4	no	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	#11765849	—	4	1	no	—
42	#11765849	—	4	1	no	—
43	C₂×C₇₁⋊C₈	—	4	1	no	—
44	C₄⋊Dic₇₁	—	2	4	no	—
45	C₄⋊Dic₇₁	—	2	2	no	—
46	C₄⋊Dic₇₁	—	4	4	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₂²⋊Dic₇₁	—	4	1	no	—
49	C₄×Dic₇₁	—	4	1	no	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	1	no	—
53	C₂×C₈×C₇₁	—	2	1	no	—
54	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
59	C₄²×C₇₁	—	2	1	no	—
60	C₄²×C₇₁	SD₁₆	1	2	yes	—
61	C₁₆×C₇₁	—	2	4	no	—
62	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	4	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
71	C₂×Dic₇₁	—	4	4	no	—
72	C₂×Dic₇₁	—	4	2	no	—
73	C₂×Dic₇₁	—	8	4	no	—
74	Dic₁₄₂	—	8	8	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
78	C₇₁⋊C₈	—	8	1	no	—
79	C₂²×C₇₁	—	2	2	no	—
80	C₂²×C₇₁	#450	1	1	yes	characteristic
81	C₂²×C₇₁	#476	1	1	yes	characteristic
82	C₂²×C₇₁	#501	1	1	yes	characteristic
83	C₄×C₇₁	—	2	4	no	—
84	C₄×C₇₁	#395	1	1	yes	characteristic
85	C₄×C₇₁	#448	1	1	yes	characteristic
86	C₄×C₇₁	#475	1	1	yes	characteristic
87	C₄×C₇₁	#498	1	1	yes	characteristic
88	Dic₇₁	—	8	4	no	—
89	Dic₇₁	—	8	2	no	—
90	#13457	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#2058	1	1	yes	characteristic
94	C₂×C₇₁	#2230	1	2	yes	—
95	C₂×#498	—	71	1	no	—
96	C₂×#365	—	71	2	no	—
97	#1945	—	71	1	no	—
98	#1386	—	71	2	no	—
99	#1376	D₇₁	1	1	yes	characteristic
100	C₇₁	#13457	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	142	1	no	—
102	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
103	#498	—	142	4	no	—
104	C₂×#108	—	71	1	no	—
105	#427	—	71	1	no	—
106	C₂×#106	—	71	2	no	—
107	C₂×#106	—	142	1	no	—
108	#424	—	142	1	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	#365	—	142	4	no	—
111	#339	—	142	1	no	—
112	C₂²×Q₈	—	284	1	no	—
113	C₂×M₁₆	—	142	1	no	—
114	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₄⋊Q₈	—	284	2	no	—
117	C₄⋊Q₈	—	142	4	no	—
118	#117	—	142	1	no	—
119	C₂×C₄⋊C₄	—	284	1	no	—
120	C₂×C₄⋊C₄	—	142	2	no	—
121	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	Dic₁₄₂	1	4	yes	complemented
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	142	2	no	—
126	#107	—	71	2	no	—
127	#106	—	71	4	no	—
128	#106	—	142	4	no	—
129	#106	—	142	2	no	—
130	#106	—	142	1	no	—
131	C₄⋊C₈	—	284	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	284	4	no	—
134	C₂×Q₈	—	568	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	284	2	no	—
137	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
138	M₁₆	—	284	1	no	—
139	M₁₆	—	284	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	284	1	no	—
143	C₂×C₈	#11765854	1	2	yes	—
144	C₂×C₈	D₄×D₇₁	1	2	yes	—
145	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
147	C₂×C₈	#11765864	1	1	yes	characteristic
148	C₄⋊C₄	—	284	4	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄⋊C₄	—	142	4	no	—
151	C₄⋊C₄	—	142	2	no	—
152	C₂²⋊C₄	—	284	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	284	1	no	—
155	C₄²	#11765854	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	#14556156	1	1	yes	characteristic
158	Q₈	—	568	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	284	4	no	—
162	C₂×C₄	—	284	2	no	—
163	C₂×C₄	—	568	4	no	—
164	C₂×C₄	#14556098	1	1	yes	characteristic
165	C₂×C₄	#14556108	1	1	yes	characteristic
166	C₂×C₄	#14556128	1	1	yes	characteristic
167	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
168	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
169	C₂×C₄	#14556161	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	568	1	no	—
172	C₈	#14556105	1	2	yes	—
173	C₈	#14556127	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#17986152	1	1	yes	characteristic
176	C₂²	#17986290	1	1	yes	characteristic
177	C₂²	#17986408	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	568	4	no	—
180	C₄	—	568	2	no	—
181	C₄	#17985920	1	1	yes	characteristic
182	C₄	#17986140	1	1	yes	characteristic
183	C₄	#17986285	1	1	yes	characteristic
184	C₄	#17986397	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#22589674	1	1	yes	characteristic
188	C₂	#22590512	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986397

C₂

yes

maximal, characteristic

C₂×#17985822

C₂

yes

maximal

#22589088

C₂

yes

maximal, characteristic

#22586446

C₂

yes

maximal

C₇₁×#1376

C₂

yes

maximal, characteristic

C₂×#14556128

—

#17986397

—

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

characteristic

C₂×#14556060

—

C₂×#14556060

C₂²

yes

—

#17986025

—

#17985822

—

#17985759

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂²×Dic₁₄₂

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556064

—

#14556062

Q₈

yes

—

—

—

D₄

yes

—

Dic₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

Q₈

yes

complemented

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#120

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#137

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

C₄⋊Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#13457

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2058

yes

characteristic

C₂×C₇₁

#2230

yes

—

—

—

—

—

D₇₁

yes

characteristic

100

C₇₁

#13457

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

142

—

102

C₂×M₃₂

C₂×D₇₁

yes

—

103

#498

—

142

—

104

C₂×#108

—

105

#427

—

106

C₂×#106

—

107

C₂×#106

—

142

—

108

#424

—

142

—

109

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

110

#365

—

142

—

111

#339

—

142

—

112

C₂²×Q₈

—

284

—

113

C₂×M₁₆

—

142

—

114

C₂²×C₈

D₂₈₄

yes

characteristic

115

C₂²×C₈

C₂²×D₇₁

yes

characteristic

116

C₄⋊Q₈

—

284

—

117

C₄⋊Q₈

—

142

—

118

#117

—

142

—

119

C₂×C₄⋊C₄

—

284

—

120

C₂×C₄⋊C₄

—

142

—

121

C₂×C₄²

D₂₈₄

yes

characteristic

122

M₃₂

—

123

M₃₂

Dic₁₄₂

yes

complemented

124

C₂×C₁₆

—

125

#108

—

142

—

126

#107

—

127

#106

—

128

#106

—

142

—

129

#106

—

142

—

130

#106

—

142

—

131

C₄⋊C₈

—

284

—

132

C₄×C₈

—

133

C₂×Q₈

—

284

—

134

C₂×Q₈

—

568

—

135

C₂²×C₄

—

136

C₂²×C₄

—

284

—

137

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

138

M₁₆

—

284

—

139

M₁₆

—

284

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

284

—

143

C₂×C₈

#11765854

yes

—

144

C₂×C₈

D₄×D₇₁

yes

—

145

C₂×C₈

Q₈×D₇₁

yes

characteristic

146

C₂×C₈

C₂×Dic₁₄₂

yes

—

147

C₂×C₈

#11765864

yes

characteristic

148

C₄⋊C₄

—

284

—

149

C₄⋊C₄

—

284

—

150

C₄⋊C₄

—

142

—

151

C₄⋊C₄

—

142

—

152

C₂²⋊C₄

—

284

—

153

C₄²

—

154

C₄²

—

284

—

155

C₄²

#11765854

yes

—

156

C₁₆

—

157

C₂³

#14556156

yes

characteristic

158

Q₈

—

568

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

284

—

162

C₂×C₄

—

284

—

163

C₂×C₄

—

568

—

164

C₂×C₄

#14556098

yes

characteristic

165

C₂×C₄

#14556108

yes

characteristic

166

C₂×C₄

#14556128

yes

characteristic

167

C₂×C₄

C₂×#11765854

yes

characteristic

168

C₂×C₄

C₂×#11765854

yes

characteristic

169

C₂×C₄

#14556161

yes

characteristic

170

C₈

—

171

C₈

—

568

—

172

C₈

#14556105

yes

—

173

C₈

#14556127

yes

—

174

C₂²

—

175

C₂²

#17986152

yes

characteristic

176

C₂²

#17986290

yes

characteristic

177

C₂²

#17986408

yes

characteristic

178

C₄

—

179

C₄

—

568

—

180

C₄

—

568

—

181

C₄

#17985920

yes

characteristic

182

C₄

#17986140

yes

characteristic

183

C₄

#17986285

yes

characteristic

184

C₄

#17986397

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#22589674

yes

characteristic

188

C₂

#22590512

yes

—

	subgroup	G/H
center	C₂²	#17986290
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1376	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
hypercenter	#1376	D₇₁
hypocenter	C₇₁	#13457
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#17986290

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1376

C₂

C₂²×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
25. #14556062	Q₈	semidirect
35. M₃₂×C₇₁	Q₈	semidirect
100. C₇₁	#13457	semidirect
123. M₃₂	Dic₁₄₂	semidirect

H ≅ G/N

type

25. #14556062

Q₈

semidirect

35. M₃₂×C₇₁

Q₈

semidirect

100. C₇₁

#13457

semidirect

123. M₃₂

Dic₁₄₂

semidirect

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

subgroup	index
C₂×#17986397	2
C₂×#17985822	2
#22589088	2
#22586446	2
C₇₁×#1376	2
#13457	71

subgroup

index

C₇₁×#1376

Subgroups — #31917101

Conjugacy classes of subgroups

Complements

Maximal subgroups