Group #31917110 — subgroups

Subgroups — #31917110

Order 18176.

318 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22598864	C₂	1	1	yes	maximal, characteristic
2	#22590114	C₂	1	2	yes	maximal
3	#22586441	C₂	1	2	yes	maximal
4	#22586420	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1946	C₂	1	1	yes	maximal, characteristic
6	#17986556	—	2	1	no	—
7	C₂×#14556068	C₂²	1	2	yes	—
8	#17986408	—	2	4	no	—
9	#17986037	—	2	1	no	—
10	#17986037	C₂²	1	2	yes	—
11	#17985823	—	2	4	no	—
12	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
13	#427×C₇₁	C₂²	1	1	yes	characteristic
14	#424×C₇₁	—	2	1	no	—
15	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
16	#339×C₇₁	—	2	1	no	—
17	#14556128	—	2	2	no	—
18	#14556120	—	2	2	no	—
19	#14556107	—	4	2	no	—
20	C₄×Dic₁₄₂	—	4	2	no	—
21	#14556087	—	2	2	no	—
22	#14556068	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
25	#14556062	—	2	2	no	—
26	#14556062	—	2	2	no	—
27	#14556060	—	2	2	no	—
28	#14556060	D₄	1	4	yes	—
29	C₂×M₁₆×C₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	#117×C₇₁	—	2	1	no	—
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	#108×C₇₁	—	2	2	no	—
35	#107×C₇₁	—	2	1	no	—
36	#106×C₇₁	Q₈	1	2	yes	complemented
37	C₄⋊C₈×C₇₁	—	4	2	no	—
38	#97×C₇₁	—	2	2	no	—
39	C₂×Dic₁₄₂	—	4	2	no	—
40	C₄⋊Dic₇₁	—	2	4	no	—
41	C₄⋊Dic₇₁	—	4	2	no	—
42	Dic₇₁⋊C₄	—	4	4	no	—
43	C₂²⋊Dic₇₁	—	4	2	no	—
44	C₄×Dic₇₁	—	4	2	no	—
45	C₇₁⋊C₁₆	—	2	4	no	—
46	C₂²×C₄×C₇₁	—	2	1	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₁	—	4	1	no	—
49	M₁₆×C₇₁	—	4	1	no	—
50	C₂×C₈×C₇₁	—	2	1	no	—
51	C₂×C₈×C₇₁	—	2	1	no	—
52	C₂×C₈×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
58	C₄⋊C₄×C₇₁	—	2	2	no	—
59	C₂²⋊C₄×C₇₁	—	4	1	no	—
60	C₄²×C₇₁	—	2	1	no	—
61	C₄²×C₇₁	—	4	1	no	—
62	C₄²×C₇₁	SD₁₆	1	2	yes	—
63	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	—	4	2	no	—
66	C₂×C₄×C₇₁	—	8	1	no	—
67	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
73	C₂×Dic₇₁	—	4	4	no	—
74	Dic₁₄₂	—	8	4	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	—	8	1	no	—
77	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
79	C₂²×C₇₁	—	4	1	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	—	4	2	no	—
84	C₄×C₇₁	—	8	2	no	—
85	C₄×C₇₁	#395	1	1	yes	characteristic
86	C₄×C₇₁	#448	1	1	yes	characteristic
87	C₄×C₇₁	#475	1	1	yes	characteristic
88	C₄×C₇₁	#498	1	1	yes	characteristic
89	Dic₇₁	—	8	4	no	—
90	#13459	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	#2058	1	1	yes	characteristic
95	#3162	—	71	1	no	—
96	#2155	—	71	2	no	—
97	#1946	D₇₁	1	1	yes	characteristic
98	#1384	—	71	2	no	—
99	#1375	—	71	1	no	—
100	C₇₁	#13459	1	1	yes	characteristic, complemented, Sylow
101	#533	—	142	1	no	—
102	C₂×M₃₂	—	71	2	no	—
103	#501	—	142	4	no	—
104	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
105	#427	—	71	2	no	—
106	#427	—	142	1	no	—
107	#427	C₂×D₇₁	1	1	yes	characteristic
108	#424	—	2	1	no	—
109	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
110	#365	—	142	4	no	—
111	#339	—	2	1	no	—
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₄⋊Q₈	—	142	2	no	—
116	#125	—	142	2	no	—
117	C₂²⋊Q₈	—	284	2	no	—
118	C₄×Q₈	—	284	2	no	—
119	#117	—	2	1	no	—
120	#117	—	142	2	no	—
121	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
122	M₃₂	—	142	2	no	—
123	M₃₂	—	142	2	no	—
124	C₂×C₁₆	—	71	4	no	—
125	#108	—	2	2	no	—
126	#107	—	2	1	no	—
127	#107	—	142	2	no	—
128	#107	—	142	2	no	—
129	#106	—	71	4	no	—
130	#106	—	142	2	no	—
131	#106	Dic₁₄₂	1	2	yes	complemented
132	C₄⋊C₈	—	4	2	no	—
133	#97	—	2	2	no	—
134	C₂×Q₈	—	284	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	M₁₆	—	4	1	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	#11765853	1	2	yes	—
143	C₂×C₈	D₄×D₇₁	1	2	yes	—
144	C₂×C₈	Q₈×D₇₁	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
146	C₂×C₈	#11765865	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	284	4	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄⋊C₄	—	142	4	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₂²⋊C₄	—	284	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	—	284	2	no	—
156	C₄²	#11765852	1	2	yes	—
157	C₁₆	—	142	4	no	—
158	C₂³	#14556153	1	1	yes	characteristic
159	Q₈	—	568	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	284	4	no	—
164	C₂×C₄	#14556096	1	1	yes	characteristic
165	C₂×C₄	#14556107	1	1	yes	characteristic
166	C₂×C₄	#14556126	1	1	yes	characteristic
167	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
168	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
169	C₂×C₄	#14556158	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₈	#14556106	1	2	yes	—
173	C₈	#14556127	1	2	yes	—
174	C₂²	—	4	1	no	—
175	C₂²	#17986149	1	1	yes	characteristic
176	C₂²	#17986289	1	1	yes	characteristic
177	C₂²	#17986409	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	568	4	no	—
181	C₄	#17985919	1	1	yes	characteristic
182	C₄	#17986137	1	1	yes	characteristic
183	C₄	#17986284	1	1	yes	characteristic
184	C₄	#17986395	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22589673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22598864

C₂

yes

maximal, characteristic

#22590114

C₂

yes

maximal

#22586441

C₂

yes

maximal

#22586420

C₂

yes

maximal, characteristic

C₇₁×#1946

C₂

yes

maximal, characteristic

#17986556

—

C₂×#14556068

C₂²

yes

—

#17986408

—

#17986037

—

#17986037

C₂²

yes

—

#17985823

—

C₂×#108×C₇₁

C₂²

yes

characteristic

#427×C₇₁

C₂²

yes

characteristic

#424×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

#339×C₇₁

—

#14556128

—

#14556120

—

#14556107

—

C₄×Dic₁₄₂

—

#14556087

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

Q₈

yes

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#108×C₇₁

—

#107×C₇₁

—

#106×C₇₁

Q₈

yes

complemented

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

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

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#13459

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1384

—

#1375

—

100

C₇₁

#13459

yes

characteristic, complemented, Sylow

101

#533

—

142

—

102

C₂×M₃₂

—

103

#501

—

142

—

104

C₂×#108

C₂×D₇₁

yes

characteristic

105

#427

—

106

#427

—

142

—

107

#427

C₂×D₇₁

yes

characteristic

108

#424

—

109

C₂×#97

C₂×D₇₁

yes

characteristic

110

#365

—

142

—

111

#339

—

112

C₂×M₁₆

—

113

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

114

C₂²×C₈

C₂²×D₇₁

yes

characteristic

115

C₄⋊Q₈

—

142

—

116

#125

—

142

—

117

C₂²⋊Q₈

—

284

—

118

C₄×Q₈

—

284

—

119

#117

—

120

#117

—

142

—

121

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

122

M₃₂

—

142

—

123

M₃₂

—

142

—

124

C₂×C₁₆

—

125

#108

—

126

#107

—

127

#107

—

142

—

128

#107

—

142

—

129

#106

—

130

#106

—

142

—

131

#106

Dic₁₄₂

yes

complemented

132

C₄⋊C₈

—

133

#97

—

134

C₂×Q₈

—

284

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#11765853

yes

—

143

C₂×C₈

D₄×D₇₁

yes

—

144

C₂×C₈

Q₈×D₇₁

yes

—

145

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

146

C₂×C₈

#11765865

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

284

—

149

C₄⋊C₄

—

284

—

150

C₄⋊C₄

—

142

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

284

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

284

—

156

C₄²

#11765852

yes

—

157

C₁₆

—

142

—

158

C₂³

#14556153

yes

characteristic

159

Q₈

—

568

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

284

—

164

C₂×C₄

#14556096

yes

characteristic

165

C₂×C₄

#14556107

yes

characteristic

166

C₂×C₄

#14556126

yes

characteristic

167

C₂×C₄

C₂×#11765852

yes

characteristic

168

C₂×C₄

C₂×#11765853

yes

characteristic

169

C₂×C₄

#14556158

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

#14556106

yes

—

173

C₈

#14556127

yes

—

174

C₂²

—

175

C₂²

#17986149

yes

characteristic

176

C₂²

#17986289

yes

characteristic

177

C₂²

#17986409

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

568

—

181

C₄

#17985919

yes

characteristic

182

C₄

#17986137

yes

characteristic

183

C₄

#17986284

yes

characteristic

184

C₄

#17986395

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

#22589673

yes

characteristic

	subgroup	G/H
center	C₄	#17986395
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1946	C₂
socle	C₂×C₇₁	#2058
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
hypercenter	#1946	D₇₁
hypocenter	C₇₁	#13459
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#17986395

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1946

C₂

C₂×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
24. C₂×C₇₁⋊C₁₆	Q₈	semidirect
36. #106×C₇₁	Q₈	semidirect
100. C₇₁	#13459	semidirect
131. #106	Dic₁₄₂	semidirect

H ≅ G/N

type

24. C₂×C₇₁⋊C₁₆

Q₈

semidirect

36. #106×C₇₁

Q₈

semidirect

100. C₇₁

#13459

semidirect

131. #106

Dic₁₄₂

semidirect

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

subgroup	index
#22598864	2
#22590114	2
#22586441	2
#22586420	2
C₇₁×#1946	2
#13459	71

subgroup

index

C₇₁×#1946

Subgroups — #31917110

Conjugacy classes of subgroups

Complements

Maximal subgroups