Group #31917106 — subgroups

Subgroups — #31917106

Order 18176.

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

normal subgroups only

This group has 324 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×#17986401	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×#14556120	—	2	1	no	—
7	#17986401	—	2	4	no	—
8	C₂×#14556064	C₂²	1	1	yes	characteristic
9	#17986037	C₂²	1	1	yes	characteristic
10	C₂×#14556062	—	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	C₂×#11765849	—	2	1	no	—
18	#14556120	—	2	4	no	—
19	#14556120	—	4	2	no	—
20	#14556087	—	2	1	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×Dic₇₁⋊C₄	—	4	2	no	—
23	#14556064	—	2	2	no	—
24	#14556062	—	2	4	no	—
25	#14556062	Q₈	1	2	yes	—
26	#14556060	—	2	2	no	—
27	#14556060	—	2	1	no	—
28	#14556060	D₄	1	4	yes	—
29	Dic₇₁⋊C₈	—	4	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	M₃₂×C₇₁	—	2	2	no	—
34	M₃₂×C₇₁	Q₈	1	4	yes	—
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₄×C₈×C₇₁	—	2	2	no	—
37	C₂²×Dic₇₁	—	4	2	no	—
38	#11765849	—	4	1	no	—
39	#11765849	—	4	1	no	—
40	C₂×C₇₁⋊C₈	—	4	1	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	2	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	Dic₇₁⋊C₄	—	4	8	no	—
45	C₂²⋊Dic₇₁	—	4	1	no	—
46	C₄×Dic₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	—	2	1	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	—	2	1	no	—
50	C₂×C₈×C₇₁	—	2	1	no	—
51	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
52	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
56	C₄²×C₇₁	—	2	1	no	—
57	C₄²×C₇₁	SD₁₆	1	2	yes	—
58	C₁₆×C₇₁	—	2	4	no	—
59	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	4	2	no	—
62	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
68	C₂×Dic₇₁	—	4	4	no	—
69	C₂×Dic₇₁	—	4	2	no	—
70	C₂×Dic₇₁	—	8	4	no	—
71	C₈×C₇₁	—	2	2	no	—
72	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
74	C₇₁⋊C₈	—	8	1	no	—
75	C₂²×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	#450	1	1	yes	characteristic
77	C₂²×C₇₁	#476	1	1	yes	characteristic
78	C₂²×C₇₁	#501	1	1	yes	characteristic
79	C₄×C₇₁	—	4	2	no	—
80	C₄×C₇₁	#395	1	1	yes	characteristic
81	C₄×C₇₁	#448	1	1	yes	characteristic
82	C₄×C₇₁	#475	1	1	yes	characteristic
83	C₄×C₇₁	#498	1	1	yes	characteristic
84	Dic₇₁	—	8	4	no	—
85	Dic₇₁	—	8	2	no	—
86	#13458	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	—	2	1	no	—
88	C₂×C₇₁	—	2	1	no	—
89	C₂×C₇₁	#2058	1	1	yes	characteristic
90	C₂×C₇₁	#2231	1	2	yes	—
91	C₂×#499	—	71	1	no	—
92	C₂×#365	—	71	2	no	—
93	#1945	—	71	1	no	—
94	#1386	—	71	2	no	—
95	#1376	D₇₁	1	1	yes	characteristic
96	C₇₁	#13458	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	142	1	no	—
98	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
99	#499	—	142	4	no	—
100	C₂×#108	—	71	1	no	—
101	#427	—	71	1	no	—
102	C₂×#107	—	142	1	no	—
103	C₂×#106	—	71	2	no	—
104	#424	—	142	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	#365	—	142	4	no	—
107	#339	—	142	1	no	—
108	C₂×M₁₆	—	142	1	no	—
109	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	#125	—	284	2	no	—
112	#125	—	142	4	no	—
113	#117	—	142	1	no	—
114	C₂×C₄⋊C₄	—	284	2	no	—
115	C₂×C₄⋊C₄	—	142	2	no	—
116	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	Dic₁₄₂	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	142	2	no	—
121	#107	—	71	2	no	—
122	#107	—	142	4	no	—
123	#106	—	71	4	no	—
124	#106	—	142	2	no	—
125	#106	—	142	1	no	—
126	C₄⋊C₈	—	284	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	284	2	no	—
130	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
131	M₁₆	—	284	1	no	—
132	M₁₆	—	284	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	284	1	no	—
136	C₂×C₈	#11765854	1	2	yes	—
137	C₂×C₈	D₄×D₇₁	1	2	yes	—
138	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
140	C₂×C₈	#11765864	1	1	yes	characteristic
141	C₄⋊C₄	—	284	8	no	—
142	C₄⋊C₄	—	284	2	no	—
143	C₄⋊C₄	—	142	4	no	—
144	C₄⋊C₄	—	142	2	no	—
145	C₂²⋊C₄	—	284	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	284	1	no	—
148	C₄²	#11765854	1	2	yes	—
149	C₁₆	—	2	4	no	—
150	C₂³	#14556156	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	284	4	no	—
154	C₂×C₄	—	284	2	no	—
155	C₂×C₄	—	568	4	no	—
156	C₂×C₄	#14556098	1	1	yes	characteristic
157	C₂×C₄	#14556108	1	1	yes	characteristic
158	C₂×C₄	#14556128	1	1	yes	characteristic
159	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
160	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
161	C₂×C₄	#14556161	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	568	1	no	—
164	C₈	#14556105	1	2	yes	—
165	C₈	#14556127	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#17986152	1	1	yes	characteristic
168	C₂²	#17986290	1	1	yes	characteristic
169	C₂²	#17986408	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	568	4	no	—
172	C₄	—	568	2	no	—
173	C₄	#17985920	1	1	yes	characteristic
174	C₄	#17986140	1	1	yes	characteristic
175	C₄	#17986285	1	1	yes	characteristic
176	C₄	#17986397	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#22589674	1	1	yes	characteristic
180	C₂	#22590517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

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×#14556120

—

#17986401

—

C₂×#14556064

C₂²

yes

characteristic

#17986037

C₂²

yes

characteristic

C₂×#14556062

—

C₂×#14556060

C₂²

yes

—

#17986025

—

#17985822

—

#17985759

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

#14556064

—

#14556062

—

#14556062

Q₈

yes

—

#14556060

—

#14556060

—

#14556060

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

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

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

—

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

—

#13458

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2058

yes

characteristic

C₂×C₇₁

#2231

yes

—

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#13458

yes

characteristic, complemented, Sylow

C₂×#125

—

142

—

C₂×M₃₂

C₂×D₇₁

yes

—

#499

—

142

—

100

C₂×#108

—

101

#427

—

102

C₂×#107

—

142

—

103

C₂×#106

—

104

#424

—

142

—

105

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

106

#365

—

142

—

107

#339

—

142

—

108

C₂×M₁₆

—

142

—

109

C₂²×C₈

D₂₈₄

yes

characteristic

110

C₂²×C₈

C₂²×D₇₁

yes

characteristic

111

#125

—

284

—

112

#125

—

142

—

113

#117

—

142

—

114

C₂×C₄⋊C₄

—

284

—

115

C₂×C₄⋊C₄

—

142

—

116

C₂×C₄²

D₂₈₄

yes

characteristic

117

M₃₂

—

118

M₃₂

Dic₁₄₂

yes

—

119

C₂×C₁₆

—

120

#108

—

142

—

121

#107

—

122

#107

—

142

—

123

#106

—

124

#106

—

142

—

125

#106

—

142

—

126

C₄⋊C₈

—

284

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

284

—

130

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

131

M₁₆

—

284

—

132

M₁₆

—

284

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

284

—

136

C₂×C₈

#11765854

yes

—

137

C₂×C₈

D₄×D₇₁

yes

—

138

C₂×C₈

Q₈×D₇₁

yes

characteristic

139

C₂×C₈

C₂×Dic₁₄₂

yes

—

140

C₂×C₈

#11765864

yes

characteristic

141

C₄⋊C₄

—

284

—

142

C₄⋊C₄

—

284

—

143

C₄⋊C₄

—

142

—

144

C₄⋊C₄

—

142

—

145

C₂²⋊C₄

—

284

—

146

C₄²

—

147

C₄²

—

284

—

148

C₄²

#11765854

yes

—

149

C₁₆

—

150

C₂³

#14556156

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

284

—

154

C₂×C₄

—

284

—

155

C₂×C₄

—

568

—

156

C₂×C₄

#14556098

yes

characteristic

157

C₂×C₄

#14556108

yes

characteristic

158

C₂×C₄

#14556128

yes

characteristic

159

C₂×C₄

C₂×#11765854

yes

characteristic

160

C₂×C₄

C₂×#11765854

yes

characteristic

161

C₂×C₄

#14556161

yes

characteristic

162

C₈

—

163

C₈

—

568

—

164

C₈

#14556105

yes

—

165

C₈

#14556127

yes

—

166

C₂²

—

167

C₂²

#17986152

yes

characteristic

168

C₂²

#17986290

yes

characteristic

169

C₂²

#17986408

yes

characteristic

170

C₄

—

171

C₄

—

568

—

172

C₄

—

568

—

173

C₄

#17985920

yes

characteristic

174

C₄

#17986140

yes

characteristic

175

C₄

#17986285

yes

characteristic

176

C₄

#17986397

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#22589674

yes

characteristic

180

C₂

#22590517

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	#13458
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
96. C₇₁	#13458	semidirect

H ≅ G/N

type

96. C₇₁

#13458

semidirect

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

subgroup	index
C₂×#17986401	2
C₂×#17985822	2
#22589088	2
#22586446	2
C₇₁×#1376	2
#13458	71

subgroup

index

C₇₁×#1376

Subgroups — #31917106

Conjugacy classes of subgroups

Complements

Maximal subgroups