Group #31902470 — subgroups

Subgroups — #31902470

Order 18176.

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

normal subgroups only

This group has 354 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	#22588494	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587116	C₂	1	2	yes	maximal, complemented
3	#22586739	C₂	1	2	yes	maximal
4	#22586169	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1963	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	#17985751	—	2	4	no	—
10	D₄⋊(C₇₁⋊C₈)	—	2	4	no	—
11	#532×C₇₁	C₂²	1	1	yes	characteristic
12	#423×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#104×C₇₁	—	2	2	no	—
14	#376×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₈	—	2	2	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	4	4	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄×C₇₁⋊C₈	D₄	1	4	yes	—
22	C₂×#41×C₇₁	—	2	1	no	—
23	C₂×M₁₆×C₇₁	—	2	1	no	—
24	C₄⋊Q₈×C₇₁	Q₈	1	1	yes	characteristic
25	#127×C₇₁	Q₈	1	1	yes	characteristic
26	#124×C₇₁	—	2	1	no	—
27	#121×C₇₁	—	2	2	no	—
28	C₄×D₄×C₇₁	—	2	2	no	—
29	C₂×C₄²×C₇₁	D₄	1	2	yes	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	#104×C₇₁	—	4	4	no	—
33	#95×C₇₁	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	4	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	4	4	no	—
37	#41×C₇₁	—	4	2	no	—
38	#41×C₇₁	—	4	2	no	—
39	C₂×Q₈×C₇₁	—	2	1	no	—
40	C₂×D₄×C₇₁	—	2	2	no	—
41	C₂×D₄×C₇₁	—	2	1	no	—
42	C₂×D₄×C₇₁	—	4	1	no	—
43	C₂²×C₄×C₇₁	—	2	2	no	—
44	C₂²×C₄×C₇₁	—	4	1	no	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	#41	1	2	yes	—
47	M₁₆×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	—	4	1	no	—
49	C₄⋊C₄×C₇₁	—	2	2	no	—
50	C₂²⋊C₄×C₇₁	—	4	2	no	—
51	C₄²×C₇₁	—	2	2	no	—
52	C₄²×C₇₁	D₈	1	4	yes	—
53	C₄²×C₇₁	C₂×D₄	1	2	yes	—
54	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₇₁	#41	1	1	yes	characteristic
56	C₂³×C₇₁	—	4	1	no	—
57	C₂³×C₇₁	#123	1	1	yes	characteristic
58	Q₈×C₇₁	—	4	2	no	—
59	D₄×C₇₁	—	4	4	no	—
60	D₄×C₇₁	—	4	2	no	—
61	D₄×C₇₁	—	8	2	no	—
62	C₂×C₄×C₇₁	—	2	4	no	—
63	C₂×C₄×C₇₁	—	4	2	no	—
64	C₂×C₄×C₇₁	—	4	4	no	—
65	C₂×C₄×C₇₁	—	4	1	no	—
66	C₂×C₄×C₇₁	—	8	1	no	—
67	C₂×C₄×C₇₁	—	8	1	no	—
68	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#123	1	2	yes	—
71	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
73	C₂×C₄×C₇₁	#137	1	2	yes	—
74	C₈×C₇₁	—	8	1	no	—
75	C₇₁⋊C₈	—	4	4	no	—
76	C₇₁⋊C₈	—	8	2	no	—
77	C₂²×C₇₁	—	2	2	no	—
78	C₂²×C₇₁	—	8	1	no	—
79	C₂²×C₇₁	—	8	2	no	—
80	C₂²×C₇₁	#399	1	1	yes	characteristic
81	C₂²×C₇₁	#480	1	2	yes	—
82	C₄×C₇₁	—	2	2	no	—
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	4	4	no	—
85	C₄×C₇₁	—	8	1	no	—
86	C₄×C₇₁	#474	1	2	yes	—
87	C₄×C₇₁	#493	1	1	yes	characteristic
88	C₄×C₇₁	#497	1	1	yes	characteristic
89	#10667	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	2	2	no	—
91	C₂×C₇₁	—	16	1	no	—
92	C₂×C₇₁	#2093	1	1	yes	characteristic
93	C₂×C₇₁	#2098	1	2	yes	—
94	#1963	D₇₁	1	1	yes	characteristic
95	#1843	—	71	1	no	—
96	#1571	—	71	2	no	—
97	#1480	—	71	2	no	—
98	#1274	—	71	1	no	—
99	C₇₁	#10667	1	1	yes	characteristic, complemented, Sylow
100	#532	C₂×D₇₁	1	1	yes	characteristic
101	#431	—	71	2	no	—
102	#423	C₂×D₇₁	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	71	2	no	—
104	C₂×C₄⋊C₈	—	142	2	no	—
105	C₂×#104	—	2	2	no	—
106	#376	C₂×D₇₁	1	1	yes	characteristic
107	#335	—	142	4	no	—
108	D₄⋊C₈	—	142	4	no	—
109	C₂×#41	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	142	2	no	—
112	C₄⋊Q₈	Dic₁₄₂	1	1	yes	characteristic
113	#127	Dic₁₄₂	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
118	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	284	4	no	—
121	C₄⋊C₈	—	142	2	no	—
122	C₄⋊C₈	—	142	2	no	—
123	C₄⋊C₈	—	142	2	no	—
124	#104	—	4	4	no	—
125	C₂²⋊C₈	—	142	2	no	—
126	C₄×C₈	—	71	4	no	—
127	#95	—	2	2	no	—
128	#41	—	4	2	no	—
129	#41	—	4	2	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	4	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂²×C₄	#11765864	1	2	yes	—
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	284	2	no	—
141	C₂×C₈	—	284	4	no	—
142	C₂×C₈	—	142	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#11765850	1	4	yes	—
147	C₄²	D₄×D₇₁	1	2	yes	—
148	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
149	C₄²	#11765865	1	1	yes	characteristic
150	C₂³	—	4	1	no	—
151	C₂³	#14556109	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	4	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#14556105	1	2	yes	—
163	C₂×C₄	#14556107	1	1	yes	characteristic
164	C₂×C₄	#14556110	1	2	yes	—
165	C₂×C₄	#14556124	1	1	yes	characteristic
166	C₂×C₄	C₂×#11765850	1	2	yes	—
167	C₂×C₄	#14556158	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	284	4	no	—
170	C₈	—	568	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#17985935	1	1	yes	characteristic
175	C₂²	#17986310	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	8	1	no	—
180	C₄	#17986279	1	2	yes	—
181	C₄	#17986375	1	1	yes	characteristic
182	C₄	#17986394	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#22589868	1	1	yes	characteristic
186	C₂	#22589895	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588494

C₂

yes

maximal, characteristic, complemented

#22587116

C₂

yes

maximal, complemented

#22586739

C₂

yes

maximal

#22586169

C₂

yes

maximal, characteristic, complemented

C₇₁×#1963

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×C₄⋊(C₇₁⋊C₈)

—

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#17985751

—

D₄⋊(C₇₁⋊C₈)

—

#532×C₇₁

C₂²

yes

characteristic

#423×C₇₁

C₂²

yes

characteristic

C₂×#104×C₇₁

—

#376×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×#41×C₇₁

—

C₂×M₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

Q₈

yes

characteristic

#127×C₇₁

Q₈

yes

characteristic

#124×C₇₁

—

#121×C₇₁

—

C₄×D₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

#104×C₇₁

—

#95×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

D₈

yes

—

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

#123

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#399

yes

characteristic

C₂²×C₇₁

#480

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#474

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2093

yes

characteristic

C₂×C₇₁

#2098

yes

—

#1963

D₇₁

yes

—

—

—

—

C₇₁

#10667

yes

characteristic, complemented, Sylow

100

#532

C₂×D₇₁

yes

characteristic

101

#431

—

102

#423

C₂×D₇₁

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

—

142

—

105

C₂×#104

—

106

#376

C₂×D₇₁

yes

characteristic

107

#335

—

142

—

108

D₄⋊C₈

—

142

—

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

142

—

112

C₄⋊Q₈

Dic₁₄₂

yes

characteristic

113

#127

Dic₁₄₂

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₇₁

yes

—

118

C₂×C₄²

C₂²×D₇₁

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

284

—

121

C₄⋊C₈

—

142

—

122

C₄⋊C₈

—

142

—

123

C₄⋊C₈

—

142

—

124

#104

—

125

C₂²⋊C₈

—

142

—

126

C₄×C₈

—

127

#95

—

128

#41

—

129

#41

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

137

C₂²×C₄

#11765864

yes

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

284

—

141

C₂×C₈

—

284

—

142

C₂×C₈

—

142

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

#11765850

yes

—

147

C₄²

D₄×D₇₁

yes

—

148

C₄²

C₂×Dic₁₄₂

yes

characteristic

149

C₄²

#11765865

yes

characteristic

150

C₂³

—

151

C₂³

#14556109

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#14556105

yes

—

163

C₂×C₄

#14556107

yes

characteristic

164

C₂×C₄

#14556110

yes

—

165

C₂×C₄

#14556124

yes

characteristic

166

C₂×C₄

C₂×#11765850

yes

—

167

C₂×C₄

#14556158

yes

—

168

C₈

—

169

C₈

—

284

—

170

C₈

—

568

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#17985935

yes

characteristic

175

C₂²

#17986310

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#17986279

yes

—

181

C₄

#17986375

yes

characteristic

182

C₄

#17986394

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#22589868

yes

characteristic

186

C₂

#22589895

yes

—

	subgroup	G/H
center	C₂×C₄	#14556124
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1963	C₂
socle	C₂²×C₇₁	#399
prime radical	#532×C₇₁	C₂²
hypercenter	#1963	D₇₁
hypocenter	C₇₁	#10667
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#14556124

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1963

C₂

socle

C₂²×C₇₁

#399

prime radical

#532×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #22588494	C₂	semidirect
2. #22587116	C₂	semidirect
4. #22586169	C₂	semidirect
99. C₇₁	#10667	semidirect

H ≅ G/N

type

1. #22588494

C₂

semidirect

2. #22587116

C₂

semidirect

4. #22586169

C₂

semidirect

99. C₇₁

#10667

semidirect

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

subgroup	index
#22588494	2
#22587116	2
#22586739	2
#22586169	2
C₇₁×#1963	2
#10667	71

subgroup

index

C₇₁×#1963

Subgroups — #31902470

Conjugacy classes of subgroups

Complements

Maximal subgroups