Group #31891158 — subgroups

Subgroups — #31891158

Order 18176.

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

normal subgroups only

This group has 228 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	#22587103	C₂	1	1	yes	maximal, characteristic
2	#22586623	C₂	1	1	yes	maximal, characteristic
3	#22586203	C₂	1	2	yes	maximal
4	#22586201	C₂	1	2	yes	maximal
5	C₇₁×#1568	C₂	1	1	yes	maximal, characteristic
6	#17986056	C₂²	1	2	yes	—
7	#17986052	C₂²	1	1	yes	characteristic
8	#17986052	C₂²	1	1	yes	characteristic
9	#17985752	—	2	1	no	—
10	#17985750	—	2	1	no	—
11	#17985720	C₄	1	2	yes	—
12	C₈×C₇₁⋊C₈	C₄	1	2	yes	—
13	#432×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#335×C₇₁	—	2	1	no	—
16	#334×C₇₁	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	#14556039	—	2	2	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₂²⋊C₈×C₇₁	—	2	2	no	—
31	#97×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	2	no	—
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	4	2	no	—
45	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₇₁	#41	1	1	yes	characteristic
52	C₄²×C₇₁	#41	1	1	yes	characteristic
53	C₂³×C₇₁	#125	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	1	no	—
55	C₂×C₄×C₇₁	—	2	1	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
64	C₈×C₇₁	—	2	1	no	—
65	C₈×C₇₁	—	2	1	no	—
66	C₈×C₇₁	—	8	2	no	—
67	C₈×C₇₁	#108	1	2	yes	—
68	C₈×C₇₁	#108	1	2	yes	—
69	C₇₁⋊C₈	—	4	2	no	—
70	C₇₁⋊C₈	—	4	2	no	—
71	C₇₁⋊C₈	—	8	2	no	—
72	C₂²×C₇₁	—	2	1	no	—
73	C₂²×C₇₁	—	2	1	no	—
74	C₂²×C₇₁	—	2	1	no	—
75	C₂²×C₇₁	#382	1	1	yes	characteristic
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	2	1	no	—
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
82	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
83	#8796	—	71	1	no	maximal, Sylow
84	C₂×C₇₁	—	4	1	no	—
85	C₂×C₇₁	#1833	1	1	yes	characteristic
86	C₂×C₇₁	#1833	1	1	yes	characteristic
87	C₂×C₇₁	#1837	1	1	yes	characteristic
88	#1568	—	71	1	no	—
89	#1568	D₇₁	1	1	yes	characteristic
90	#1450	—	71	1	no	—
91	#1287	—	71	2	no	—
92	#1287	—	71	2	no	—
93	C₇₁	#8796	1	1	yes	characteristic, complemented, Sylow
94	#432	—	71	2	no	—
95	#432	C₂×D₇₁	1	2	yes	—
96	#431	—	71	1	no	—
97	#431	—	71	1	no	—
98	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
99	#335	—	2	1	no	—
100	#335	—	142	1	no	—
101	#334	—	2	1	no	—
102	#334	—	142	1	no	—
103	#322	—	71	2	no	—
104	C₈²	—	71	2	no	—
105	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
106	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	142	2	no	—
110	C₄⋊C₈	—	142	1	no	—
111	C₄⋊C₈	—	142	1	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	142	2	no	—
114	C₂²⋊C₈	—	142	1	no	—
115	C₂²⋊C₈	—	142	1	no	—
116	#97	—	2	2	no	—
117	#97	—	142	2	no	—
118	C₄×C₈	—	71	2	no	—
119	C₄×C₈	—	71	2	no	—
120	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
121	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
125	C₂²×C₄	#11765864	1	1	yes	characteristic
126	C₂²×C₄	#11765866	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	284	2	no	—
131	C₂×C₈	—	142	2	no	—
132	C₂×C₈	—	142	2	no	—
133	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
136	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
139	C₄²	#11765864	1	1	yes	characteristic
140	C₄²	#11765866	1	1	yes	characteristic
141	C₂³	#14556119	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#14556086	1	1	yes	characteristic
148	C₂×C₄	#14556086	1	1	yes	characteristic
149	C₂×C₄	#14556115	1	1	yes	characteristic
150	C₂×C₄	#14556117	1	1	yes	characteristic
151	C₂×C₄	#14556119	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	8	2	no	—
155	C₈	—	284	2	no	—
156	C₈	—	284	2	no	—
157	C₈	—	568	2	no	—
158	C₈	#14556063	1	2	yes	—
159	C₈	#14556063	1	2	yes	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	#17985864	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	C₂×#14556063	1	1	yes	characteristic
170	C₄	C₂×#14556063	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#22588463	1	1	yes	characteristic
173	C₂	#22588465	1	1	yes	characteristic
174	C₂	#22588477	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22587103

C₂

yes

maximal, characteristic

#22586623

C₂

yes

maximal, characteristic

#22586203

C₂

yes

maximal

#22586201

C₂

yes

maximal

C₇₁×#1568

C₂

yes

maximal, characteristic

#17986056

C₂²

yes

—

#17986052

C₂²

yes

characteristic

#17986052

C₂²

yes

—

—

C₄

yes

—

C₈×C₇₁⋊C₈

C₄

yes

—

#432×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#335×C₇₁

—

#334×C₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×Q₈

yes

C₂²×C₄×C₇₁

yes

C₂²×C₄×C₇₁

yes

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

C₂³×C₇₁

yes

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#108

yes

—

C₈×C₇₁

#108

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#382

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

—

—

—

C₇₁

#8796

yes

characteristic, complemented, Sylow

#432

—

#432

C₂×D₇₁

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#335

—

100

#335

—

142

—

101

#334

—

102

#334

—

142

—

103

#322

—

104

C₈²

—

105

C₂²×C₈

Dic₁₄₂

yes

characteristic

106

C₂²×C₈

Dic₁₄₂

yes

characteristic

107

C₂×C₄²

C₂²×D₇₁

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

142

—

110

C₄⋊C₈

—

142

—

111

C₄⋊C₈

—

142

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

142

—

114

C₂²⋊C₈

—

142

—

115

C₂²⋊C₈

—

142

—

116

#97

—

117

#97

—

142

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₄×D₇₁

yes

characteristic

121

C₄×C₈

C₄×D₇₁

yes

characteristic

122

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

123

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

124

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

125

C₂²×C₄

#11765864

yes

characteristic

126

C₂²×C₄

#11765866

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

284

—

131

C₂×C₈

—

142

—

132

C₂×C₈

—

142

—

133

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

134

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

135

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

136

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

137

C₄²

C₂×C₄×D₇₁

yes

characteristic

138

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

139

C₄²

#11765864

yes

characteristic

140

C₄²

#11765866

yes

characteristic

141

C₂³

#14556119

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

147

C₂×C₄

#14556086

yes

characteristic

148

C₂×C₄

#14556086

yes

characteristic

149

C₂×C₄

#14556115

yes

characteristic

150

C₂×C₄

#14556117

yes

characteristic

151

C₂×C₄

#14556119

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

284

—

156

C₈

—

284

—

157

C₈

—

568

—

158

C₈

#14556063

yes

—

159

C₈

#14556063

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#17985864

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

C₂×#14556063

yes

characteristic

170

C₄

C₂×#14556063

yes

characteristic

171

C₂

—

172

C₂

#22588463

yes

characteristic

173

C₂

#22588465

yes

characteristic

174

C₂

#22588477

yes

characteristic

	subgroup	G/H
center	C₂²	#17985864
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1568	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#125
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1568	D₇₁
hypocenter	C₇₁	#8796
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#17985864

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1568

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
93. C₇₁	#8796	semidirect

H ≅ G/N

type

93. C₇₁

#8796

semidirect

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

subgroup	index
#22587103	2
#22586623	2
#22586203	2
#22586201	2
C₇₁×#1568	2
#8796	71

subgroup

index

C₇₁×#1568

Subgroups — #31891158

Conjugacy classes of subgroups

Complements

Maximal subgroups