Group #31916046 — subgroups

Subgroups — #31916046

Order 18176.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22590096	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
3	#22586448	C₂	1	4	yes	maximal
4	C₂×#499×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556064	C₂²	1	2	yes	—
7	C₄×#11765849	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
9	#17985764	—	2	1	no	—
10	C₂×#125×C₇₁	—	2	1	no	—
11	#499×C₇₁	—	2	2	no	—
12	#499×C₇₁	C₄	1	4	yes	—
13	C₂×#107×C₇₁	C₂²	1	2	yes	—
14	C₂×#106×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765849	—	2	2	no	—
17	#14556068	—	2	4	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	#14556064	—	2	2	no	—
20	#14556064	Q₈	1	4	yes	—
21	#14556039	—	4	1	no	—
22	C₄×C₇₁⋊C₈	—	4	1	no	—
23	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
25	#125×C₇₁	—	2	4	no	—
26	#125×C₇₁	—	4	1	no	—
27	#125×C₇₁	—	4	1	no	—
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
31	#107×C₇₁	—	2	2	no	—
32	#107×C₇₁	C₂×C₄	1	4	yes	—
33	#106×C₇₁	—	2	4	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	#11765849	—	2	4	no	—
37	#11765849	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₇₁⋊C₁₆	—	4	2	no	—
40	C₂²×C₄×C₇₁	—	4	2	no	—
41	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₇₁	#41	1	2	yes	—
49	C₄⋊C₄×C₇₁	—	2	4	no	—
50	C₄⋊C₄×C₇₁	—	4	4	no	—
51	C₄⋊C₄×C₇₁	—	4	4	no	—
52	C₄⋊C₄×C₇₁	—	4	2	no	—
53	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₁	D₈	1	1	yes	characteristic
55	C₄²×C₇₁	D₈	1	1	yes	characteristic
56	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	4	2	no	—
58	C₂×C₄×C₇₁	—	4	4	no	—
59	C₂×C₄×C₇₁	—	8	4	no	—
60	C₂×C₄×C₇₁	#102	1	2	yes	—
61	C₂×C₄×C₇₁	#102	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#137	1	2	yes	—
66	C₈×C₇₁	—	2	2	no	—
67	C₈×C₇₁	C₄×Q₈	1	2	yes	—
68	C₈×C₇₁	#124	1	1	yes	characteristic
69	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₇₁⋊C₈	—	4	4	no	—
71	C₂²×C₇₁	—	2	2	no	—
72	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
73	C₂²×C₇₁	#475	1	2	yes	—
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	8	4	no	—
77	C₄×C₇₁	#391	1	1	yes	characteristic
78	C₄×C₇₁	#419	1	1	yes	characteristic
79	C₄×C₇₁	#474	1	2	yes	—
80	#13280	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	2	no	—
82	C₂×C₇₁	#1988	1	1	yes	characteristic
83	C₂×C₇₁	#2148	1	2	yes	—
84	C₂×#499	D₇₁	1	1	yes	characteristic
85	#2150	—	71	1	no	—
86	C₄×#108	—	71	1	no	—
87	#1387	—	71	4	no	—
88	C₇₁	#13280	1	1	yes	characteristic, complemented, Sylow
89	C₂×#125	—	2	1	no	—
90	C₂×M₃₂	—	71	2	no	—
91	#499	—	2	2	no	—
92	#499	Dic₇₁	1	4	yes	—
93	C₂×#108	—	71	2	no	—
94	C₂×#107	C₂×D₇₁	1	2	yes	—
95	C₂×#106	—	2	1	no	—
96	C₄×M₁₆	—	142	1	no	—
97	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
98	C₄⋊C₁₆	—	142	2	no	—
99	#341	—	142	1	no	—
100	C₂×M₁₆	—	142	2	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
109	M₃₂	—	142	4	no	—
110	C₂×C₁₆	—	142	2	no	—
111	#108	—	71	4	no	—
112	#108	—	142	2	no	—
113	#107	—	2	2	no	—
114	#107	C₂×Dic₇₁	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	284	1	no	—
117	C₄×C₈	—	284	1	no	—
118	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
119	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11765853	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
123	M₁₆	—	284	2	no	—
124	M₁₆	—	142	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	284	2	no	—
127	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
128	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₇₁	1	2	yes	—
131	C₂×C₈	#11765866	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
137	C₄²	#11765850	1	1	yes	characteristic
138	C₄²	#11765850	1	1	yes	characteristic
139	C₁₆	—	284	2	no	—
140	C₂³	C₂×#11765853	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#14556049	1	2	yes	—
145	C₂×C₄	#14556049	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
147	C₂×C₄	#14556104	1	2	yes	—
148	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
149	C₂×C₄	#14556159	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	284	4	no	—
152	C₈	Q₈×Dic₇₁	1	2	yes	—
153	C₈	#14556117	1	1	yes	characteristic
154	C₈	#14556126	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14556049	1	1	yes	characteristic
157	C₂²	#17986281	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#17985906	1	1	yes	characteristic
162	C₄	#17986008	1	1	yes	characteristic
163	C₄	#17986276	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#22589293	1	1	yes	characteristic
166	C₂	#22590089	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590096

C₂

yes

maximal, characteristic

C₄×#14556064

C₂

yes

maximal, characteristic

#22586448

C₂

yes

maximal

C₂×#499×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556064

C₂²

yes

—

C₄×#11765849

—

C₄⋊(C₇₁⋊C₁₆)

—

#17985764

—

C₂×#125×C₇₁

—

#499×C₇₁

—

#499×C₇₁

C₄

yes

—

C₂×#107×C₇₁

C₂²

yes

—

C₂×#106×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

Q₈

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#125×C₇₁

—

#125×C₇₁

—

#125×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#107×C₇₁

C₂×C₄

yes

—

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₂³×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#102

yes

characteristic

C₂²×C₇₁

#475

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#13280

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1988

yes

characteristic

C₂×C₇₁

#2148

yes

—

C₂×#499

D₇₁

yes

—

—

—

C₇₁

#13280

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₇₁

yes

—

C₂×#108

—

C₂×#107

C₂×D₇₁

yes

—

C₂×#106

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

#341

—

142

—

100

C₂×M₁₆

—

142

—

101

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

102

C₂²×C₈

C₂²×D₇₁

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

109

M₃₂

—

142

—

110

C₂×C₁₆

—

142

—

111

#108

—

112

#108

—

142

—

113

#107

—

114

#107

C₂×Dic₇₁

yes

—

115

#106

—

116

#97

—

284

—

117

C₄×C₈

—

284

—

118

C₄×C₈

C₂×Dic₇₁

yes

—

119

C₄×C₈

C₂²⋊D₇₁

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

#11765853

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

123

M₁₆

—

284

—

124

M₁₆

—

142

—

125

C₂×C₈

—

126

C₂×C₈

—

284

—

127

C₂×C₈

C₂²⋊Dic₇₁

yes

—

128

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

129

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

130

C₂×C₈

Q₈×D₇₁

yes

—

131

C₂×C₈

#11765866

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂²⋊Dic₇₁

yes

—

137

C₄²

#11765850

yes

characteristic

138

C₄²

#11765850

yes

characteristic

139

C₁₆

—

284

—

140

C₂³

C₂×#11765853

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14556049

yes

—

145

C₂×C₄

#14556049

yes

—

146

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

147

C₂×C₄

#14556104

yes

—

148

C₂×C₄

C₂×#11765850

yes

characteristic

149

C₂×C₄

#14556159

yes

—

150

C₈

—

151

C₈

—

284

—

152

C₈

Q₈×Dic₇₁

yes

—

153

C₈

#14556117

yes

characteristic

154

C₈

#14556126

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#14556049

yes

characteristic

157

C₂²

#17986281

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#17985906

yes

characteristic

162

C₄

#17986008

yes

characteristic

163

C₄

#17986276

yes

—

164

C₂

—

165

C₂

#22589293

yes

characteristic

166

C₂

#22590089

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#499×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#499	D₇₁
hypocenter	C₇₁	#13280
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#14556049

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#499×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#102

prime radical

C₂³×C₇₁

C₂×SD₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
88. C₇₁	#13280	semidirect

H ≅ G/N

type

88. C₇₁

#13280

semidirect

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

subgroup	index
#22590096	2
C₄×#14556064	2
#22586448	2
C₂×#499×C₇₁	2
#13280	71

subgroup

index

C₂×#499×C₇₁

Subgroups — #31916046

Conjugacy classes of subgroups

Complements

Maximal subgroups