Group #31884302 — subgroups

Subgroups — #31884302

Order 18176.

210 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22590143	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	#17985775	C₄	1	2	yes	—
9	#17985721	—	2	1	no	—
10	#17985721	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₁	—	2	1	no	—
13	C₂²⋊C₁₆×C₇₁	—	2	1	no	—
14	#346×C₇₁	C₄	1	2	yes	—
15	C₂²×C₇₁⋊C₈	—	2	1	no	—
16	C₂²×C₇₁⋊C₈	C₈	1	2	yes	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₄×C₇₁⋊C₈	—	2	1	no	—
19	C₄×C₇₁⋊C₈	—	2	1	no	—
20	C₄×C₇₁⋊C₈	—	2	1	no	—
21	C₄×C₇₁⋊C₈	C₈	1	2	yes	—
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	—	2	2	no	—
26	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	—	2	4	no	—
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₂×C₇₁⋊C₈	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₇₁⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	4	1	no	—
39	C₂×C₈×C₇₁	—	4	1	no	—
40	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₁	C₄²	1	1	yes	characteristic
45	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
48	C₁₆×C₇₁	—	4	2	no	—
49	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	2	1	no	—
64	C₇₁⋊C₈	—	2	4	no	—
65	C₇₁⋊C₈	—	4	2	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	#336	1	1	yes	characteristic
69	C₂²×C₇₁	#364	1	2	yes	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
75	C₄×C₇₁	#338	1	1	yes	characteristic
76	C₄×C₇₁	#349	1	2	yes	—
77	#7146	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	2	2	no	—
79	C₂×C₇₁	#1275	1	1	yes	characteristic
80	C₂×C₇₁	#1376	1	1	yes	characteristic
81	C₂×C₇₁	#1377	1	1	yes	characteristic
82	#2163	—	71	1	no	—
83	#2163	D₇₁	1	1	yes	characteristic
84	C₂×#322	—	71	1	no	—
85	C₇₁	#7146	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄×C₈	—	71	1	no	—
87	C₂×C₄×C₈	—	71	1	no	—
88	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	—	142	1	no	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₂²⋊C₁₆	—	142	1	no	—
93	#346	—	71	2	no	—
94	#346	Dic₇₁	1	2	yes	—
95	#322	—	142	1	no	—
96	#322	—	142	1	no	—
97	C₂²×C₈	—	71	2	no	—
98	C₂²×C₈	—	142	1	no	—
99	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	142	2	no	—
104	C₄×C₈	—	71	2	no	—
105	C₄×C₈	—	142	1	no	—
106	C₄×C₈	—	142	1	no	—
107	C₄×C₈	—	142	1	no	—
108	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
109	C₄×C₈	D₂₈₄	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
114	C₂²×C₄	#11765848	1	1	yes	characteristic
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	284	2	no	—
118	C₂×C₈	—	142	2	no	—
119	C₂×C₈	—	142	4	no	—
120	C₂×C₈	—	142	2	no	—
121	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
123	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
125	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
126	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
127	C₄²	C₈×D₇₁	1	1	yes	characteristic
128	C₄²	#11765848	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₁₆	—	284	2	no	—
131	C₂³	Dic₇₁⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
137	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
138	C₂×C₄	#14556038	1	1	yes	characteristic
139	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	284	2	no	—
147	C₈	—	142	4	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#17985753	1	1	yes	characteristic
151	C₂²	#17985819	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#17985755	1	1	yes	characteristic
157	C₄	C₇₁⋊#338	1	1	yes	characteristic
158	C₄	#17985780	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#22586170	1	1	yes	characteristic
161	C₂	#22586421	1	1	yes	characteristic
162	C₂	#22586423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal, characteristic

C₂×#17985721

C₂

yes

maximal, characteristic

C₇₁×#2163

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

—

C₂²⋊(C₇₁⋊C₁₆)

—

#17985775

C₄

yes

—

#17985721

—

#17985721

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₂²⋊C₁₆×C₇₁

—

#346×C₇₁

C₄

yes

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₈

yes

—

C₂×C₇₁⋊C₁₆

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₈

yes

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×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

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₄²

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

M₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

characteristic

C₂×C₄×C₇₁

#97

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#336

yes

characteristic

C₂²×C₇₁

#364

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#7146

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

C₂×#322

—

C₇₁

#7146

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

142

—

#346

—

#346

Dic₇₁

yes

—

#322

—

142

—

#322

—

142

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

Dic₁₄₂

yes

characteristic

100

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

101

C₂×C₄²

C₄×D₇₁

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

142

—

104

C₄×C₈

—

105

C₄×C₈

—

142

—

106

C₄×C₈

—

142

—

107

C₄×C₈

—

142

—

108

C₄×C₈

C₄×D₇₁

yes

characteristic

109

C₄×C₈

D₂₈₄

yes

characteristic

110

C₄×C₈

C₂×Dic₇₁

yes

characteristic

111

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

112

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

113

C₂²×C₄

C₈×D₇₁

yes

characteristic

114

C₂²×C₄

#11765848

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

284

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

—

142

—

121

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

122

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

123

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

124

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

125

C₄²

C₄×Dic₇₁

yes

characteristic

126

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

127

C₄²

C₈×D₇₁

yes

characteristic

128

C₄²

#11765848

yes

characteristic

129

C₁₆

—

130

C₁₆

—

284

—

131

C₂³

Dic₇₁⋊C₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₇₁⋊#95

yes

characteristic

137

C₂×C₄

C₈×Dic₇₁

yes

characteristic

138

C₂×C₄

#14556038

yes

characteristic

139

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

140

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

141

C₂×C₄

Dic₇₁⋊C₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

284

—

147

C₈

—

142

—

148

C₂²

—

149

C₂²

—

150

C₂²

#17985753

yes

characteristic

151

C₂²

#17985819

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#17985755

yes

characteristic

157

C₄

C₇₁⋊#338

yes

characteristic

158

C₄

#17985780

yes

—

159

C₂

—

160

C₂

#22586170

yes

characteristic

161

C₂

#22586421

yes

characteristic

162

C₂

#22586423

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#2163	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#2163	D₇₁
hypocenter	C₇₁	#7146
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

C₇₁⋊#95

commutator

C₂×C₄×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₇₁×#2163

C₂

socle

C₂²×C₇₁

#336

prime radical

C₂³×C₇₁

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
85. C₇₁	#7146	semidirect

H ≅ G/N

type

85. C₇₁

#7146

semidirect

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

subgroup	index
#22590143	2
C₂×#17985721	2
C₇₁×#2163	2
#7146	71

subgroup

index

#22590143

C₂×#17985721

C₇₁×#2163

#7146

Subgroups — #31884302

Conjugacy classes of subgroups

Complements

Maximal subgroups