Group #31915986 — subgroups

Subgroups — #31915986

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	#22597274	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	#22586430	C₂	1	4	yes	maximal
4	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
5	#17986556	C₄	1	1	yes	characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	#17986129	—	2	2	no	—
8	C₈×Dic₁₄₂	—	2	2	no	—
9	#17986025	C₄	1	2	yes	—
10	#17986023	C₄	1	1	yes	characteristic
11	#17985954	C₂²	1	2	yes	—
12	#17985775	—	2	2	no	—
13	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	—	2	2	no	—
16	C₂×#11765849	C₂×C₄	1	2	yes	—
17	#14556128	C₈	1	2	yes	—
18	#14556120	C₈	1	2	yes	—
19	#14556107	—	2	2	no	—
20	C₄×Dic₁₄₂	—	2	2	no	—
21	#14556087	C₂×C₄	1	2	yes	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	Dic₇₁⋊C₈	—	2	4	no	—
27	#14556038	—	2	2	no	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₁	—	2	4	no	—
33	C₂×C₁₆×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	D₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×Dic₁₄₂	—	2	2	no	—
37	#11765849	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₄⋊Dic₇₁	—	2	2	no	—
40	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
41	Dic₇₁⋊C₄	—	2	4	no	—
42	C₂²⋊Dic₇₁	—	2	2	no	—
43	C₄×Dic₇₁	—	2	2	no	—
44	C₇₁⋊C₁₆	—	2	4	no	—
45	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₁	#41	1	2	yes	—
53	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₁	—	4	2	no	—
57	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	4	2	no	—
59	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#131	1	2	yes	—
65	C₂×Dic₇₁	—	2	4	no	—
66	Dic₁₄₂	—	4	4	no	—
67	C₈×C₇₁	—	2	2	no	—
68	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₇₁	#124	1	1	yes	characteristic
70	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₇₁⋊C₈	—	4	2	no	—
72	C₂²×C₇₁	—	2	2	no	—
73	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₁	#446	1	2	yes	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	2	2	no	—
77	C₄×C₇₁	#391	1	1	yes	characteristic
78	C₄×C₇₁	#408	1	1	yes	characteristic
79	C₄×C₇₁	C₈×Q₈	1	2	yes	—
80	Dic₇₁	—	4	4	no	—
81	#13267	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	2	no	—
83	C₂×C₇₁	#1984	1	1	yes	characteristic
84	C₂×C₇₁	#2117	1	2	yes	—
85	#2960	—	71	1	no	—
86	#2150	D₇₁	1	1	yes	characteristic
87	C₂×#346	—	71	1	no	—
88	#1379	—	71	4	no	—
89	C₇₁	#13267	1	1	yes	characteristic, complemented, Sylow
90	#533	—	71	1	no	—
91	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
92	C₂²×C₁₆	—	71	2	no	—
93	#446	—	142	2	no	—
94	C₈×Q₈	—	142	2	no	—
95	#424	—	71	2	no	—
96	#423	—	71	1	no	—
97	#405	—	71	2	no	—
98	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	2	no	—
100	#346	—	142	2	no	—
101	C₂×M₁₆	—	71	2	no	—
102	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₄⋊Q₈	—	71	2	no	—
105	#125	—	71	2	no	—
106	C₂²⋊Q₈	—	142	2	no	—
107	C₄×Q₈	—	142	2	no	—
108	#117	—	71	2	no	—
109	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	71	4	no	—
113	C₂×C₁₆	—	142	2	no	—
114	C₂×C₁₆	—	142	2	no	—
115	C₄⋊C₈	—	142	4	no	—
116	C₄⋊C₈	—	142	2	no	—
117	#97	—	142	2	no	—
118	C₄×C₈	—	142	2	no	—
119	C₄×C₈	D₂₈₄	1	2	yes	—
120	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
121	C₂×Q₈	—	142	2	no	—
122	C₂²×C₄	#11765848	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
124	M₁₆	—	142	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	142	2	no	—
127	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₇₁	1	2	yes	—
131	C₂×C₈	#11765864	1	2	yes	—
132	C₄⋊C₄	—	71	4	no	—
133	C₄⋊C₄	—	142	4	no	—
134	C₄⋊C₄	—	142	2	no	—
135	C₂²⋊C₄	—	142	2	no	—
136	C₄²	—	142	2	no	—
137	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
138	C₄²	C₈×D₇₁	1	1	yes	characteristic
139	C₄²	C₈×D₇₁	1	1	yes	characteristic
140	C₁₆	—	4	2	no	—
141	C₁₆	—	142	4	no	—
142	C₂³	C₂×#11765848	1	1	yes	characteristic
143	Q₈	—	284	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	142	4	no	—
146	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14556093	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
151	C₂×C₄	#14556136	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	284	2	no	—
154	C₈	#14556108	1	2	yes	—
155	C₈	#14556115	1	1	yes	characteristic
156	C₈	#14556126	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
159	C₂²	#17986126	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	284	4	no	—
163	C₄	#17985904	1	1	yes	characteristic
164	C₄	#17985967	1	1	yes	characteristic
165	C₄	#17986123	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#22589275	1	1	yes	characteristic
168	C₂	#22589969	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22597274

C₂

yes

maximal, characteristic

C₂×#17985775

C₂

yes

maximal, characteristic

#22586430

C₂

yes

maximal

C₇₁×#2150

C₂

yes

maximal, characteristic

#17986556

C₄

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

#17986129

—

C₈×Dic₁₄₂

—

#17986025

C₄

yes

—

#17986023

C₄

yes

characteristic

#17985954

C₂²

yes

—

#17985775

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₂×#11765849

C₂×C₄

yes

—

#14556128

C₈

yes

—

#14556120

C₈

yes

—

#14556107

—

C₄×Dic₁₄₂

—

#14556087

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₈

yes

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

#131

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

#446

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₈×Q₈

yes

—

Dic₇₁

—

#13267

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1984

yes

characteristic

C₂×C₇₁

#2117

yes

—

#2960

—

#2150

D₇₁

yes

characteristic

C₂×#346

—

#1379

—

C₇₁

#13267

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

C₂×D₇₁

yes

—

C₂²×C₁₆

—

#446

—

142

—

C₈×Q₈

—

142

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

100

#346

—

142

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₇₁

yes

characteristic

103

C₂²×C₈

C₂²×D₇₁

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

142

—

107

C₄×Q₈

—

142

—

108

#117

—

109

C₂×C₄²

C₄×D₇₁

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

142

—

114

C₂×C₁₆

—

142

—

115

C₄⋊C₈

—

142

—

116

C₄⋊C₈

—

142

—

117

#97

—

142

—

118

C₄×C₈

—

142

—

119

C₄×C₈

D₂₈₄

yes

—

120

C₄×C₈

C₂²⋊D₇₁

yes

—

121

C₂×Q₈

—

142

—

122

C₂²×C₄

#11765848

yes

—

123

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

124

M₁₆

—

142

—

125

C₂×C₈

—

126

C₂×C₈

—

142

—

127

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

128

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

129

C₂×C₈

C₂×D₂₈₄

yes

characteristic

130

C₂×C₈

Q₈×D₇₁

yes

—

131

C₂×C₈

#11765864

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

142

—

134

C₄⋊C₄

—

142

—

135

C₂²⋊C₄

—

142

—

136

C₄²

—

142

—

137

C₄²

(C₂×D₇₁)⋊C₄

yes

—

138

C₄²

C₈×D₇₁

yes

characteristic

139

C₄²

C₈×D₇₁

yes

characteristic

140

C₁₆

—

141

C₁₆

—

142

—

142

C₂³

C₂×#11765848

yes

characteristic

143

Q₈

—

284

—

144

C₂×C₄

—

145

C₂×C₄

—

142

—

146

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

149

C₂×C₄

#14556093

yes

—

150

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

151

C₂×C₄

#14556136

yes

—

152

C₈

—

153

C₈

—

284

—

154

C₈

#14556108

yes

—

155

C₈

#14556115

yes

characteristic

156

C₈

#14556126

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₇₁)⋊C₈

yes

characteristic

159

C₂²

#17986126

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

284

—

163

C₄

#17985904

yes

characteristic

164

C₄

#17985967

yes

characteristic

165

C₄

#17986123

yes

—

166

C₂

—

167

C₂

#22589275

yes

characteristic

168

C₂

#22589969

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2150	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#13267
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2150

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
24. C₂×C₇₁⋊C₁₆	Q₈	semidirect
89. C₇₁	#13267	semidirect

H ≅ G/N

type

24. C₂×C₇₁⋊C₁₆

Q₈

semidirect

89. C₇₁

#13267

semidirect

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

subgroup	index
#22597274	2
C₂×#17985775	2
#22586430	2
C₇₁×#2150	2
#13267	71

subgroup

index

C₇₁×#2150

Subgroups — #31915986

Conjugacy classes of subgroups

Complements

Maximal subgroups