Group #31915522 — subgroups

Subgroups — #31915522

Order 18176.

262 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
3	#22589130	C₂	1	2	yes	maximal
4	#22586452	C₂	1	2	yes	maximal
5	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
6	#17986047	—	2	2	no	—
7	C₂×#14556064	C₂²	1	2	yes	—
8	C₂×#14556064	C₂²	1	2	yes	—
9	#17985829	—	2	1	no	—
10	#17985829	—	2	1	no	—
11	#17985829	C₄	1	4	yes	complemented
12	#17985769	—	2	2	no	—
13	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	#405×C₇₁	C₂²	1	1	yes	characteristic
16	#346×C₇₁	—	2	1	no	—
17	C₄×C₁₆×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	C₂×#11765849	—	2	2	no	—
20	#14556064	—	2	2	no	—
21	#14556064	—	2	2	no	—
22	#14556064	—	2	2	no	—
23	#14556064	C₂×C₄	1	4	yes	—
24	#14556063	—	4	4	no	—
25	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	#117×C₇₁	D₄	1	1	yes	characteristic
28	M₃₂×C₇₁	—	2	1	no	—
29	M₃₂×C₇₁	Q₈	1	2	yes	—
30	C₂×C₁₆×C₇₁	—	2	1	no	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
35	#97×C₇₁	—	2	1	no	—
36	C₄×C₈×C₇₁	—	2	1	no	—
37	#11765849	—	2	4	no	—
38	#11765849	—	4	2	no	—
39	#11765849	—	4	2	no	—
40	#11765849	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₇₁	—	2	1	no	—
45	M₁₆×C₇₁	D₈	1	2	yes	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₇₁	—	2	1	no	—
54	C₂²⋊C₄×C₇₁	Q₁₆	1	2	yes	—
55	C₄²×C₇₁	—	2	1	no	—
56	C₁₆×C₇₁	—	2	2	no	—
57	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
67	C₈×C₇₁	—	2	2	no	—
68	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
70	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₈×C₇₁	#125	1	1	yes	characteristic
72	C₇₁⋊C₈	—	4	4	no	—
73	C₇₁⋊C₈	—	8	2	no	—
74	C₂²×C₇₁	—	4	1	no	—
75	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
76	C₂²×C₇₁	#467	1	1	yes	characteristic
77	C₂²×C₇₁	#500	1	1	yes	characteristic
78	C₄×C₇₁	—	4	2	no	—
79	C₄×C₇₁	#389	1	1	yes	characteristic
80	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
81	C₄×C₇₁	#466	1	1	yes	characteristic
82	C₄×C₇₁	#499	1	1	yes	characteristic
83	#13173	—	71	1	no	maximal, Sylow
84	C₂×C₇₁	—	2	1	no	—
85	C₂×C₇₁	—	2	1	no	—
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	#1942	1	1	yes	characteristic
88	C₂×#368	—	71	1	no	—
89	C₂×#368	—	71	1	no	—
90	#2108	D₇₁	1	1	yes	characteristic
91	#1953	—	71	2	no	—
92	#1388	—	71	2	no	—
93	C₇₁	#13173	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
95	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
96	#430	—	142	2	no	—
97	C₂×#108	—	71	2	no	—
98	C₂×#108	—	71	2	no	—
99	#405	C₂×D₇₁	1	1	yes	characteristic
100	#368	—	71	4	no	—
101	#368	—	142	1	no	—
102	#368	—	142	1	no	—
103	#346	—	2	1	no	—
104	C₄×C₁₆	—	2	1	no	—
105	#344	—	142	2	no	—
106	C₂×M₁₆	—	142	2	no	—
107	C₂×M₁₆	—	142	2	no	—
108	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
110	#117	D₂₈₄	1	1	yes	characteristic
111	M₃₂	—	2	1	no	—
112	M₃₂	Dic₁₄₂	1	2	yes	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
115	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
116	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
118	#108	—	284	4	no	—
119	#108	—	71	4	no	—
120	#108	—	142	2	no	—
121	#108	—	142	2	no	—
122	#108	—	142	2	no	—
123	#97	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
126	M₁₆	—	2	1	no	—
127	M₁₆	—	284	2	no	—
128	M₁₆	—	284	2	no	—
129	M₁₆	—	284	2	no	—
130	M₁₆	—	142	4	no	—
131	M₁₆	D₅₆₈	1	2	yes	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	284	2	no	—
134	C₂×C₈	—	284	2	no	—
135	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
139	C₂×C₈	#11765865	1	1	yes	characteristic
140	C₂×C₈	#11765865	1	1	yes	characteristic
141	C₄⋊C₄	—	2	1	no	—
142	C₂²⋊C₄	Dic₂₈₄	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
146	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
147	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
150	C₂×C₄	#14556102	1	1	yes	characteristic
151	C₂×C₄	#14556128	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
153	C₂×C₄	#14556150	1	1	yes	characteristic
154	C₂×C₄	#14556150	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	284	4	no	—
157	C₈	—	568	2	no	—
158	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
159	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
160	C₈	#14556107	1	1	yes	characteristic
161	C₈	#14556120	1	1	yes	characteristic
162	C₂²	—	4	1	no	—
163	C₂²	C₄×Dic₂₈₄	1	1	yes	characteristic
164	C₂²	#17986240	1	1	yes	characteristic
165	C₂²	#17986404	1	1	yes	characteristic
166	C₄	—	4	2	no	—
167	C₄	#17985897	1	1	yes	characteristic
168	C₄	C₄×D₅₆₈	1	1	yes	characteristic
169	C₄	#17986235	1	1	yes	characteristic
170	C₄	#17986401	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#22589073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal, characteristic

C₂×#17985829

C₂

yes

maximal, characteristic

#22589130

C₂

yes

maximal

#22586452

C₂

yes

maximal

C₇₁×#2108

C₂

yes

maximal, characteristic

#17986047

—

C₂×#14556064

C₂²

yes

—

C₂×#14556064

C₂²

yes

—

#17985829

—

#17985829

—

#17985829

C₄

yes

complemented

#17985769

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

#405×C₇₁

C₂²

yes

characteristic

#346×C₇₁

—

C₄×C₁₆×C₇₁

—

—

—

—

—

—

C₂×C₄

yes

—

#14556063

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

Q₈

yes

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

Q₈

yes

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

Q₁₆

yes

—

C₄²×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

characteristic

C₈×C₇₁

#125

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

#389

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1953

—

#1388

—

C₇₁

#13173

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₁

yes

characteristic

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#430

—

142

—

C₂×#108

—

C₂×#108

—

#405

C₂×D₇₁

yes

characteristic

100

#368

—

101

#368

—

142

—

102

#368

—

142

—

103

#346

—

104

C₄×C₁₆

—

105

#344

—

142

—

106

C₂×M₁₆

—

142

—

107

C₂×M₁₆

—

142

—

108

C₂×M₁₆

D₂₈₄

yes

characteristic

109

C₂²×C₈

C₂²×D₇₁

yes

characteristic

110

#117

D₂₈₄

yes

characteristic

111

M₃₂

—

112

M₃₂

Dic₁₄₂

yes

—

113

C₂×C₁₆

—

114

C₂×C₁₆

Dic₁₄₂

yes

—

115

C₂×C₁₆

C₄×D₇₁

yes

—

116

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

118

#108

—

284

—

119

#108

—

120

#108

—

142

—

121

#108

—

142

—

122

#108

—

142

—

123

#97

—

124

C₄×C₈

—

125

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

126

M₁₆

—

127

M₁₆

—

284

—

128

M₁₆

—

284

—

129

M₁₆

—

284

—

130

M₁₆

—

142

—

131

M₁₆

D₅₆₈

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

284

—

134

C₂×C₈

—

284

—

135

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

136

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

137

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

138

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

139

C₂×C₈

#11765865

yes

characteristic

140

C₂×C₈

#11765865

yes

characteristic

141

C₄⋊C₄

—

142

C₂²⋊C₄

Dic₂₈₄

yes

—

143

C₄²

—

144

C₁₆

—

145

C₁₆

Dic₇₁⋊C₄

yes

—

146

C₁₆

Dic₇₁⋊C₄

yes

—

147

C₂³

C₂×Dic₂₈₄

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

C₄×D₂₈₄

yes

characteristic

150

C₂×C₄

#14556102

yes

characteristic

151

C₂×C₄

#14556128

yes

characteristic

152

C₂×C₄

C₂×D₅₆₈

yes

characteristic

153

C₂×C₄

#14556150

yes

characteristic

154

C₂×C₄

#14556150

yes

characteristic

155

C₈

—

156

C₈

—

284

—

157

C₈

—

568

—

158

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

159

C₈

C₄×Dic₁₄₂

yes

characteristic

160

C₈

#14556107

yes

characteristic

161

C₈

#14556120

yes

characteristic

162

C₂²

—

163

C₂²

C₄×Dic₂₈₄

yes

characteristic

164

C₂²

#17986240

yes

characteristic

165

C₂²

#17986404

yes

characteristic

166

C₄

—

167

C₄

#17985897

yes

characteristic

168

C₄

C₄×D₅₆₈

yes

characteristic

169

C₄

#17986235

yes

characteristic

170

C₄

#17986401

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#22589073

yes

characteristic

	subgroup	G/H
center	C₄	#17986235
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂×C₇₁	#1942
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#13173
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#17986235

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2108

C₂

socle

C₂×C₇₁

#1942

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108

N	H ≅ G/N	type
11. #17985829	C₄	semidirect
93. C₇₁	#13173	semidirect

H ≅ G/N

type

11. #17985829

C₄

semidirect

93. C₇₁

#13173

semidirect

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

subgroup	index
C₂×#17985829	2
#22589130	2
#22586452	2
C₇₁×#2108	2
#13173	71

subgroup

index

C₇₁×#2108

Subgroups — #31915522

Conjugacy classes of subgroups

Complements

Maximal subgroups