Group #31915500 — subgroups

Subgroups — #31915500

Order 18176.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22590124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22589086	C₂	1	2	yes	maximal
4	#22586437	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	2	yes	—
8	#17986027	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985827	C₄	1	2	yes	complemented
11	#17985825	C₄	1	2	yes	complemented
12	C₁₆×Dic₇₁	—	2	2	no	—
13	#17985760	—	2	2	no	—
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₁	C₂²	1	2	yes	—
16	#368×C₇₁	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
18	#14556087	—	2	2	no	—
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
23	#14556062	C₂×C₄	1	4	yes	—
24	#14556060	—	2	2	no	—
25	Dic₇₁⋊C₈	—	4	4	no	—
26	C₈×Dic₇₁	—	2	1	no	—
27	C₈×Dic₇₁	—	2	1	no	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
34	#108×C₇₁	—	2	2	no	—
35	#108×C₇₁	—	2	2	no	—
36	C₂²×Dic₇₁	—	2	1	no	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
40	C₄⋊Dic₇₁	—	2	4	no	—
41	C₂²⋊Dic₇₁	—	4	2	no	—
42	C₄×Dic₇₁	—	2	1	no	—
43	C₄×Dic₇₁	—	4	2	no	—
44	C₄×Dic₇₁	Q₁₆	1	2	yes	—
45	C₇₁⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₁	—	4	2	no	—
48	M₁₆×C₇₁	—	4	2	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
56	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₁	#135	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
65	C₂×Dic₇₁	—	2	4	no	—
66	C₂×Dic₇₁	—	2	2	no	—
67	C₂×Dic₇₁	—	4	4	no	—
68	C₈×C₇₁	—	8	2	no	—
69	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₈×C₇₁	#125	1	1	yes	characteristic
73	C₇₁⋊C₈	—	2	2	no	—
74	C₂²×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₇₁	#467	1	1	yes	characteristic
77	C₂²×C₇₁	#499	1	1	yes	characteristic
78	C₄×C₇₁	#389	1	1	yes	characteristic
79	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₇₁	#466	1	1	yes	characteristic
81	C₄×C₇₁	#500	1	1	yes	characteristic
82	Dic₇₁	—	2	4	no	—
83	Dic₇₁	—	8	4	no	—
84	#13169	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	—	2	1	no	—
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	#1942	1	1	yes	characteristic
88	C₂×C₇₁	#2178	1	2	yes	—
89	C₂×#368	D₇₁	1	1	yes	characteristic
90	#2158	—	71	1	no	—
91	C₂×C₄×C₁₆	—	71	1	no	—
92	#1945	—	71	2	no	—
93	#1382	—	71	2	no	—
94	C₇₁	#13169	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	71	1	no	—
96	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
97	C₂×#108	C₂×D₇₁	1	2	yes	—
98	#427	—	71	2	no	—
99	#424	—	142	2	no	—
100	C₂×C₄×C₈	—	71	1	no	—
101	#368	—	2	2	no	—
102	#367	—	71	2	no	—
103	#366	—	71	2	no	—
104	C₄×C₁₆	—	142	2	no	—
105	#339	—	142	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	71	1	no	—
108	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	#117	—	142	2	no	—
110	C₂×C₄²	—	71	1	no	—
111	C₂×C₁₆	—	71	2	no	—
112	C₂×C₁₆	—	142	1	no	—
113	C₂×C₁₆	—	142	1	no	—
114	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
115	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	71	4	no	—
121	#106	—	142	2	no	—
122	C₄⋊C₈	—	284	4	no	—
123	C₄×C₈	—	142	1	no	—
124	C₄×C₈	—	142	1	no	—
125	C₂²×C₄	—	142	1	no	—
126	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	71	2	no	—
131	C₂×C₈	—	142	1	no	—
132	C₂×C₈	—	142	1	no	—
133	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
137	C₂×C₈	#11765864	1	1	yes	characteristic
138	C₂×C₈	#11765866	1	1	yes	characteristic
139	C₄⋊C₄	—	142	4	no	—
140	C₂²⋊C₄	—	284	2	no	—
141	C₄²	—	284	2	no	—
142	C₄²	—	71	2	no	—
143	C₄²	—	142	1	no	—
144	C₁₆	—	142	2	no	—
145	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
146	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
147	C₂³	#14556151	1	1	yes	characteristic
148	C₂×C₄	—	284	4	no	—
149	C₂×C₄	—	142	4	no	—
150	C₂×C₄	—	142	2	no	—
151	C₂×C₄	#14556090	1	1	yes	characteristic
152	C₂×C₄	#14556099	1	1	yes	characteristic
153	C₂×C₄	#14556127	1	1	yes	characteristic
154	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
156	C₂×C₄	#14556147	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	142	2	no	—
159	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
160	C₈	#14556093	1	1	yes	characteristic
161	C₈	#14556104	1	1	yes	characteristic
162	C₈	#14556119	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#17986081	1	1	yes	characteristic
165	C₂²	#17986237	1	1	yes	characteristic
166	C₂²	#17986400	1	1	yes	characteristic
167	C₄	—	568	4	no	—
168	C₄	—	142	4	no	—
169	C₄	#17985895	1	1	yes	characteristic
170	C₄	#17986093	1	1	yes	characteristic
171	C₄	#17986232	1	1	yes	characteristic
172	C₄	#17986403	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#22589071	1	1	yes	characteristic
176	C₂	#22590220	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590124

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#22589086

C₂

yes

maximal

#22586437

C₂

yes

maximal

C₂×#368×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17986037

C₂²

yes

—

#17986027

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985827

C₄

yes

complemented

#17985825

C₄

yes

complemented

C₁₆×Dic₇₁

—

#17985760

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

#368×C₇₁

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

#14556087

—

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

Q₈

yes

—

#14556062

C₂×C₄

yes

—

#14556060

—

Dic₇₁⋊C₈

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

Q₈

yes

—

#108×C₇₁

—

#108×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

Q₁₆

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

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

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

#135

yes

characteristic

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

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

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

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×D₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₄×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#13169

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1942

yes

characteristic

C₂×C₇₁

#2178

yes

—

C₂×#368

D₇₁

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₇₁

#13169

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

C₂×D₇₁

yes

—

#427

—

#424

—

142

—

100

C₂×C₄×C₈

—

101

#368

—

102

#367

—

103

#366

—

104

C₄×C₁₆

—

142

—

105

#339

—

142

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₇₁

yes

characteristic

109

#117

—

142

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

142

—

113

C₂×C₁₆

—

142

—

114

C₂×C₁₆

Dic₁₄₂

yes

—

115

C₂×C₁₆

C₄×D₇₁

yes

characteristic

116

C₂×C₁₆

C₄×D₇₁

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₇₁

yes

—

118

#108

—

119

#108

—

120

#107

—

121

#106

—

142

—

122

C₄⋊C₈

—

284

—

123

C₄×C₈

—

142

—

124

C₄×C₈

—

142

—

125

C₂²×C₄

—

142

—

126

C₂²×C₄

D₄×D₇₁

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

142

—

132

C₂×C₈

—

142

—

133

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

134

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

135

C₂×C₈

Q₈×D₇₁

yes

characteristic

136

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

137

C₂×C₈

#11765864

yes

characteristic

138

C₂×C₈

#11765866

yes

characteristic

139

C₄⋊C₄

—

142

—

140

C₂²⋊C₄

—

284

—

141

C₄²

—

284

—

142

C₄²

—

143

C₄²

—

142

—

144

C₁₆

—

142

—

145

C₁₆

Dic₇₁⋊C₄

yes

—

146

C₁₆

Dic₇₁⋊C₄

yes

—

147

C₂³

#14556151

yes

characteristic

148

C₂×C₄

—

284

—

149

C₂×C₄

—

142

—

150

C₂×C₄

—

142

—

151

C₂×C₄

#14556090

yes

characteristic

152

C₂×C₄

#14556099

yes

characteristic

153

C₂×C₄

#14556127

yes

characteristic

154

C₂×C₄

D₈×D₇₁

yes

characteristic

155

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

156

C₂×C₄

#14556147

yes

characteristic

157

C₈

—

158

C₈

—

142

—

159

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

160

C₈

#14556093

yes

characteristic

161

C₈

#14556104

yes

characteristic

162

C₈

#14556119

yes

characteristic

163

C₂²

—

164

C₂²

#17986081

yes

characteristic

165

C₂²

#17986237

yes

characteristic

166

C₂²

#17986400

yes

characteristic

167

C₄

—

568

—

168

C₄

—

142

—

169

C₄

#17985895

yes

characteristic

170

C₄

#17986093

yes

characteristic

171

C₄

#17986232

yes

characteristic

172

C₄

#17986403

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#22589071

yes

characteristic

176

C₂

#22590220

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	#13169
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#368×C₇₁

C₂

socle

C₂²×C₇₁

C₄×D₈

prime radical

C₂³×C₇₁

#135

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #17985827	C₄	semidirect
11. #17985825	C₄	semidirect
94. C₇₁	#13169	semidirect

H ≅ G/N

type

10. #17985827

C₄

semidirect

11. #17985825

C₄

semidirect

94. C₇₁

#13169

semidirect

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

subgroup	index
#22590124	2
C₂×C₁₆×Dic₇₁	2
#22589086	2
#22586437	2
C₂×#368×C₇₁	2
#13169	71

subgroup

index

C₂×C₁₆×Dic₇₁

C₂×#368×C₇₁

Subgroups — #31915500

Conjugacy classes of subgroups

Complements

Maximal subgroups