Group #31915515 — subgroups

Subgroups — #31915515

Order 18176.

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

normal subgroups only

This group has 274 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	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
2	#22589936	C₂	1	1	yes	maximal, characteristic
3	#22589123	C₂	1	2	yes	maximal
4	#22586448	C₂	1	2	yes	maximal
5	C₂×#367×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	#17986039	—	2	2	no	—
9	#17985954	C₂²	1	1	yes	characteristic
10	#17985829	C₄	1	4	yes	complemented
11	#17985774	—	2	1	no	—
12	C₁₆×Dic₇₁	—	2	1	no	—
13	#17985768	—	2	2	no	—
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#107×C₇₁	C₂²	1	2	yes	—
16	#367×C₇₁	—	2	1	no	—
17	#367×C₇₁	—	2	1	no	—
18	C₂×#11765849	—	2	2	no	—
19	C₂×#11765849	D₄	1	1	yes	characteristic
20	#14556087	D₄	1	1	yes	characteristic
21	#14556068	—	2	1	no	—
22	#14556068	Q₈	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	#14556064	—	2	2	no	—
25	#14556064	C₂×C₄	1	4	yes	—
26	#14556061	—	4	2	no	—
27	#14556059	—	4	2	no	—
28	#14556038	—	2	1	no	—
29	C₈×Dic₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
36	#107×C₇₁	—	2	2	no	—
37	#107×C₇₁	—	2	2	no	—
38	#11765849	—	2	1	no	—
39	#11765849	—	2	4	no	—
40	#11765849	—	4	2	no	—
41	#11765849	Q₁₆	1	2	yes	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₁	D₈	1	2	yes	—
45	C₂²⋊Dic₇₁	—	2	1	no	—
46	C₄×Dic₇₁	—	2	1	no	—
47	C₇₁⋊C₁₆	—	2	2	no	—
48	C₂²×C₄×C₇₁	—	4	2	no	—
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₇₁	—	4	2	no	—
57	C₄⋊C₄×C₇₁	—	4	2	no	—
58	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂³×C₇₁	#135	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	8	2	no	—
62	C₂×C₄×C₇₁	—	16	2	no	—
63	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
69	C₂×Dic₇₁	—	2	2	no	—
70	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₈×C₇₁	#125	1	1	yes	characteristic
74	C₇₁⋊C₈	—	2	2	no	—
75	C₇₁⋊C₈	—	4	4	no	—
76	C₂²×C₇₁	—	2	2	no	—
77	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₇₁	#466	1	1	yes	characteristic
79	C₂²×C₇₁	#499	1	1	yes	characteristic
80	C₄×C₇₁	—	16	2	no	—
81	C₄×C₇₁	#389	1	1	yes	characteristic
82	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
83	C₄×C₇₁	#467	1	1	yes	characteristic
84	C₄×C₇₁	#500	1	1	yes	characteristic
85	Dic₇₁	—	4	2	no	—
86	#13172	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	—	2	1	no	—
88	C₂×C₇₁	—	2	1	no	—
89	C₂×C₇₁	#1942	1	1	yes	characteristic
90	C₂×C₇₁	#2223	1	2	yes	—
91	C₂×#368	—	71	1	no	—
92	C₂×#367	D₇₁	1	1	yes	characteristic
93	#2108	—	71	1	no	—
94	#1952	—	71	2	no	—
95	#1387	—	71	2	no	—
96	C₇₁	#13172	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₃₂	—	71	1	no	—
98	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
99	C₂×#108	—	71	2	no	—
100	#428	—	142	2	no	—
101	C₂×#107	C₂×D₇₁	1	2	yes	—
102	#405	—	71	1	no	—
103	#368	—	71	4	no	—
104	#367	—	2	1	no	—
105	#367	—	2	1	no	—
106	#346	—	142	1	no	—
107	C₄×C₁₆	—	142	1	no	—
108	#343	—	142	2	no	—
109	C₂×M₁₆	—	71	1	no	—
110	C₂×M₁₆	—	142	2	no	—
111	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	#117	—	71	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	M₃₂	—	71	2	no	—
115	M₃₂	—	142	1	no	—
116	C₂×C₁₆	—	142	1	no	—
117	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
118	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
119	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
120	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
121	#108	—	71	4	no	—
122	#108	—	142	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	284	2	no	—
126	#106	—	284	2	no	—
127	#97	—	142	1	no	—
128	C₄×C₈	—	142	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
131	M₁₆	—	284	2	no	—
132	M₁₆	—	71	2	no	—
133	M₁₆	—	142	4	no	—
134	M₁₆	—	142	1	no	—
135	C₂×C₈	—	284	2	no	—
136	C₂×C₈	—	142	1	no	—
137	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
141	C₂×C₈	#11765864	1	1	yes	characteristic
142	C₂×C₈	#11765866	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	71	2	no	—
146	C₂²⋊C₄	—	142	1	no	—
147	C₄²	—	142	1	no	—
148	C₁₆	—	142	2	no	—
149	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
150	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
151	C₂³	#14556147	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	—	142	2	no	—
155	C₂×C₄	#14556090	1	1	yes	characteristic
156	C₂×C₄	#14556099	1	1	yes	characteristic
157	C₂×C₄	#14556127	1	1	yes	characteristic
158	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
159	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
160	C₂×C₄	#14556151	1	1	yes	characteristic
161	C₈	—	284	4	no	—
162	C₈	—	142	2	no	—
163	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
164	C₈	#14556093	1	1	yes	characteristic
165	C₈	#14556104	1	1	yes	characteristic
166	C₈	#14556119	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	#17986093	1	1	yes	characteristic
169	C₂²	#17986232	1	1	yes	characteristic
170	C₂²	#17986400	1	1	yes	characteristic
171	C₄	—	16	2	no	—
172	C₄	—	284	2	no	—
173	C₄	#17985895	1	1	yes	characteristic
174	C₄	#17986081	1	1	yes	characteristic
175	C₄	#17986237	1	1	yes	characteristic
176	C₄	#17986403	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#22589071	1	1	yes	characteristic
180	C₂	#22590467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal, characteristic

#22589936

C₂

yes

maximal, characteristic

#22589123

C₂

yes

maximal

#22586448

C₂

yes

maximal

C₂×#367×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

#17986039

—

#17985954

C₂²

yes

characteristic

#17985829

C₄

yes

complemented

#17985774

—

C₁₆×Dic₇₁

—

#17985768

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#107×C₇₁

C₂²

yes

—

#367×C₇₁

—

#367×C₇₁

—

C₂×#11765849

—

C₂×#11765849

D₄

yes

characteristic

#14556087

D₄

yes

characteristic

#14556068

—

#14556068

Q₈

yes

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556061

—

#14556059

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

Q₈

yes

—

#107×C₇₁

—

#107×C₇₁

—

—

—

—

Q₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

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

—

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

#135

yes

characteristic

C₂×C₄×C₇₁

—

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

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

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

Dic₇₁

—

#13172

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1942

yes

characteristic

C₂×C₇₁

#2223

yes

—

C₂×#368

—

C₂×#367

D₇₁

yes

—

—

—

C₇₁

#13172

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

100

#428

—

142

—

101

C₂×#107

C₂×D₇₁

yes

—

102

#405

—

103

#368

—

104

#367

—

105

#367

—

106

#346

—

142

—

107

C₄×C₁₆

—

142

—

108

#343

—

142

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

142

—

111

C₂²×C₈

C₂²×D₇₁

yes

characteristic

112

#117

—

113

C₂×C₄⋊C₄

—

114

M₃₂

—

115

M₃₂

—

142

—

116

C₂×C₁₆

—

142

—

117

C₂×C₁₆

Dic₁₄₂

yes

—

118

C₂×C₁₆

C₄×D₇₁

yes

—

119

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

120

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

121

#108

—

122

#108

—

142

—

123

#107

—

124

#107

—

125

#107

—

284

—

126

#106

—

284

—

127

#97

—

142

—

128

C₄×C₈

—

142

—

129

C₂²×C₄

—

130

C₂²×C₄

D₄×D₇₁

yes

characteristic

131

M₁₆

—

284

—

132

M₁₆

—

133

M₁₆

—

142

—

134

M₁₆

—

142

—

135

C₂×C₈

—

284

—

136

C₂×C₈

—

142

—

137

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

138

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

139

C₂×C₈

Q₈×D₇₁

yes

characteristic

140

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

141

C₂×C₈

#11765864

yes

characteristic

142

C₂×C₈

#11765866

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

142

—

147

C₄²

—

142

—

148

C₁₆

—

142

—

149

C₁₆

Dic₇₁⋊C₄

yes

—

150

C₁₆

Dic₇₁⋊C₄

yes

—

151

C₂³

#14556147

yes

characteristic

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

142

—

155

C₂×C₄

#14556090

yes

characteristic

156

C₂×C₄

#14556099

yes

characteristic

157

C₂×C₄

#14556127

yes

characteristic

158

C₂×C₄

D₈×D₇₁

yes

characteristic

159

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

160

C₂×C₄

#14556151

yes

characteristic

161

C₈

—

284

—

162

C₈

—

142

—

163

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

164

C₈

#14556093

yes

characteristic

165

C₈

#14556104

yes

characteristic

166

C₈

#14556119

yes

characteristic

167

C₂²

—

168

C₂²

#17986093

yes

characteristic

169

C₂²

#17986232

yes

characteristic

170

C₂²

#17986400

yes

characteristic

171

C₄

—

172

C₄

—

284

—

173

C₄

#17985895

yes

characteristic

174

C₄

#17986081

yes

characteristic

175

C₄

#17986237

yes

characteristic

176

C₄

#17986403

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#22589071

yes

characteristic

180

C₂

#22590467

yes

—

	subgroup	G/H
center	C₂²	#17986232
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#367×C₇₁	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#367	D₇₁
hypocenter	C₇₁	#13172
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂²

#17986232

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#367×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #17985829	C₄	semidirect
96. C₇₁	#13172	semidirect

H ≅ G/N

type

10. #17985829

C₄

semidirect

96. C₇₁

#13172

semidirect

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

subgroup	index
C₂×#17985829	2
#22589936	2
#22589123	2
#22586448	2
C₂×#367×C₇₁	2
#13172	71

subgroup

index

C₂×#367×C₇₁

Subgroups — #31915515

Conjugacy classes of subgroups

Complements

Maximal subgroups