Group #31816858 — subgroups

Subgroups — #31816858

Order 17792.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#17984871	C₂	1	4	yes	maximal, complemented
2	#22581501	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581344	C₂	1	4	yes	maximal, complemented
5	#22581333	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581332	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581325	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14555768	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555784	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₃₉	C₂²	1	2	yes	—
12	#17984871	—	2	16	no	—
13	C₂×Dic₅₅₆	C₂²	1	4	yes	—
14	C₂×#14555775	C₂²	1	4	yes	—
15	C₂×#14555773	C₂²	1	4	yes	—
16	C₂×#14555768	C₂²	1	2	yes	—
17	#17984842	C₂²	1	1	yes	characteristic
18	#17984842	C₂²	1	1	yes	characteristic
19	#17984841	—	2	2	no	—
20	#17984840	C₂²	1	2	yes	—
21	#17984829	C₂²	1	2	yes	—
22	#17984813	—	2	2	no	—
23	D₄×Dic₁₃₉	—	2	2	no	—
24	C₄⋊C₄×D₁₃₉	—	2	2	no	—
25	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
26	#17984752	C₂²	1	2	yes	—
27	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₃₉	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
30	#127×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
32	#14555784	—	2	8	no	—
33	C₂×Dic₂₇₈	—	2	2	no	—
34	C₂×Dic₂₇₈	C₂³	1	2	yes	—
35	C₂×Dic₂₇₈	C₂³	1	2	yes	—
36	Q₈×D₁₃₉	—	2	8	no	—
37	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
38	C₂²×Dic₁₃₉	—	2	2	no	—
39	C₂×C₄×D₁₃₉	—	2	2	no	—
40	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
41	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
42	Dic₅₅₆	—	2	8	no	—
43	#14555775	—	2	8	no	—
44	#14555773	—	2	8	no	—
45	#14555768	D₄	1	8	yes	complemented
46	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
47	C₄⋊Dic₁₃₉	—	2	2	no	—
48	C₄⋊Dic₁₃₉	—	2	2	no	—
49	Dic₁₃₉⋊C₄	—	2	4	no	—
50	C₂²⋊Dic₁₃₉	—	2	4	no	—
51	C₄×Dic₁₃₉	D₄	1	2	yes	—
52	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
54	C₂×D₄×C₁₃₉	—	2	2	no	—
55	C₂×D₄×C₁₃₉	C₂³	1	2	yes	—
56	SD₁₆×C₁₃₉	—	2	8	no	—
57	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₃₉	—	2	2	no	—
59	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
61	Dic₂₇₈	—	2	4	no	—
62	Dic₂₇₈	—	2	4	no	—
63	Dic₂₇₈	—	4	4	no	—
64	C₂²⋊D₁₃₉	—	4	4	no	—
65	C₂×Dic₁₃₉	—	2	2	no	—
66	C₂×Dic₁₃₉	—	2	2	no	—
67	C₂×Dic₁₃₉	—	4	4	no	—
68	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₃₉	—	2	4	no	—
71	C₄×D₁₃₉	—	4	4	no	—
72	C₄×D₁₃₉	C₂×D₄	1	4	yes	—
73	C₁₃₉⋊C₈	C₂×D₄	1	4	yes	—
74	C₂³×C₁₃₉	—	2	2	no	—
75	Q₈×C₁₃₉	—	2	4	no	—
76	D₄×C₁₃₉	—	2	4	no	—
77	D₄×C₁₃₉	—	4	4	no	—
78	C₂×C₄×C₁₃₉	—	2	2	no	—
79	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₃₉	C₂×D₄	1	4	yes	—
82	C₂²×C₁₃₉	—	4	2	no	—
83	C₂²×C₁₃₉	—	4	4	no	—
84	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₃₉	—	2	2	no	—
86	C₂×D₁₃₉	#127	1	2	yes	—
87	C₄×C₁₃₉	—	4	2	no	—
88	C₄×C₁₃₉	#137	1	4	yes	complemented
89	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
90	Dic₁₃₉	—	2	2	no	—
91	Dic₁₃₉	—	4	2	no	—
92	Dic₁₃₉	—	4	2	no	—
93	Dic₁₃₉	#127	1	2	yes	—
94	C₂×C₁₃₉	—	8	2	no	—
95	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₃₉	C₂×#137	1	2	yes	—
97	D₁₃₉	—	2	2	no	—
98	C₁₃₉	#3152	1	1	yes	characteristic, complemented, Sylow
99	#3152	—	139	1	no	maximal, Sylow
100	C₂×#137	—	139	4	no	—
101	#532	—	139	1	no	—
102	C₂×C₄⋊Q₈	—	139	1	no	—
103	#497	—	139	4	no	—
104	#494	—	139	1	no	—
105	#494	—	139	1	no	—
106	#492	—	139	1	no	—
107	#492	D₁₃₉	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	139	1	no	—
109	C₂×#41	—	139	2	no	—
110	C₂²×Q₈	—	139	2	no	—
111	#137	—	278	16	no	—
112	C₂×Q₁₆	—	139	4	no	—
113	C₂×Q₁₆	—	139	4	no	—
114	C₂×SD₁₆	—	139	4	no	—
115	C₂×SD₁₆	C₂×D₁₃₉	1	4	yes	—
116	C₂×M₁₆	—	139	2	no	—
117	C₄⋊Q₈	—	278	2	no	—
118	C₄⋊Q₈	—	139	2	no	—
119	C₄⋊Q₈	—	139	1	no	—
120	C₄⋊Q₈	—	139	1	no	—
121	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
122	#127	C₂×D₁₃₉	1	1	yes	characteristic
123	#124	—	139	2	no	—
124	#121	—	278	2	no	—
125	C₄×D₄	—	278	2	no	—
126	C₂×C₄⋊C₄	—	278	2	no	—
127	C₂×C₄²	—	139	1	no	—
128	#97	—	139	2	no	—
129	C₄×C₈	—	139	1	no	—
130	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
131	#41	—	278	8	no	—
132	C₂×Q₈	—	278	2	no	—
133	C₂×Q₈	—	278	8	no	—
134	C₂×Q₈	—	139	2	no	—
135	C₂×Q₈	—	139	2	no	—
136	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	278	2	no	—
139	C₂×D₄	C₂²×D₁₃₉	1	2	yes	—
140	C₂²×C₄	—	278	2	no	—
141	C₂²×C₄	—	278	2	no	—
142	C₂²×C₄	—	139	2	no	—
143	C₂²×C₄	—	139	1	no	—
144	Q₁₆	—	278	8	no	—
145	Q₁₆	—	278	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	278	8	no	—
148	M₁₆	—	139	8	no	—
149	C₂×C₈	—	139	2	no	—
150	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	278	4	no	—
153	C₄⋊C₄	—	278	2	no	—
154	C₄⋊C₄	—	278	2	no	—
155	C₂²⋊C₄	—	278	4	no	—
156	C₄²	—	139	2	no	—
157	C₄²	—	139	1	no	—
158	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	139	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	278	4	no	—
163	Q₈	—	278	4	no	—
164	Q₈	—	556	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	556	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	278	4	no	—
170	C₂×C₄	—	278	2	no	—
171	C₂×C₄	—	278	2	no	—
172	C₂×C₄	—	556	4	no	—
173	C₂×C₄	—	556	4	no	—
174	C₂×C₄	—	139	2	no	—
175	C₂×C₄	—	139	1	no	—
176	C₂×C₄	—	139	4	no	—
177	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
178	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
179	C₈	—	139	4	no	—
180	C₈	D₄×D₁₃₉	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	278	2	no	—
184	C₂²	—	139	2	no	—
185	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	278	2	no	—
188	C₄	—	556	2	no	—
189	C₄	—	556	2	no	—
190	C₄	—	139	2	no	—
191	C₄	#17984871	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	278	2	no	—
195	C₂	#127×D₁₃₉	1	1	yes	characteristic
196	C₂	C₂×#17984871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984871

C₂

yes

maximal, complemented

#22581501

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₃₉

C₂

yes

maximal, characteristic, complemented

#22581344

C₂

yes

maximal, complemented

#22581333

C₂

yes

maximal, characteristic, complemented

#22581332

C₂

yes

maximal, characteristic, complemented

#22581325

C₂

yes

maximal, characteristic, complemented

C₄×#14555768

C₂

yes

maximal, characteristic, complemented

#492×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555784

C₂²

yes

—

C₂×Q₈×D₁₃₉

C₂²

yes

—

#17984871

—

C₂×Dic₅₅₆

C₂²

yes

—

C₂×#14555775

C₂²

yes

—

C₂×#14555773

C₂²

yes

—

C₂×#14555768

C₂²

yes

—

#17984842

C₂²

yes

characteristic

#17984842

C₂²

yes

characteristic

#17984841

—

#17984840

C₂²

yes

—

#17984829

C₂²

yes

—

#17984813

—

D₄×Dic₁₃₉

—

C₄⋊C₄×D₁₃₉

—

C₄²×D₁₃₉

C₂²

yes

characteristic

#17984752

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

#14555784

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×Dic₂₇₈

C₂³

yes

—

Q₈×D₁₃₉

—

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

characteristic

Dic₅₅₆

—

#14555775

—

#14555773

—

#14555768

D₄

yes

complemented

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

characteristic

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂³

yes

—

SD₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂×D₄

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂×Dic₁₃₉

C₂×D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

C₂×D₄

yes

—

C₁₃₉⋊C₈

C₂×D₄

yes

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

#127

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#137

yes

complemented

C₄×C₁₃₉

C₂²×D₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

#127

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#127

yes

characteristic

C₂×C₁₃₉

C₂×#137

yes

—

D₁₃₉

—

C₁₃₉

#3152

yes

characteristic, complemented, Sylow

#3152

—

139

maximal, Sylow

100

C₂×#137

—

139

—

101

#532

—

139

—

102

C₂×C₄⋊Q₈

—

139

—

103

#497

—

139

—

104

#494

—

139

—

105

#494

—

139

—

106

#492

—

139

—

107

#492

D₁₃₉

yes

characteristic, complemented

108

C₄×M₁₆

—

139

—

109

C₂×#41

—

139

—

110

C₂²×Q₈

—

139

—

111

#137

—

278

—

112

C₂×Q₁₆

—

139

—

113

C₂×Q₁₆

—

139

—

114

C₂×SD₁₆

—

139

—

115

C₂×SD₁₆

C₂×D₁₃₉

yes

—

116

C₂×M₁₆

—

139

—

117

C₄⋊Q₈

—

278

—

118

C₄⋊Q₈

—

139

—

119

C₄⋊Q₈

—

139

—

120

C₄⋊Q₈

—

139

—

121

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

122

#127

C₂×D₁₃₉

yes

characteristic

123

#124

—

139

—

124

#121

—

278

—

125

C₄×D₄

—

278

—

126

C₂×C₄⋊C₄

—

278

—

127

C₂×C₄²

—

139

—

128

#97

—

139

—

129

C₄×C₈

—

139

—

130

C₄×C₈

C₂×D₁₃₉

yes

characteristic

131

#41

—

278

—

132

C₂×Q₈

—

278

—

133

C₂×Q₈

—

278

—

134

C₂×Q₈

—

139

—

135

C₂×Q₈

—

139

—

136

C₂×Q₈

C₂²×D₁₃₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

278

—

139

C₂×D₄

C₂²×D₁₃₉

yes

—

140

C₂²×C₄

—

278

—

141

C₂²×C₄

—

278

—

142

C₂²×C₄

—

139

—

143

C₂²×C₄

—

139

—

144

Q₁₆

—

278

—

145

Q₁₆

—

278

—

146

SD₁₆

—

147

SD₁₆

—

278

—

148

M₁₆

—

139

—

149

C₂×C₈

—

139

—

150

C₂×C₈

C₂²×D₁₃₉

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

278

—

153

C₄⋊C₄

—

278

—

154

C₄⋊C₄

—

278

—

155

C₂²⋊C₄

—

278

—

156

C₄²

—

139

—

157

C₄²

—

139

—

158

C₄²

C₂²×D₁₃₉

yes

characteristic

159

C₂³

—

160

C₂³

—

139

—

161

Q₈

—

162

Q₈

—

278

—

163

Q₈

—

278

—

164

Q₈

—

556

—

165

D₄

—

166

D₄

—

167

D₄

—

556

—

168

C₂×C₄

—

169

C₂×C₄

—

278

—

170

C₂×C₄

—

278

—

171

C₂×C₄

—

278

—

172

C₂×C₄

—

556

—

173

C₂×C₄

—

556

—

174

C₂×C₄

—

139

—

175

C₂×C₄

—

139

—

176

C₂×C₄

—

139

—

177

C₂×C₄

D₄×D₁₃₉

yes

—

178

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

179

C₈

—

139

—

180

C₈

D₄×D₁₃₉

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

278

—

184

C₂²

—

139

—

185

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

186

C₄

—

187

C₄

—

278

—

188

C₄

—

556

—

189

C₄

—

556

—

190

C₄

—

139

—

191

C₄

#17984871

yes

complemented

192

C₄

C₂×D₄×D₁₃₉

yes

—

193

C₂

—

194

C₂

—

278

—

195

C₂

#127×D₁₃₉

yes

characteristic

196

C₂

C₂×#17984871

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#492×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581501	C₂
hypercenter	#492	D₁₃₉
hypocenter	C₁₃₉	#3152
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#492×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#17984871	C₂	semidirect
3. C₄⋊Q₈×D₁₃₉	C₂	semidirect
4. #22581344	C₂	semidirect
5. #22581333	C₂	semidirect
6. #22581332	C₂	semidirect
7. #22581325	C₂	semidirect
8. C₄×#14555768	C₂	semidirect
9. #492×C₁₃₉	C₂	semidirect
45. #14555768	D₄	semidirect
88. C₄×C₁₃₉	#137	semidirect
98. C₁₃₉	#3152	semidirect
107. #492	D₁₃₉	semidirect
191. C₄	#17984871	semidirect

H ≅ G/N

type

1. C₂×#17984871

C₂

semidirect

3. C₄⋊Q₈×D₁₃₉

C₂

semidirect

4. #22581344

C₂

semidirect

5. #22581333

C₂

semidirect

6. #22581332

C₂

semidirect

7. #22581325

C₂

semidirect

8. C₄×#14555768

C₂

semidirect

9. #492×C₁₃₉

C₂

semidirect

45. #14555768

D₄

semidirect

88. C₄×C₁₃₉

#137

semidirect

98. C₁₃₉

#3152

semidirect

107. #492

D₁₃₉

semidirect

191. C₄

#17984871

semidirect

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

subgroup	index
C₂×#17984871	2
#22581501	2
C₄⋊Q₈×D₁₃₉	2
#22581344	2
#22581333	2
#22581332	2
#22581325	2
C₄×#14555768	2
#492×C₁₃₉	2
#3152	139

subgroup

index

C₄⋊Q₈×D₁₃₉

#492×C₁₃₉

139

Subgroups — #31816858

Conjugacy classes of subgroups

Complements

Maximal subgroups