Group #31816899 — subgroups

Subgroups — #31816899

Order 17792.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984872	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581524	C₂	1	2	yes	maximal, complemented
4	#22581344	C₂	1	4	yes	maximal, complemented
5	#22581336	C₂	1	2	yes	maximal, complemented
6	#22581331	C₂	1	2	yes	maximal, complemented
7	#22580908	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
10	#17984872	—	2	8	no	—
11	C₂×Dic₅₅₆	—	2	4	no	—
12	C₂×Dic₅₅₆	C₂²	1	2	yes	—
13	C₂×Dic₅₅₆	C₂²	1	2	yes	—
14	C₂×#14555775	C₂²	1	4	yes	—
15	C₂×#14555773	C₂²	1	4	yes	—
16	C₂×#14555769	C₂²	1	1	yes	characteristic
17	#17984840	C₂²	1	4	yes	—
18	#17984829	C₂²	1	4	yes	—
19	#17984823	—	2	4	no	—
20	#17984821	—	2	4	no	—
21	#17984801	C₂²	1	1	yes	characteristic
22	#17984752	C₂²	1	2	yes	—
23	C₈×Dic₁₃₉	C₂²	1	2	yes	—
24	C₂×#41×C₁₃₉	C₂²	1	2	yes	—
25	#135×C₁₃₉	—	2	4	no	—
26	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆×C₁₃₉	C₂²	1	2	yes	—
28	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×Dic₂₇₈	—	2	2	no	—
31	C₂×Dic₂₇₈	—	2	4	no	—
32	C₂×Dic₂₇₈	—	2	4	no	—
33	C₂×Dic₂₇₈	C₂³	1	4	yes	—
34	C₂²×Dic₁₃₉	—	2	2	no	—
35	Dic₅₅₆	—	2	4	no	—
36	Dic₅₅₆	—	2	4	no	—
37	#14555775	—	2	8	no	—
38	#14555773	—	2	8	no	—
39	#14555769	D₄	1	4	yes	—
40	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₁₃₉	D₄	1	2	yes	—
42	Dic₁₃₉⋊C₄	—	2	8	no	—
43	C₂²⋊Dic₁₃₉	—	2	8	no	—
44	C₂²⋊Dic₁₃₉	D₄	1	2	yes	—
45	C₄×Dic₁₃₉	C₂³	1	2	yes	—
46	#41×C₁₃₉	—	2	4	no	—
47	#41×C₁₃₉	—	2	4	no	—
48	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
49	C₂×D₄×C₁₃₉	—	2	2	no	—
50	C₂×D₄×C₁₃₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃₉	—	2	2	no	—
52	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
53	Q₁₆×C₁₃₉	—	2	2	no	—
54	SD₁₆×C₁₃₉	—	2	4	no	—
55	D₈×C₁₃₉	—	2	2	no	—
56	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
57	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
59	Dic₂₇₈	—	2	4	no	—
60	Dic₂₇₈	—	2	4	no	—
61	Dic₂₇₈	—	4	4	no	—
62	C₂×Dic₁₃₉	—	2	4	no	—
63	C₂×Dic₁₃₉	—	4	4	no	—
64	C₂×Dic₁₃₉	C₂×D₄	1	4	yes	—
65	C₁₃₉⋊C₈	C₂×D₄	1	4	yes	—
66	C₂³×C₁₃₉	—	2	2	no	—
67	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₁₃₉	—	2	2	no	—
69	Q₈×C₁₃₉	—	2	2	no	—
70	D₄×C₁₃₉	—	2	2	no	—
71	D₄×C₁₃₉	—	2	2	no	—
72	D₄×C₁₃₉	—	4	4	no	—
73	C₂×C₄×C₁₃₉	—	2	2	no	—
74	C₂×C₄×C₁₃₉	—	4	4	no	—
75	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
81	C₂²×C₁₃₉	—	2	2	no	—
82	C₂²×C₁₃₉	—	4	2	no	—
83	C₂²×C₁₃₉	—	4	4	no	—
84	C₂²×C₁₃₉	#127	1	2	yes	—
85	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₁₃₉	—	4	2	no	—
87	C₄×C₁₃₉	#127	1	2	yes	—
88	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
90	Dic₁₃₉	—	2	4	no	—
91	Dic₁₃₉	—	4	4	no	—
92	C₂×C₁₃₉	—	2	2	no	—
93	C₂×C₁₃₉	—	8	2	no	—
94	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
95	C₂×C₁₃₉	#578	1	2	yes	—
96	C₁₃₉	#3156	1	1	yes	characteristic, complemented, Sylow
97	#3156	—	139	1	no	maximal, Sylow
98	C₂×#137	—	139	2	no	—
99	C₂×#135	D₁₃₉	1	1	yes	characteristic
100	C₂²×Q₁₆	—	139	1	no	—
101	#536	—	139	2	no	—
102	#497	—	139	4	no	—
103	#495	—	139	2	no	—
104	#494	—	139	2	no	—
105	#405	—	139	1	no	—
106	C₂×#41	C₂×D₁₃₉	1	2	yes	—
107	C₂²×Q₈	—	139	2	no	—
108	#137	—	278	8	no	—
109	#135	—	2	4	no	—
110	C₂×Q₁₆	—	278	4	no	—
111	C₂×Q₁₆	—	139	4	no	—
112	C₂×Q₁₆	—	139	2	no	—
113	C₂×Q₁₆	—	139	2	no	—
114	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
115	C₂×SD₁₆	—	139	4	no	—
116	C₂×SD₁₆	C₂×D₁₃₉	1	2	yes	—
117	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
118	C₂×M₁₆	—	139	1	no	—
119	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	139	4	no	—
121	#124	—	139	4	no	—
122	#123	—	278	4	no	—
123	C₂²⋊Q₈	—	278	4	no	—
124	#117	—	139	1	no	—
125	#97	—	139	2	no	—
126	C₄×C₈	—	139	2	no	—
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	278	2	no	—
130	C₂×Q₈	—	278	4	no	—
131	C₂×Q₈	—	278	4	no	—
132	C₂×Q₈	—	139	4	no	—
133	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂²×D₁₃₉	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	278	2	no	—
138	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	Q₁₆	—	278	8	no	—
141	Q₁₆	—	278	4	no	—
142	Q₁₆	—	278	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	278	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	—	139	4	no	—
147	C₂×C₈	—	139	2	no	—
148	C₂×C₈	C₂²⋊D₁₃₉	1	4	yes	—
149	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
151	C₄⋊C₄	—	278	8	no	—
152	C₄⋊C₄	—	139	2	no	—
153	C₂²⋊C₄	—	278	8	no	—
154	C₂²⋊C₄	—	139	2	no	—
155	C₄²	—	139	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	278	4	no	—
161	Q₈	—	278	4	no	—
162	Q₈	—	556	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	278	4	no	—
169	C₂×C₄	—	556	4	no	—
170	C₂×C₄	—	139	4	no	—
171	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
172	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
174	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
175	C₈	—	139	4	no	—
176	C₈	C₂×C₂²⋊D₁₃₉	1	2	yes	—
177	C₈	C₂×C₂²⋊D₁₃₉	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	2	no	—
181	C₂²	#17984838	1	2	yes	—
182	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	—	278	4	no	—
185	C₄	—	556	4	no	—
186	C₄	#17984838	1	2	yes	—
187	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
188	C₄	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×#17984838	1	1	yes	characteristic
192	C₂	#22581767	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984872

C₂

yes

maximal, complemented

C₂²×Dic₅₅₆

C₂

yes

maximal, characteristic, complemented

#22581524

C₂

yes

maximal, complemented

#22581344

C₂

yes

maximal, complemented

#22581336

C₂

yes

maximal, complemented

#22581331

C₂

yes

maximal, complemented

#22580908

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

#17984872

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

C₂²

yes

—

C₂×Dic₅₅₆

C₂²

yes

—

C₂×#14555775

C₂²

yes

—

C₂×#14555773

C₂²

yes

—

C₂×#14555769

C₂²

yes

characteristic

#17984840

C₂²

yes

—

#17984829

C₂²

yes

—

#17984823

—

#17984821

—

#17984801

C₂²

yes

characteristic

#17984752

C₂²

yes

—

C₈×Dic₁₃₉

C₂²

yes

—

C₂×#41×C₁₃₉

C₂²

yes

—

#135×C₁₃₉

—

C₂×Q₁₆×C₁₃₉

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₉

C₂²

yes

—

C₂×D₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂²×Dic₁₃₉

—

Dic₅₅₆

—

Dic₅₅₆

—

#14555775

—

#14555773

—

#14555769

D₄

yes

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

#41×C₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

Q₁₆×C₁₃₉

—

SD₁₆×C₁₃₉

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₁₃₉⋊C₈

C₂×D₄

yes

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

#127

yes

—

C₄×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

C₂²×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#127

yes

characteristic

C₂×C₁₃₉

#578

yes

—

C₁₃₉

#3156

yes

characteristic, complemented, Sylow

#3156

—

139

maximal, Sylow

C₂×#137

—

139

—

C₂×#135

D₁₃₉

yes

characteristic

100

C₂²×Q₁₆

—

139

—

101

#536

—

139

—

102

#497

—

139

—

103

#495

—

139

—

104

#494

—

139

—

105

#405

—

139

—

106

C₂×#41

C₂×D₁₃₉

yes

—

107

C₂²×Q₈

—

139

—

108

#137

—

278

—

109

#135

—

110

C₂×Q₁₆

—

278

—

111

C₂×Q₁₆

—

139

—

112

C₂×Q₁₆

—

139

—

113

C₂×Q₁₆

—

139

—

114

C₂×Q₁₆

C₂×D₁₃₉

yes

characteristic

115

C₂×SD₁₆

—

139

—

116

C₂×SD₁₆

C₂×D₁₃₉

yes

—

117

C₂×D₈

C₂×D₁₃₉

yes

characteristic

118

C₂×M₁₆

—

139

—

119

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

120

C₄⋊Q₈

—

139

—

121

#124

—

139

—

122

#123

—

278

—

123

C₂²⋊Q₈

—

278

—

124

#117

—

139

—

125

#97

—

139

—

126

C₄×C₈

—

139

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

278

—

130

C₂×Q₈

—

278

—

131

C₂×Q₈

—

278

—

132

C₂×Q₈

—

139

—

133

C₂×Q₈

C₂²×D₁₃₉

yes

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₁₃₉

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

278

—

138

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

278

—

141

Q₁₆

—

278

—

142

Q₁₆

—

278

—

143

SD₁₆

—

144

SD₁₆

—

278

—

145

D₈

—

146

M₁₆

—

139

—

147

C₂×C₈

—

139

—

148

C₂×C₈

C₂²⋊D₁₃₉

yes

—

149

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

150

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

151

C₄⋊C₄

—

278

—

152

C₄⋊C₄

—

139

—

153

C₂²⋊C₄

—

278

—

154

C₂²⋊C₄

—

139

—

155

C₄²

—

139

—

156

C₂³

—

157

C₂³

D₄×D₁₃₉

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

278

—

161

Q₈

—

278

—

162

Q₈

—

556

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

278

—

169

C₂×C₄

—

556

—

170

C₂×C₄

—

139

—

171

C₂×C₄

D₄×D₁₃₉

yes

characteristic

172

C₂×C₄

D₄×D₁₃₉

yes

—

173

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

174

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

175

C₈

—

139

—

176

C₈

C₂×C₂²⋊D₁₃₉

yes

—

177

C₈

C₂×C₂²⋊D₁₃₉

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#17984838

yes

—

182

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

183

C₄

—

184

C₄

—

278

—

185

C₄

—

556

—

186

C₄

#17984838

yes

—

187

C₄

C₂×D₄×D₁₃₉

yes

characteristic

188

C₄

C₂²×C₂²⋊D₁₃₉

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#17984838

yes

characteristic

192

C₂

#22581767

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting, prime radical	C₂×#135×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	C₂×#135	D₁₃₉
hypocenter	C₁₃₉	#3156
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting, prime radical

C₂×#135×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#17984872	C₂	semidirect
2. C₂²×Dic₅₅₆	C₂	semidirect
3. #22581524	C₂	semidirect
4. #22581344	C₂	semidirect
5. #22581336	C₂	semidirect
6. #22581331	C₂	semidirect
7. #22580908	C₂	semidirect
96. C₁₃₉	#3156	semidirect

H ≅ G/N

type

1. C₂×#17984872

C₂

semidirect

2. C₂²×Dic₅₅₆

C₂

semidirect

3. #22581524

C₂

semidirect

4. #22581344

C₂

semidirect

5. #22581336

C₂

semidirect

6. #22581331

C₂

semidirect

7. #22580908

C₂

semidirect

96. C₁₃₉

#3156

semidirect

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

subgroup	index
C₂×#17984872	2
C₂²×Dic₅₅₆	2
#22581524	2
#22581344	2
#22581336	2
#22581331	2
#22580908	2
C₂×#135×C₁₃₉	2
#3156	139

subgroup

index

C₂²×Dic₅₅₆

C₂×#135×C₁₃₉

139

Subgroups — #31816899

Conjugacy classes of subgroups

Complements

Maximal subgroups