Group #31805096 — subgroups

Subgroups — #31805096

Order 17792.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22581501	C₂	1	1	yes	maximal, characteristic
2	#22581009	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580701	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580701	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580697	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14555784	C₄	1	2	yes	—
9	C₂×C₈×D₁₃₉	—	2	1	no	—
10	#17984842	C₂²	1	1	yes	characteristic
11	#17984829	C₄	1	2	yes	—
12	#17984813	—	2	1	no	—
13	#17984813	—	2	1	no	—
14	D₄×Dic₁₃₉	—	2	1	no	—
15	D₄×Dic₁₃₉	—	2	1	no	—
16	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₁₃₉⋊C₈	—	2	1	no	—
20	(C₂×D₁₃₉)⋊C₈	—	2	1	no	—
21	#17984752	—	2	1	no	—
22	#127×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	#14555784	—	4	4	no	—
26	C₂×Dic₂₇₈	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
28	C₂²×Dic₁₃₉	—	2	2	no	—
29	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
32	C₂×C₁₃₉⋊C₈	—	2	1	no	—
33	C₂×C₁₃₉⋊C₈	—	2	1	no	—
34	C₈×D₁₃₉	—	4	2	no	—
35	C₄⋊Dic₁₃₉	—	2	1	no	—
36	C₄⋊Dic₁₃₉	—	2	1	no	—
37	C₂²⋊Dic₁₃₉	—	2	2	no	—
38	C₂²⋊Dic₁₃₉	—	2	2	no	—
39	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
41	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃₉	—	2	1	no	—
43	C₂×D₄×C₁₃₉	—	2	1	no	—
44	C₂×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₁₃₉	—	2	1	no	—
46	C₂×C₈×C₁₃₉	—	2	1	no	—
47	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
48	C₂²×D₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
49	Dic₂₇₈	—	4	2	no	—
50	C₂²⋊D₁₃₉	—	4	4	no	—
51	C₂×Dic₁₃₉	—	2	2	no	—
52	C₂×Dic₁₃₉	—	4	4	no	—
53	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×D₁₃₉	—	2	2	no	—
57	C₄×D₁₃₉	—	2	1	no	—
58	C₄×D₁₃₉	—	2	1	no	—
59	C₄×D₁₃₉	—	2	1	no	—
60	C₄×D₁₃₉	—	2	1	no	—
61	C₁₃₉⋊C₈	—	4	1	no	—
62	C₁₃₉⋊C₈	—	4	1	no	—
63	C₂³×C₁₃₉	—	2	2	no	—
64	D₄×C₁₃₉	—	4	2	no	—
65	D₄×C₁₃₉	—	4	1	no	—
66	D₄×C₁₃₉	—	4	1	no	—
67	D₄×C₁₃₉	—	4	1	no	—
68	D₄×C₁₃₉	—	4	1	no	—
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
72	C₈×C₁₃₉	—	4	1	no	—
73	C₈×C₁₃₉	—	4	1	no	—
74	C₂²×C₁₃₉	—	4	2	no	—
75	C₂²×C₁₃₉	—	4	2	no	—
76	C₂²×C₁₃₉	—	4	2	no	—
77	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×D₁₃₉	—	2	1	no	—
79	C₂×D₁₃₉	—	2	1	no	—
80	C₂×D₁₃₉	#100	1	2	yes	—
81	C₄×C₁₃₉	—	2	1	no	—
82	C₄×C₁₃₉	#135	1	1	yes	characteristic
83	C₄×C₁₃₉	#135	1	1	yes	characteristic
84	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₁₃₉	#137	1	1	yes	characteristic
86	Dic₁₃₉	—	2	1	no	—
87	Dic₁₃₉	—	2	1	no	—
88	Dic₁₃₉	—	2	1	no	—
89	Dic₁₃₉	—	4	2	no	—
90	C₂×C₁₃₉	—	8	2	no	—
91	C₂×C₁₃₉	C₂×#100	1	1	yes	characteristic
92	C₂×C₁₃₉	#416	1	1	yes	characteristic
93	C₂×C₁₃₉	#417	1	1	yes	characteristic
94	D₁₃₉	—	2	2	no	—
95	C₁₃₉	#1551	1	1	yes	characteristic, complemented, Sylow
96	#1551	—	139	1	no	maximal, Sylow
97	#532	—	139	1	no	—
98	#432	—	139	1	no	—
99	C₂×C₄⋊C₈	—	139	1	no	—
100	#333	—	139	1	no	—
101	#333	—	139	1	no	—
102	#331	—	139	1	no	—
103	#331	D₁₃₉	1	1	yes	characteristic, complemented
104	C₂×#41	—	139	2	no	—
105	C₂²×C₈	—	278	1	no	—
106	C₄⋊Q₈	—	139	1	no	—
107	#127	C₂×D₁₃₉	1	1	yes	characteristic
108	#124	—	139	2	no	—
109	#121	—	278	1	no	—
110	#121	—	278	1	no	—
111	C₄×D₄	—	278	1	no	—
112	C₄×D₄	—	278	1	no	—
113	C₂×C₄²	—	139	1	no	—
114	C₄⋊C₈	—	278	1	no	—
115	C₄⋊C₈	—	139	1	no	—
116	C₄⋊C₈	—	139	1	no	—
117	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
118	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
119	C₂²⋊C₈	—	278	1	no	—
120	#97	—	278	1	no	—
121	#41	—	556	4	no	—
122	C₂×Q₈	—	139	2	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	278	2	no	—
126	C₂×D₄	C₄×D₁₃₉	1	2	yes	—
127	C₂²×C₄	—	278	2	no	—
128	C₂²×C₄	—	139	1	no	—
129	C₂²×C₄	—	139	1	no	—
130	C₂²×C₄	—	139	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	278	1	no	—
134	C₂×C₈	—	278	1	no	—
135	C₂×C₈	—	556	2	no	—
136	C₄⋊C₄	—	278	1	no	—
137	C₄⋊C₄	—	278	1	no	—
138	C₂²⋊C₄	—	278	2	no	—
139	C₂²⋊C₄	—	278	2	no	—
140	C₄²	—	139	1	no	—
141	C₄²	—	139	1	no	—
142	C₄²	—	139	1	no	—
143	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
144	C₂³	—	2	2	no	—
145	C₂³	—	139	1	no	—
146	Q₈	—	556	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	556	4	no	—
153	C₂×C₄	—	278	2	no	—
154	C₂×C₄	—	278	1	no	—
155	C₂×C₄	—	278	1	no	—
156	C₂×C₄	—	278	1	no	—
157	C₂×C₄	—	278	1	no	—
158	C₂×C₄	—	278	2	no	—
159	C₂×C₄	—	556	4	no	—
160	C₂×C₄	—	139	1	no	—
161	C₂×C₄	—	139	1	no	—
162	C₂×C₄	—	139	1	no	—
163	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
164	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
165	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	556	1	no	—
169	C₈	—	556	1	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	278	1	no	—
174	C₂²	—	278	1	no	—
175	C₂²	—	139	2	no	—
176	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	278	1	no	—
179	C₄	—	278	1	no	—
180	C₄	—	278	1	no	—
181	C₄	—	556	2	no	—
182	C₄	#17984861	1	1	yes	characteristic
183	C₄	#17984863	1	1	yes	characteristic
184	C₄	#17984866	1	1	yes	characteristic
185	C₄	#17984871	1	1	yes	characteristic
186	C₂	—	8	2	no	—
187	C₂	—	278	2	no	—
188	C₂	#100×D₁₃₉	1	1	yes	characteristic
189	C₂	#22580946	1	1	yes	characteristic
190	C₂	#22580951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581501

C₂

yes

maximal, characteristic

#22581009

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₁₃₉

C₂

yes

maximal, characteristic, complemented

#22580701

C₂

yes

maximal, characteristic, complemented

#22580701

C₂

yes

maximal, characteristic, complemented

#22580697

C₂

yes

maximal, characteristic, complemented

#331×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555784

C₄

yes

—

C₂×C₈×D₁₃₉

—

#17984842

C₂²

yes

characteristic

#17984829

C₄

yes

—

#17984813

—

#17984813

—

D₄×Dic₁₃₉

—

D₄×Dic₁₃₉

—

C₄²×D₁₃₉

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

—

(C₂×D₁₃₉)⋊C₈

—

#17984752

—

#127×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

characteristic

#14555784

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₈×D₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂²⋊C₄

yes

characteristic

Dic₂₇₈

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#100

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#1551

yes

characteristic, complemented, Sylow

#1551

—

139

maximal, Sylow

#532

—

139

—

#432

—

139

—

C₂×C₄⋊C₈

—

139

—

100

#333

—

139

—

101

#333

—

139

—

102

#331

—

139

—

103

#331

D₁₃₉

yes

characteristic, complemented

104

C₂×#41

—

139

—

105

C₂²×C₈

—

278

—

106

C₄⋊Q₈

—

139

—

107

#127

C₂×D₁₃₉

yes

characteristic

108

#124

—

139

—

109

#121

—

278

—

110

#121

—

278

—

111

C₄×D₄

—

278

—

112

C₄×D₄

—

278

—

113

C₂×C₄²

—

139

—

114

C₄⋊C₈

—

278

—

115

C₄⋊C₈

—

139

—

116

C₄⋊C₈

—

139

—

117

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

118

C₄⋊C₈

C₂×D₁₃₉

yes

characteristic

119

C₂²⋊C₈

—

278

—

120

#97

—

278

—

121

#41

—

556

—

122

C₂×Q₈

—

139

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

278

—

126

C₂×D₄

C₄×D₁₃₉

yes

—

127

C₂²×C₄

—

278

—

128

C₂²×C₄

—

139

—

129

C₂²×C₄

—

139

—

130

C₂²×C₄

—

139

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

278

—

134

C₂×C₈

—

278

—

135

C₂×C₈

—

556

—

136

C₄⋊C₄

—

278

—

137

C₄⋊C₄

—

278

—

138

C₂²⋊C₄

—

278

—

139

C₂²⋊C₄

—

278

—

140

C₄²

—

139

—

141

C₄²

—

139

—

142

C₄²

—

139

—

143

C₄²

C₂²×D₁₃₉

yes

characteristic

144

C₂³

—

145

C₂³

—

139

—

146

Q₈

—

556

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

556

—

153

C₂×C₄

—

278

—

154

C₂×C₄

—

278

—

155

C₂×C₄

—

278

—

156

C₂×C₄

—

278

—

157

C₂×C₄

—

278

—

158

C₂×C₄

—

278

—

159

C₂×C₄

—

556

—

160

C₂×C₄

—

139

—

161

C₂×C₄

—

139

—

162

C₂×C₄

—

139

—

163

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

164

C₂×C₄

D₄×D₁₃₉

yes

characteristic

165

C₂×C₄

D₄×D₁₃₉

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

—

556

—

169

C₈

—

556

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

278

—

174

C₂²

—

278

—

175

C₂²

—

139

—

176

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

177

C₄

—

178

C₄

—

278

—

179

C₄

—

278

—

180

C₄

—

278

—

181

C₄

—

556

—

182

C₄

#17984861

yes

characteristic

183

C₄

#17984863

yes

characteristic

184

C₄

#17984866

yes

characteristic

185

C₄

#17984871

yes

characteristic

186

C₂

—

187

C₂

—

278

—

188

C₂

#100×D₁₃₉

yes

characteristic

189

C₂

#22580946

yes

characteristic

190

C₂

#22580951

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#331×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	#22581501	C₂
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#331	D₁₃₉
hypocenter	C₁₃₉	#1551
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#331×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

#22581501

C₂

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #22581009	C₂	semidirect
3. C₄⋊C₈×D₁₃₉	C₂	semidirect
4. #22580701	C₂	semidirect
5. #22580701	C₂	semidirect
6. #22580697	C₂	semidirect
7. #331×C₁₃₉	C₂	semidirect
95. C₁₃₉	#1551	semidirect
103. #331	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581009

C₂

semidirect

3. C₄⋊C₈×D₁₃₉

C₂

semidirect

4. #22580701

C₂

semidirect

5. #22580701

C₂

semidirect

6. #22580697

C₂

semidirect

7. #331×C₁₃₉

C₂

semidirect

95. C₁₃₉

#1551

semidirect

103. #331

D₁₃₉

semidirect

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

subgroup	index
#22581501	2
#22581009	2
C₄⋊C₈×D₁₃₉	2
#22580701	2
#22580697	2
#331×C₁₃₉	2
#1551	139

subgroup

index

C₄⋊C₈×D₁₃₉

#331×C₁₃₉

139

Subgroups — #31805096

Conjugacy classes of subgroups

Complements

Maximal subgroups