Group #31817851 — subgroups

Subgroups — #31817851

Order 17792.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22581565	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581558	C₂	1	1	yes	maximal, characteristic
3	#22581261	C₂	1	2	yes	maximal, complemented
4	#22581254	C₂	1	2	yes	maximal, complemented
5	#22581199	C₂	1	2	yes	maximal, complemented
6	#22581182	C₂	1	2	yes	maximal, complemented
7	#22581159	C₂	1	2	yes	maximal, complemented
8	#22581154	C₂	1	2	yes	maximal, complemented
9	#424×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555774	C₂²	1	2	yes	—
11	C₂×#14555774	C₂²	1	2	yes	—
12	#17984842	C₂²	1	1	yes	characteristic
13	#17984839	C₂²	1	1	yes	characteristic
14	#17984835	—	2	2	no	—
15	#17984834	C₂²	1	1	yes	characteristic
16	#17984834	C₂²	1	1	yes	characteristic
17	#17984832	—	2	2	no	—
18	#17984827	—	2	2	no	—
19	#17984821	C₂²	1	2	yes	—
20	#17984816	C₂²	1	2	yes	—
21	#17984815	—	2	2	no	—
22	C₄×Dic₂₇₈	C₂²	1	2	yes	—
23	C₄×D₅₅₆	C₂²	1	2	yes	—
24	#17984776	C₂²	1	2	yes	—
25	#17984774	C₂²	1	2	yes	—
26	#17984767	C₂²	1	2	yes	—
27	#17984767	C₂²	1	2	yes	—
28	#17984764	C₂²	1	2	yes	—
29	#17984764	C₂²	1	2	yes	—
30	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
32	#117×C₁₃₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
35	C₂×Dic₂₇₈	C₂³	1	2	yes	—
36	C₂×D₅₅₆	—	2	2	no	—
37	C₂×D₅₅₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
39	C₂×C₄×D₁₃₉	—	2	2	no	—
40	#14555774	—	4	2	no	—
41	#14555774	—	4	2	no	—
42	C₄⋊Dic₁₃₉	—	2	2	no	—
43	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
44	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
45	Dic₁₃₉⋊C₄	—	2	2	no	—
46	Dic₁₃₉⋊C₄	—	2	2	no	—
47	Dic₁₃₉⋊C₄	—	2	2	no	—
48	Dic₁₃₉⋊C₄	—	2	2	no	—
49	C₂²⋊Dic₁₃₉	—	2	2	no	—
50	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
51	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
52	C₄×Dic₁₃₉	—	2	2	no	—
53	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
54	M₁₆×C₁₃₉	—	4	1	no	—
55	C₂×C₈×C₁₃₉	—	4	1	no	—
56	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
57	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
59	C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
60	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
61	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
62	C₂²×D₁₃₉	—	2	2	no	—
63	Dic₂₇₈	—	2	2	no	—
64	Dic₂₇₈	—	2	2	no	—
65	D₅₅₆	—	2	2	no	—
66	D₅₅₆	—	2	2	no	—
67	D₅₅₆	—	8	2	no	—
68	C₂²⋊D₁₃₉	—	8	2	no	—
69	C₂×Dic₁₃₉	—	2	2	no	—
70	C₂×Dic₁₃₉	—	2	2	no	—
71	C₂×Dic₁₃₉	—	2	2	no	—
72	C₄×D₁₃₉	—	8	2	no	—
73	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	—	2	1	no	—
75	C₂×C₄×C₁₃₉	—	2	1	no	—
76	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
80	C₈×C₁₃₉	—	2	2	no	—
81	C₈×C₁₃₉	—	2	2	no	—
82	C₂²×C₁₃₉	—	2	1	no	—
83	C₂²×C₁₃₉	—	2	2	no	—
84	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₃₉	—	4	2	no	—
86	C₂×D₁₃₉	—	4	2	no	—
87	C₂×D₁₃₉	—	4	2	no	—
88	C₄×C₁₃₉	—	2	1	no	—
89	C₄×C₁₃₉	—	2	2	no	—
90	C₄×C₁₃₉	—	2	2	no	—
91	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
93	Dic₁₃₉	—	4	2	no	—
94	Dic₁₃₉	—	4	2	no	—
95	Dic₁₃₉	—	4	2	no	—
96	C₂×C₁₃₉	—	4	1	no	—
97	C₂×C₁₃₉	#537	1	1	yes	characteristic
98	C₂×C₁₃₉	#577	1	2	yes	—
99	D₁₃₉	—	8	2	no	—
100	C₁₃₉	#3260	1	1	yes	characteristic, complemented, Sylow
101	#3260	—	139	1	no	maximal, Sylow
102	#541	—	139	1	no	—
103	#540	—	139	1	no	—
104	#479	—	139	2	no	—
105	#478	—	139	2	no	—
106	#468	—	139	2	no	—
107	#465	—	139	2	no	—
108	#461	—	139	2	no	—
109	#460	—	139	2	no	—
110	#424	D₁₃₉	1	1	yes	characteristic, complemented
111	C₂×SD₁₆	—	139	2	no	—
112	C₂×SD₁₆	—	139	2	no	—
113	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
115	C₄⋊Q₈	—	139	1	no	—
116	#127	—	139	1	no	—
117	#126	—	278	2	no	—
118	#125	—	278	2	no	—
119	#125	—	139	1	no	—
120	#125	—	139	1	no	—
121	#123	—	278	2	no	—
122	C₂²⋊Q₈	—	139	2	no	—
123	#121	—	278	2	no	—
124	#121	—	139	2	no	—
125	C₄×Q₈	—	139	2	no	—
126	C₄×D₄	—	139	2	no	—
127	#117	C₂×D₁₃₉	1	1	yes	characteristic
128	#107	—	139	2	no	—
129	#106	—	139	2	no	—
130	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
131	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
132	#103	—	139	2	no	—
133	#103	—	139	2	no	—
134	#102	—	139	2	no	—
135	#102	—	139	2	no	—
136	C₂×Q₈	—	139	2	no	—
137	C₂×D₄	—	278	2	no	—
138	C₂×D₄	—	278	2	no	—
139	C₂×D₄	—	139	2	no	—
140	C₂²×C₄	—	278	2	no	—
141	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
142	SD₁₆	—	556	2	no	—
143	SD₁₆	—	556	2	no	—
144	M₁₆	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
147	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
148	C₄⋊C₄	—	278	2	no	—
149	C₄⋊C₄	—	278	2	no	—
150	C₄⋊C₄	—	278	2	no	—
151	C₄⋊C₄	—	278	2	no	—
152	C₄⋊C₄	—	278	2	no	—
153	C₄⋊C₄	—	139	2	no	—
154	C₄⋊C₄	—	139	2	no	—
155	C₄⋊C₄	D₅₅₆	1	2	yes	—
156	C₂²⋊C₄	—	278	2	no	—
157	C₂²⋊C₄	—	278	2	no	—
158	C₂²⋊C₄	—	278	2	no	—
159	C₂²⋊C₄	D₅₅₆	1	2	yes	—
160	C₄²	—	278	2	no	—
161	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
162	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
163	C₂³	—	278	2	no	—
164	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
165	Q₈	—	278	2	no	—
166	Q₈	—	278	2	no	—
167	D₄	—	278	2	no	—
168	D₄	—	278	2	no	—
169	D₄	—	1112	2	no	—
170	D₄	—	1112	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	278	2	no	—
174	C₂×C₄	—	278	2	no	—
175	C₂×C₄	—	278	2	no	—
176	C₂×C₄	—	1112	2	no	—
177	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
178	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
179	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	556	2	no	—
186	C₂²	—	556	2	no	—
187	C₂²	—	556	2	no	—
188	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	556	2	no	—
193	C₄	—	556	2	no	—
194	C₄	—	556	2	no	—
195	C₄	#17984887	1	1	yes	characteristic
196	C₄	#17984890	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	1112	2	no	—
199	C₂	#22581534	1	1	yes	characteristic
200	C₂	#22581765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581565

C₂

yes

maximal, characteristic, complemented

#22581558

C₂

yes

maximal, characteristic

#22581261

C₂

yes

maximal, complemented

#22581254

C₂

yes

maximal, complemented

#22581199

C₂

yes

maximal, complemented

#22581182

C₂

yes

maximal, complemented

#22581159

C₂

yes

maximal, complemented

#22581154

C₂

yes

maximal, complemented

#424×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555774

C₂²

yes

—

C₂×#14555774

C₂²

yes

—

#17984842

C₂²

yes

characteristic

#17984839

C₂²

yes

characteristic

#17984835

—

#17984834

C₂²

yes

characteristic

#17984834

C₂²

yes

—

—

C₂²

yes

—

#17984816

C₂²

yes

—

#17984815

—

C₄×Dic₂₇₈

C₂²

yes

—

C₄×D₅₅₆

C₂²

yes

—

#17984776

C₂²

yes

—

#17984774

C₂²

yes

—

#17984767

C₂²

yes

—

#17984767

C₂²

yes

—

#17984764

C₂²

yes

—

#17984764

C₂²

yes

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂³

yes

—

C₂×C₂²⋊D₁₃₉

—

C₂×C₄×D₁₃₉

—

#14555774

—

#14555774

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#537

yes

characteristic

C₂×C₁₃₉

#577

yes

—

D₁₃₉

—

100

C₁₃₉

#3260

yes

characteristic, complemented, Sylow

101

#3260

—

139

maximal, Sylow

102

#541

—

139

—

103

#540

—

139

—

104

#479

—

139

—

105

#478

—

139

—

106

#468

—

139

—

107

#465

—

139

—

108

#461

—

139

—

109

#460

—

139

—

110

#424

D₁₃₉

yes

characteristic, complemented

111

C₂×SD₁₆

—

139

—

112

C₂×SD₁₆

—

139

—

113

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

114

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

115

C₄⋊Q₈

—

139

—

116

#127

—

139

—

117

#126

—

278

—

118

#125

—

278

—

119

#125

—

139

—

120

#125

—

139

—

121

#123

—

278

—

122

C₂²⋊Q₈

—

139

—

123

#121

—

278

—

124

#121

—

139

—

125

C₄×Q₈

—

139

—

126

C₄×D₄

—

139

—

127

#117

C₂×D₁₃₉

yes

characteristic

128

#107

—

139

—

129

#106

—

139

—

130

C₄⋊C₈

C₂×D₁₃₉

yes

—

131

C₄⋊C₈

C₂×D₁₃₉

yes

—

132

#103

—

139

—

133

#103

—

139

—

134

#102

—

139

—

135

#102

—

139

—

136

C₂×Q₈

—

139

—

137

C₂×D₄

—

278

—

138

C₂×D₄

—

278

—

139

C₂×D₄

—

139

—

140

C₂²×C₄

—

278

—

141

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

142

SD₁₆

—

556

—

143

SD₁₆

—

556

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

C₂²×D₁₃₉

yes

—

147

C₂×C₈

C₂²×D₁₃₉

yes

—

148

C₄⋊C₄

—

278

—

149

C₄⋊C₄

—

278

—

150

C₄⋊C₄

—

278

—

151

C₄⋊C₄

—

278

—

152

C₄⋊C₄

—

278

—

153

C₄⋊C₄

—

139

—

154

C₄⋊C₄

—

139

—

155

C₄⋊C₄

D₅₅₆

yes

—

156

C₂²⋊C₄

—

278

—

157

C₂²⋊C₄

—

278

—

158

C₂²⋊C₄

—

278

—

159

C₂²⋊C₄

D₅₅₆

yes

—

160

C₄²

—

278

—

161

C₄²

C₂²×D₁₃₉

yes

characteristic

162

C₄²

C₂²×D₁₃₉

yes

characteristic

163

C₂³

—

278

—

164

C₂³

C₂×D₅₅₆

yes

characteristic

165

Q₈

—

278

—

166

Q₈

—

278

—

167

D₄

—

278

—

168

D₄

—

278

—

169

D₄

—

1112

—

170

D₄

—

1112

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

278

—

174

C₂×C₄

—

278

—

175

C₂×C₄

—

278

—

176

C₂×C₄

—

1112

—

177

C₂×C₄

C₂×D₅₅₆

yes

—

178

C₂×C₄

C₂×D₅₅₆

yes

—

179

C₂×C₄

C₂×D₅₅₆

yes

characteristic

180

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

556

—

186

C₂²

—

556

—

187

C₂²

—

556

—

188

C₂²

C₂²×D₅₅₆

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

556

—

193

C₄

—

556

—

194

C₄

—

556

—

195

C₄

#17984887

yes

characteristic

196

C₄

#17984890

yes

characteristic

197

C₂

—

198

C₂

—

1112

—

199

C₂

#22581534

yes

characteristic

200

C₂

#22581765

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#424×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581558	C₂
hypercenter	#424	D₁₃₉
hypocenter	C₁₃₉	#3260
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#424×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #22581565	C₂	semidirect
3. #22581261	C₂	semidirect
4. #22581254	C₂	semidirect
5. #22581199	C₂	semidirect
6. #22581182	C₂	semidirect
7. #22581159	C₂	semidirect
8. #22581154	C₂	semidirect
9. #424×C₁₃₉	C₂	semidirect
100. C₁₃₉	#3260	semidirect
110. #424	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581565

C₂

semidirect

3. #22581261

C₂

semidirect

4. #22581254

C₂

semidirect

5. #22581199

C₂

semidirect

6. #22581182

C₂

semidirect

7. #22581159

C₂

semidirect

8. #22581154

C₂

semidirect

9. #424×C₁₃₉

C₂

semidirect

100. C₁₃₉

#3260

semidirect

110. #424

D₁₃₉

semidirect

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

subgroup	index
#22581565	2
#22581558	2
#22581261	2
#22581254	2
#22581199	2
#22581182	2
#22581159	2
#22581154	2
#424×C₁₃₉	2
#3260	139

subgroup

index

#424×C₁₃₉

139

Subgroups — #31817851

Conjugacy classes of subgroups

Complements

Maximal subgroups