Group #31817875 — subgroups

Subgroups — #31817875

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	#22581574	C₂	1	2	yes	maximal, complemented
2	#22581244	C₂	1	2	yes	maximal, complemented
3	#22581239	C₂	1	2	yes	maximal, complemented
4	#22581211	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581206	C₂	1	1	yes	maximal, characteristic
6	#22581182	C₂	1	2	yes	maximal, complemented
7	#22581176	C₂	1	2	yes	maximal, complemented
8	#22581169	C₂	1	2	yes	maximal, complemented
9	#424×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
11	C₂×#14555774	C₂²	1	2	yes	—
12	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
13	#17984842	C₂²	1	2	yes	—
14	#17984837	—	2	2	no	—
15	#17984830	C₂²	1	2	yes	—
16	#17984828	—	2	2	no	—
17	#17984824	—	2	2	no	—
18	#17984821	C₂²	1	2	yes	—
19	#17984820	—	2	2	no	—
20	#17984816	C₂²	1	2	yes	—
21	C₄×Dic₂₇₈	C₂²	1	2	yes	—
22	C₄×D₅₅₆	C₂²	1	2	yes	—
23	#17984774	C₂²	1	2	yes	—
24	#17984774	C₂²	1	1	yes	characteristic
25	#17984774	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
36	C₂×Dic₂₇₈	C₂³	1	2	yes	—
37	C₂×D₅₅₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
39	C₂×C₄×D₁₃₉	—	2	2	no	—
40	Dic₅₅₆	—	4	1	no	—
41	#14555774	—	4	2	no	—
42	D₁₁₁₂	—	4	1	no	—
43	C₄⋊Dic₁₃₉	—	2	2	no	—
44	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
45	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
46	Dic₁₃₉⋊C₄	—	2	2	no	—
47	Dic₁₃₉⋊C₄	—	2	2	no	—
48	C₂²⋊Dic₁₃₉	—	2	2	no	—
49	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
50	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
51	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
52	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
53	C₄×Dic₁₃₉	—	2	2	no	—
54	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
55	M₁₆×C₁₃₉	—	4	1	no	—
56	C₂×C₈×C₁₃₉	—	4	1	no	—
57	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
61	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
63	C₄²×C₁₃₉	C₂³	1	2	yes	—
64	C₂²×D₁₃₉	—	2	2	no	—
65	Dic₂₇₈	—	2	2	no	—
66	Dic₂₇₈	—	2	2	no	—
67	Dic₂₇₈	—	8	2	no	—
68	D₅₅₆	—	2	2	no	—
69	D₅₅₆	—	2	2	no	—
70	C₂²⋊D₁₃₉	—	8	2	no	—
71	C₂×Dic₁₃₉	—	2	2	no	—
72	C₂×Dic₁₃₉	—	2	2	no	—
73	C₂×Dic₁₃₉	—	2	2	no	—
74	C₄×D₁₃₉	—	8	2	no	—
75	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₉	—	2	1	no	—
77	C₂×C₄×C₁₃₉	—	2	1	no	—
78	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₃₉	—	2	2	no	—
83	C₈×C₁₃₉	—	2	1	no	—
84	C₈×C₁₃₉	—	2	1	no	—
85	C₂²×C₁₃₉	—	2	1	no	—
86	C₂²×C₁₃₉	—	2	2	no	—
87	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₁₃₉	—	4	2	no	—
89	C₂×D₁₃₉	—	4	2	no	—
90	C₂×D₁₃₉	—	4	2	no	—
91	C₄×C₁₃₉	—	2	1	no	—
92	C₄×C₁₃₉	—	2	2	no	—
93	C₄×C₁₃₉	—	2	2	no	—
94	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₁₃₉	—	4	2	no	—
97	Dic₁₃₉	—	4	2	no	—
98	Dic₁₃₉	—	4	2	no	—
99	C₂×C₁₃₉	—	4	1	no	—
100	C₂×C₁₃₉	#537	1	1	yes	characteristic
101	C₂×C₁₃₉	#577	1	2	yes	—
102	D₁₃₉	—	8	2	no	—
103	C₁₃₉	#3263	1	1	yes	characteristic, complemented, Sylow
104	#3263	—	139	1	no	maximal, Sylow
105	#542	—	139	2	no	—
106	#476	—	139	2	no	—
107	#475	—	139	2	no	—
108	#470	—	139	1	no	—
109	#469	—	139	1	no	—
110	#465	—	139	2	no	—
111	#464	—	139	2	no	—
112	#463	—	139	2	no	—
113	#424	D₁₃₉	1	1	yes	characteristic, complemented
114	C₂×Q₁₆	—	139	1	no	—
115	C₂×SD₁₆	—	139	2	no	—
116	C₂×D₈	—	139	1	no	—
117	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
119	C₄⋊Q₈	—	139	2	no	—
120	#126	—	278	2	no	—
121	#124	—	278	2	no	—
122	#124	—	139	2	no	—
123	#123	—	278	2	no	—
124	C₂²⋊Q₈	—	278	2	no	—
125	C₂²⋊Q₈	—	139	2	no	—
126	#121	—	139	2	no	—
127	C₄×Q₈	—	139	2	no	—
128	C₄×D₄	—	139	2	no	—
129	#117	C₂×D₁₃₉	1	1	yes	characteristic
130	#106	—	139	2	no	—
131	#106	—	139	1	no	—
132	#106	—	139	1	no	—
133	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
134	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
135	#103	—	139	2	no	—
136	#103	—	139	2	no	—
137	#102	—	139	2	no	—
138	#102	—	139	2	no	—
139	C₂×Q₈	—	278	2	no	—
140	C₂×Q₈	—	139	2	no	—
141	C₂×D₄	—	278	2	no	—
142	C₂×D₄	—	139	2	no	—
143	C₂²×C₄	—	278	2	no	—
144	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
145	Q₁₆	—	556	1	no	—
146	SD₁₆	—	556	2	no	—
147	D₈	—	556	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
152	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
153	C₄⋊C₄	—	278	2	no	—
154	C₄⋊C₄	—	278	2	no	—
155	C₄⋊C₄	—	278	2	no	—
156	C₄⋊C₄	—	139	2	no	—
157	C₄⋊C₄	—	139	2	no	—
158	C₄⋊C₄	D₅₅₆	1	2	yes	—
159	C₂²⋊C₄	—	278	2	no	—
160	C₂²⋊C₄	—	278	2	no	—
161	C₂²⋊C₄	—	278	2	no	—
162	C₂²⋊C₄	—	278	2	no	—
163	C₂²⋊C₄	—	278	2	no	—
164	C₂²⋊C₄	D₅₅₆	1	1	yes	characteristic
165	C₂²⋊C₄	D₅₅₆	1	1	yes	characteristic
166	C₄²	—	278	2	no	—
167	C₄²	C₂²×D₁₃₉	1	2	yes	—
168	C₂³	—	278	2	no	—
169	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
170	Q₈	—	278	2	no	—
171	Q₈	—	278	2	no	—
172	Q₈	—	1112	2	no	—
173	D₄	—	278	2	no	—
174	D₄	—	278	2	no	—
175	D₄	—	1112	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	278	2	no	—
179	C₂×C₄	—	278	2	no	—
180	C₂×C₄	—	278	2	no	—
181	C₂×C₄	—	1112	2	no	—
182	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
183	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
184	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	556	2	no	—
192	C₂²	—	556	2	no	—
193	C₂²	—	556	2	no	—
194	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	556	2	no	—
199	C₄	—	556	2	no	—
200	C₄	—	556	2	no	—
201	C₄	#17984887	1	1	yes	characteristic
202	C₄	#17984890	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	1112	2	no	—
205	C₂	#22581534	1	1	yes	characteristic
206	C₂	#22581765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581574

C₂

yes

maximal, complemented

#22581244

C₂

yes

maximal, complemented

#22581239

C₂

yes

maximal, complemented

#22581211

C₂

yes

maximal, characteristic, complemented

#22581206

C₂

yes

maximal, characteristic

#22581182

C₂

yes

maximal, complemented

#22581176

C₂

yes

maximal, complemented

#22581169

C₂

yes

maximal, complemented

#424×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₅₆

C₂²

yes

characteristic

C₂×#14555774

C₂²

yes

—

C₂×D₁₁₁₂

C₂²

yes

characteristic

#17984842

C₂²

yes

—

#17984837

—

#17984830

C₂²

yes

—

#17984828

—

#17984824

—

#17984821

C₂²

yes

—

#17984820

—

#17984816

C₂²

yes

—

C₄×Dic₂₇₈

C₂²

yes

—

C₄×D₅₅₆

C₂²

yes

—

#17984774

C₂²

yes

—

#17984774

C₂²

yes

characteristic

#17984774

C₂²

yes

characteristic

#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

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×D₅₅₆

C₂³

yes

—

C₂×C₂²⋊D₁₃₉

—

C₂×C₄×D₁₃₉

—

Dic₅₅₆

—

#14555774

—

D₁₁₁₂

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

D₄

yes

characteristic

C₂²⋊C₄×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

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

—

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

—

100

C₂×C₁₃₉

#537

yes

characteristic

101

C₂×C₁₃₉

#577

yes

—

102

D₁₃₉

—

103

C₁₃₉

#3263

yes

characteristic, complemented, Sylow

104

#3263

—

139

maximal, Sylow

105

#542

—

139

—

106

#476

—

139

—

107

#475

—

139

—

108

#470

—

139

—

109

#469

—

139

—

110

#465

—

139

—

111

#464

—

139

—

112

#463

—

139

—

113

#424

D₁₃₉

yes

characteristic, complemented

114

C₂×Q₁₆

—

139

—

115

C₂×SD₁₆

—

139

—

116

C₂×D₈

—

139

—

117

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

118

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

119

C₄⋊Q₈

—

139

—

120

#126

—

278

—

121

#124

—

278

—

122

#124

—

139

—

123

#123

—

278

—

124

C₂²⋊Q₈

—

278

—

125

C₂²⋊Q₈

—

139

—

126

#121

—

139

—

127

C₄×Q₈

—

139

—

128

C₄×D₄

—

139

—

129

#117

C₂×D₁₃₉

yes

characteristic

130

#106

—

139

—

131

#106

—

139

—

132

#106

—

139

—

133

C₄⋊C₈

C₂×D₁₃₉

yes

—

134

C₄⋊C₈

C₂×D₁₃₉

yes

—

135

#103

—

139

—

136

#103

—

139

—

137

#102

—

139

—

138

#102

—

139

—

139

C₂×Q₈

—

278

—

140

C₂×Q₈

—

139

—

141

C₂×D₄

—

278

—

142

C₂×D₄

—

139

—

143

C₂²×C₄

—

278

—

144

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

145

Q₁₆

—

556

—

146

SD₁₆

—

556

—

147

D₈

—

556

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

151

C₂×C₈

C₂²×D₁₃₉

yes

—

152

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

153

C₄⋊C₄

—

278

—

154

C₄⋊C₄

—

278

—

155

C₄⋊C₄

—

278

—

156

C₄⋊C₄

—

139

—

157

C₄⋊C₄

—

139

—

158

C₄⋊C₄

D₅₅₆

yes

—

159

C₂²⋊C₄

—

278

—

160

C₂²⋊C₄

—

278

—

161

C₂²⋊C₄

—

278

—

162

C₂²⋊C₄

—

278

—

163

C₂²⋊C₄

—

278

—

164

C₂²⋊C₄

D₅₅₆

yes

characteristic

165

C₂²⋊C₄

D₅₅₆

yes

characteristic

166

C₄²

—

278

—

167

C₄²

C₂²×D₁₃₉

yes

—

168

C₂³

—

278

—

169

C₂³

C₂×D₅₅₆

yes

characteristic

170

Q₈

—

278

—

171

Q₈

—

278

—

172

Q₈

—

1112

—

173

D₄

—

278

—

174

D₄

—

278

—

175

D₄

—

1112

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

278

—

179

C₂×C₄

—

278

—

180

C₂×C₄

—

278

—

181

C₂×C₄

—

1112

—

182

C₂×C₄

C₂×D₅₅₆

yes

—

183

C₂×C₄

C₂×D₅₅₆

yes

—

184

C₂×C₄

C₂×D₅₅₆

yes

characteristic

185

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

556

—

192

C₂²

—

556

—

193

C₂²

—

556

—

194

C₂²

C₂²×D₅₅₆

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

556

—

199

C₄

—

556

—

200

C₄

—

556

—

201

C₄

#17984887

yes

characteristic

202

C₄

#17984890

yes

characteristic

203

C₂

—

204

C₂

—

1112

—

205

C₂

#22581534

yes

characteristic

206

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	#22581206	C₂
hypercenter	#424	D₁₃₉
hypocenter	C₁₃₉	#3263
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. #22581574	C₂	semidirect
2. #22581244	C₂	semidirect
3. #22581239	C₂	semidirect
4. #22581211	C₂	semidirect
6. #22581182	C₂	semidirect
7. #22581176	C₂	semidirect
8. #22581169	C₂	semidirect
9. #424×C₁₃₉	C₂	semidirect
103. C₁₃₉	#3263	semidirect
113. #424	D₁₃₉	semidirect

H ≅ G/N

type

1. #22581574

C₂

semidirect

2. #22581244

C₂

semidirect

3. #22581239

C₂

semidirect

4. #22581211

C₂

semidirect

6. #22581182

C₂

semidirect

7. #22581176

C₂

semidirect

8. #22581169

C₂

semidirect

9. #424×C₁₃₉

C₂

semidirect

103. C₁₃₉

#3263

semidirect

113. #424

D₁₃₉

semidirect

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

subgroup	index
#22581574	2
#22581244	2
#22581239	2
#22581211	2
#22581206	2
#22581182	2
#22581176	2
#22581169	2
#424×C₁₃₉	2
#3263	139

subgroup

index

#424×C₁₃₉

139

Subgroups — #31817875

Conjugacy classes of subgroups

Complements

Maximal subgroups