Group #31816898 — subgroups

Subgroups — #31816898

Order 17792.

532 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 165 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×#17984875	C₂	1	2	yes	maximal, complemented
2	C₂×#17984864	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581529	C₂	1	2	yes	maximal, complemented
5	#22581347	C₂	1	4	yes	maximal, complemented
6	#22581337	C₂	1	2	yes	maximal, complemented
7	#22581333	C₂	1	2	yes	maximal, complemented
8	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555785	C₂²	1	2	yes	—
10	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
11	#17984875	—	2	8	no	—
12	#17984864	—	2	4	no	—
13	C₂×Dic₅₅₆	—	2	4	no	—
14	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₅₆	C₂²	1	2	yes	—
16	C₂×Dic₅₅₆	C₂²	1	4	yes	—
17	C₂×Dic₅₅₆	C₂²	1	2	yes	—
18	C₂×#14555774	C₂²	1	4	yes	—
19	C₂×#14555774	C₂²	1	2	yes	—
20	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
21	#17984842	C₂²	1	4	yes	—
22	#17984830	C₂²	1	4	yes	—
23	#17984826	—	2	4	no	—
24	#17984822	—	2	4	no	—
25	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
27	#117×C₁₃₉	C₂²	1	1	yes	characteristic
28	#97×C₁₃₉	C₂²	1	2	yes	—
29	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
30	#14555785	—	2	4	no	—
31	#14555785	—	2	4	no	—
32	C₂×Dic₂₇₈	—	2	2	no	—
33	C₂×Dic₂₇₈	—	2	4	no	—
34	C₂×Dic₂₇₈	—	2	4	no	—
35	C₂×Dic₂₇₈	C₂³	1	4	yes	—
36	C₂×Dic₂₇₈	C₂³	1	2	yes	—
37	C₂×D₅₅₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
39	C₂²×Dic₁₃₉	—	2	2	no	—
40	C₂×C₄×D₁₃₉	—	2	2	no	—
41	Dic₅₅₆	—	2	4	no	—
42	Dic₅₅₆	—	2	4	no	—
43	Dic₅₅₆	—	2	2	no	—
44	Dic₅₅₆	—	2	8	no	—
45	#14555774	—	2	4	no	—
46	#14555774	—	2	8	no	—
47	D₁₁₁₂	—	2	2	no	—
48	C₄⋊Dic₁₃₉	—	2	8	no	—
49	(C₂×D₁₃₉)⋊C₄	—	2	8	no	—
50	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₁₃₉	D₄	1	4	yes	—
52	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
53	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
57	C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
58	C₄²×C₁₃₉	C₂³	1	2	yes	—
59	C₂²×D₁₃₉	—	2	2	no	—
60	Dic₂₇₈	—	2	4	no	—
61	Dic₂₇₈	—	2	2	no	—
62	Dic₂₇₈	—	2	4	no	—
63	Dic₂₇₈	—	2	2	no	—
64	Dic₂₇₈	—	4	4	no	—
65	D₅₅₆	—	2	2	no	—
66	D₅₅₆	—	2	2	no	—
67	C₂²⋊D₁₃₉	—	4	4	no	—
68	C₂×Dic₁₃₉	—	2	4	no	—
69	C₂×Dic₁₃₉	—	2	2	no	—
70	C₂×Dic₁₃₉	—	4	4	no	—
71	C₄×D₁₃₉	—	4	4	no	—
72	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
78	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
79	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₈×C₁₃₉	C₂×D₄	1	4	yes	—
81	C₂²×C₁₃₉	—	2	2	no	—
82	C₂²×C₁₃₉	#127	1	2	yes	—
83	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×D₁₃₉	—	4	2	no	—
85	C₂×D₁₃₉	—	4	4	no	—
86	C₄×C₁₃₉	—	2	4	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	—	4	2	no	—
91	Dic₁₃₉	—	4	4	no	—
92	C₂×C₁₃₉	—	2	2	no	—
93	C₂×C₁₃₉	C₂×#127	1	1	yes	characteristic
94	C₂×C₁₃₉	#578	1	2	yes	—
95	D₁₃₉	—	8	2	no	—
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	—	139	1	no	—
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	D₁₃₉	1	1	yes	characteristic, complemented
106	C₂×#41	—	139	2	no	—
107	C₂²×Q₈	—	139	2	no	—
108	#137	—	278	8	no	—
109	#135	—	278	4	no	—
110	C₂×Q₁₆	—	278	4	no	—
111	C₂×Q₁₆	—	139	2	no	—
112	C₂×Q₁₆	—	139	2	no	—
113	C₂×Q₁₆	—	139	1	no	—
114	C₂×Q₁₆	—	139	4	no	—
115	C₂×SD₁₆	—	139	2	no	—
116	C₂×SD₁₆	—	139	4	no	—
117	C₂×D₈	—	139	1	no	—
118	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
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	C₂×D₁₃₉	1	1	yes	characteristic
125	#97	C₂×D₁₃₉	1	2	yes	—
126	C₄×C₈	C₂×D₁₃₉	1	2	yes	—
127	#41	—	278	4	no	—
128	#41	—	278	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	—	139	2	no	—
134	C₂×D₄	—	278	2	no	—
135	C₂×D₄	—	139	2	no	—
136	C₂²×C₄	—	278	2	no	—
137	C₂²×C₄	—	278	2	no	—
138	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
139	Q₁₆	—	278	4	no	—
140	Q₁₆	—	278	4	no	—
141	Q₁₆	—	278	2	no	—
142	Q₁₆	—	278	8	no	—
143	SD₁₆	—	278	4	no	—
144	SD₁₆	—	278	8	no	—
145	D₈	—	278	2	no	—
146	M₁₆	D₅₅₆	1	4	yes	—
147	C₂×C₈	D₅₅₆	1	4	yes	—
148	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
149	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
150	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
151	C₄⋊C₄	—	278	8	no	—
152	C₄⋊C₄	D₅₅₆	1	2	yes	—
153	C₂²⋊C₄	—	278	8	no	—
154	C₂²⋊C₄	D₅₅₆	1	2	yes	—
155	C₄²	C₂²×D₁₃₉	1	2	yes	—
156	C₂³	—	278	2	no	—
157	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
158	Q₈	—	278	4	no	—
159	Q₈	—	278	2	no	—
160	Q₈	—	278	4	no	—
161	Q₈	—	278	2	no	—
162	Q₈	—	556	4	no	—
163	D₄	—	278	2	no	—
164	D₄	—	278	2	no	—
165	D₄	—	556	4	no	—
166	C₂×C₄	—	278	4	no	—
167	C₂×C₄	—	278	2	no	—
168	C₂×C₄	—	556	4	no	—
169	C₂×C₄	—	556	4	no	—
170	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
171	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
173	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
174	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
175	C₈	C₂×D₅₅₆	1	2	yes	—
176	C₈	C₂×D₅₅₆	1	2	yes	—
177	C₈	C₂×D₅₅₆	1	4	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	556	4	no	—
180	C₂²	—	556	2	no	—
181	C₂²	#17984839	1	2	yes	—
182	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	556	2	no	—
185	C₄	—	556	4	no	—
186	C₄	#17984839	1	2	yes	—
187	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
188	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	1112	2	no	—
191	C₂	C₂×#17984839	1	1	yes	characteristic
192	C₂	#22581770	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984875

C₂

yes

maximal, complemented

C₂×#17984864

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₆

C₂

yes

maximal, characteristic, complemented

#22581529

C₂

yes

maximal, complemented

#22581347

C₂

yes

maximal, complemented

#22581337

C₂

yes

maximal, complemented

#22581333

C₂

yes

maximal, complemented

#405×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555785

C₂²

yes

—

C₂²×Dic₂₇₈

C₂²

yes

—

#17984875

—

#17984864

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

C₂²

yes

characteristic

C₂×Dic₅₅₆

C₂²

yes

—

C₂×Dic₅₅₆

C₂²

yes

—

C₂×Dic₅₅₆

C₂²

yes

—

C₂×#14555774

C₂²

yes

—

C₂×#14555774

C₂²

yes

—

C₂×D₁₁₁₂

C₂²

yes

characteristic

#17984842

C₂²

yes

—

#17984830

C₂²

yes

—

#17984826

—

#17984822

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

—

#14555785

—

#14555785

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×D₅₅₆

C₂³

yes

—

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

—

Dic₅₅₆

—

Dic₅₅₆

—

Dic₅₅₆

—

Dic₅₅₆

—

#14555774

—

#14555774

—

D₁₁₁₂

—

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

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

—

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

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×D₄

yes

—

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

—

C₈×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#127

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

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×#127

yes

characteristic

C₂×C₁₃₉

#578

yes

—

D₁₃₉

—

C₁₃₉

#3156

yes

characteristic, complemented, Sylow

#3156

—

139

maximal, Sylow

C₂×#137

—

139

—

C₂×#135

—

139

—

100

C₂²×Q₁₆

—

139

—

101

#536

—

139

—

102

#497

—

139

—

103

#495

—

139

—

104

#494

—

139

—

105

#405

D₁₃₉

yes

characteristic, complemented

106

C₂×#41

—

139

—

107

C₂²×Q₈

—

139

—

108

#137

—

278

—

109

#135

—

278

—

110

C₂×Q₁₆

—

278

—

111

C₂×Q₁₆

—

139

—

112

C₂×Q₁₆

—

139

—

113

C₂×Q₁₆

—

139

—

114

C₂×Q₁₆

—

139

—

115

C₂×SD₁₆

—

139

—

116

C₂×SD₁₆

—

139

—

117

C₂×D₈

—

139

—

118

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

119

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

120

C₄⋊Q₈

—

139

—

121

#124

—

139

—

122

#123

—

278

—

123

C₂²⋊Q₈

—

278

—

124

#117

C₂×D₁₃₉

yes

characteristic

125

#97

C₂×D₁₃₉

yes

—

126

C₄×C₈

C₂×D₁₃₉

yes

—

127

#41

—

278

—

128

#41

—

278

—

129

C₂×Q₈

—

278

—

130

C₂×Q₈

—

278

—

131

C₂×Q₈

—

278

—

132

C₂×Q₈

—

139

—

133

C₂×Q₈

—

139

—

134

C₂×D₄

—

278

—

135

C₂×D₄

—

139

—

136

C₂²×C₄

—

278

—

137

C₂²×C₄

—

278

—

138

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

139

Q₁₆

—

278

—

140

Q₁₆

—

278

—

141

Q₁₆

—

278

—

142

Q₁₆

—

278

—

143

SD₁₆

—

278

—

144

SD₁₆

—

278

—

145

D₈

—

278

—

146

M₁₆

D₅₅₆

yes

—

147

C₂×C₈

D₅₅₆

yes

—

148

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

149

C₂×C₈

C₂²×D₁₃₉

yes

—

150

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

151

C₄⋊C₄

—

278

—

152

C₄⋊C₄

D₅₅₆

yes

—

153

C₂²⋊C₄

—

278

—

154

C₂²⋊C₄

D₅₅₆

yes

—

155

C₄²

C₂²×D₁₃₉

yes

—

156

C₂³

—

278

—

157

C₂³

C₂×D₅₅₆

yes

characteristic

158

Q₈

—

278

—

159

Q₈

—

278

—

160

Q₈

—

278

—

161

Q₈

—

278

—

162

Q₈

—

556

—

163

D₄

—

278

—

164

D₄

—

278

—

165

D₄

—

556

—

166

C₂×C₄

—

278

—

167

C₂×C₄

—

278

—

168

C₂×C₄

—

556

—

169

C₂×C₄

—

556

—

170

C₂×C₄

C₂×D₅₅₆

yes

—

171

C₂×C₄

C₂×D₅₅₆

yes

characteristic

172

C₂×C₄

C₂×D₅₅₆

yes

—

173

C₂×C₄

C₂×D₅₅₆

yes

—

174

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

175

C₈

C₂×D₅₅₆

yes

—

176

C₈

C₂×D₅₅₆

yes

—

177

C₈

C₂×D₅₅₆

yes

—

178

C₂²

—

179

C₂²

—

556

—

180

C₂²

—

556

—

181

C₂²

#17984839

yes

—

182

C₂²

C₂²×D₅₅₆

yes

characteristic

183

C₄

—

184

C₄

—

556

—

185

C₄

—

556

—

186

C₄

#17984839

yes

—

187

C₄

C₂²×D₅₅₆

yes

characteristic

188

C₄

C₂²×D₅₅₆

yes

characteristic

189

C₂

—

190

C₂

—

1112

—

191

C₂

C₂×#17984839

yes

characteristic

192

C₂

#22581770

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#405×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984864	C₂
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#3156
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#405×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
1. C₂×#17984875	C₂	semidirect
3. C₂²×Dic₅₅₆	C₂	semidirect
4. #22581529	C₂	semidirect
5. #22581347	C₂	semidirect
6. #22581337	C₂	semidirect
7. #22581333	C₂	semidirect
8. #405×C₁₃₉	C₂	semidirect
96. C₁₃₉	#3156	semidirect
105. #405	D₁₃₉	semidirect

H ≅ G/N

type

1. C₂×#17984875

C₂

semidirect

3. C₂²×Dic₅₅₆

C₂

semidirect

4. #22581529

C₂

semidirect

5. #22581347

C₂

semidirect

6. #22581337

C₂

semidirect

7. #22581333

C₂

semidirect

8. #405×C₁₃₉

C₂

semidirect

96. C₁₃₉

#3156

semidirect

105. #405

D₁₃₉

semidirect

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

subgroup	index
C₂×#17984875	2
C₂×#17984864	2
C₂²×Dic₅₅₆	2
#22581529	2
#22581347	2
#22581337	2
#22581333	2
#405×C₁₃₉	2
#3156	139

subgroup

index

C₂²×Dic₅₅₆

#405×C₁₃₉

139

Subgroups — #31816898

Conjugacy classes of subgroups

Complements

Maximal subgroups