Group #31816183 — subgroups

Subgroups — C₂×#22581262 #31816183

Order 17792.

384 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 157 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×#17984833	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22581262	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984776	C₂	1	1	yes	maximal, characteristic
5	C₂×#17984773	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×#17984766	C₂	1	1	yes	maximal, characteristic
8	C₂×#103×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	#17984833	—	2	2	no	—
11	#17984833	C₂²	1	4	yes	—
12	Q₈×Dic₁₃₉	—	2	2	no	—
13	Q₈×Dic₁₃₉	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
15	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
16	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
19	C₂×C₄×Dic₁₃₉	—	2	1	no	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
21	#17984776	C₂²	1	4	yes	—
22	#17984773	C₂²	1	4	yes	—
23	Dic₁₃₉⋊C₈	C₂²	1	4	yes	—
24	#17984766	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
28	#103×C₁₃₉	C₂²	1	4	yes	—
29	C₂²×Dic₁₃₉	—	2	1	no	—
30	C₂²×Dic₁₃₉	—	2	1	no	—
31	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
32	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
33	C₂×C₁₃₉⋊C₈	—	2	2	no	—
34	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
35	C₄⋊Dic₁₃₉	—	2	4	no	—
36	C₄⋊Dic₁₃₉	—	2	4	no	—
37	C₄⋊Dic₁₃₉	—	2	1	no	—
38	C₄⋊Dic₁₃₉	—	2	1	no	—
39	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
40	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
41	Dic₁₃₉⋊C₄	—	2	4	no	—
42	C₄×Dic₁₃₉	—	2	4	no	—
43	C₄×Dic₁₃₉	D₄	1	2	yes	—
44	C₄×Dic₁₃₉	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₃₉	—	2	1	no	—
46	C₂×Q₈×C₁₃₉	—	2	2	no	—
47	C₂×Q₈×C₁₃₉	Q₈	1	4	yes	—
48	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃₉	—	2	1	no	—
50	C₂²×C₄×C₁₃₉	—	2	1	no	—
51	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃₉	—	2	2	no	—
53	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₃₉	—	2	1	no	—
55	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
56	C₂×Dic₁₃₉	—	2	2	no	—
57	C₂×Dic₁₃₉	—	2	2	no	—
58	C₂×Dic₁₃₉	—	2	2	no	—
59	C₂×Dic₁₃₉	—	2	2	no	—
60	C₂×Dic₁₃₉	—	4	2	no	—
61	C₂×Dic₁₃₉	—	4	2	no	—
62	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
63	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
64	C₁₃₉⋊C₈	—	2	2	no	—
65	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₃₉	—	2	4	no	—
67	Q₈×C₁₃₉	—	2	2	no	—
68	Q₈×C₁₃₉	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₁₃₉	—	2	4	no	—
70	C₂×C₄×C₁₃₉	—	2	2	no	—
71	C₂×C₄×C₁₃₉	—	2	2	no	—
72	C₂×C₄×C₁₃₉	—	4	2	no	—
73	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₃₉	#41	1	2	yes	—
76	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
77	C₈×C₁₃₉	—	2	2	no	—
78	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₁₃₉	#135	1	2	yes	—
80	C₂²×C₁₃₉	#137	1	2	yes	—
81	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₃₉	—	2	4	no	—
83	C₄×C₁₃₉	—	4	2	no	—
84	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
85	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
87	Dic₁₃₉	—	2	2	no	—
88	Dic₁₃₉	—	2	2	no	—
89	Dic₁₃₉	—	4	2	no	—
90	Dic₁₃₉	—	4	2	no	—
91	C₂×C₁₃₉	#479	1	4	yes	complemented
92	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
94	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
95	C₁₃₉	C₂×#479	1	1	yes	characteristic, complemented, Sylow
96	C₂×#479	—	139	1	no	maximal, Sylow
97	C₂×#125	—	139	1	no	—
98	C₂×C₄×Q₈	—	139	1	no	—
99	#479	—	139	8	no	—
100	C₂×#107	—	139	1	no	—
101	C₂×#106	—	139	1	no	—
102	C₂×C₄⋊C₈	—	139	1	no	—
103	C₂×#103	—	139	1	no	—
104	C₂×#103	D₁₃₉	1	1	yes	characteristic
105	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
106	C₂²×C₈	—	139	1	no	—
107	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
108	#125	—	278	2	no	—
109	#125	—	139	4	no	—
110	C₄×Q₈	—	278	2	no	—
111	C₄×Q₈	—	139	4	no	—
112	C₂×C₄⋊C₄	—	278	1	no	—
113	C₂×C₄⋊C₄	—	278	1	no	—
114	C₂×C₄⋊C₄	—	278	1	no	—
115	C₂×C₄⋊C₄	—	139	1	no	—
116	C₂×C₄⋊C₄	—	139	1	no	—
117	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
118	C₂×C₄²	—	278	1	no	—
119	C₂×C₄²	—	139	1	no	—
120	#107	—	139	4	no	—
121	#106	—	139	4	no	—
122	C₄⋊C₈	—	139	4	no	—
123	#103	—	139	4	no	—
124	#103	C₂×D₁₃₉	1	4	yes	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	Dic₂₇₈	1	4	yes	—
128	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	278	1	no	—
132	C₂²×C₄	—	278	1	no	—
133	C₂²×C₄	—	139	1	no	—
134	C₂²×C₄	—	139	1	no	—
135	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	278	2	no	—
138	C₂×C₈	—	139	2	no	—
139	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	278	4	no	—
142	C₄⋊C₄	—	278	4	no	—
143	C₄⋊C₄	—	278	4	no	—
144	C₄⋊C₄	—	278	1	no	—
145	C₄⋊C₄	—	278	1	no	—
146	C₄⋊C₄	—	139	2	no	—
147	C₄⋊C₄	—	139	2	no	—
148	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
149	C₄²	—	278	4	no	—
150	C₄²	—	139	2	no	—
151	C₄²	—	139	2	no	—
152	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
153	Q₈	—	2	4	no	—
154	Q₈	—	2	2	no	—
155	Q₈	C₂×Dic₂₇₈	1	4	yes	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	278	2	no	—
161	C₂×C₄	—	278	2	no	—
162	C₂×C₄	—	278	2	no	—
163	C₂×C₄	—	278	2	no	—
164	C₂×C₄	—	556	2	no	—
165	C₂×C₄	—	556	2	no	—
166	C₂×C₄	—	139	2	no	—
167	C₂×C₄	—	139	2	no	—
168	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
170	C₂×C₄	#14555784	1	2	yes	—
171	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	278	2	no	—
174	C₂²	#17984819	1	2	yes	—
175	C₂²	#17984865	1	2	yes	—
176	C₂²	#17984871	1	2	yes	—
177	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	4	2	no	—
180	C₄	—	278	2	no	—
181	C₄	—	278	2	no	—
182	C₄	—	556	2	no	—
183	C₄	—	556	2	no	—
184	C₄	#17984819	1	2	yes	—
185	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
186	C₄	C₂×#14555784	1	1	yes	characteristic
187	C₂	#22581262	1	4	yes	direct product factor
188	C₂	C₂×#17984819	1	1	yes	characteristic
189	C₂	C₂×#17984865	1	1	yes	characteristic
190	C₂	C₂×#17984871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984833

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic

#22581262

C₂

yes

maximal, direct product factor

C₂×#17984776

C₂

yes

maximal, characteristic

C₂×#17984773

C₂

yes

maximal, characteristic

C₂×Dic₁₃₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#17984766

C₂

yes

maximal, characteristic

C₂×#103×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984833

—

#17984833

C₂²

yes

—

Q₈×Dic₁₃₉

—

Q₈×Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×Dic₁₃₉⋊C₄

—

C₂×C₄×Dic₁₃₉

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

#17984776

C₂²

yes

—

#17984773

C₂²

yes

—

Dic₁₃₉⋊C₈

C₂²

yes

—

#17984766

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

#103×C₁₃₉

C₂²

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

D₄

yes

characteristic

C₂²×Dic₁₃₉

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

Q₈

yes

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₂²×C₁₃₉

#135

yes

—

C₂²×C₁₃₉

#137

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

C₄×C₁₃₉

C₂×#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#479

yes

complemented

C₂×C₁₃₉

C₂×C₂²⋊Q₈

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

C₂×#479

—

139

maximal, Sylow

C₂×#125

—

139

—

C₂×C₄×Q₈

—

139

—

#479

—

139

—

100

C₂×#107

—

139

—

101

C₂×#106

—

139

—

102

C₂×C₄⋊C₈

—

139

—

103

C₂×#103

—

139

—

104

C₂×#103

D₁₃₉

yes

characteristic

105

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

106

C₂²×C₈

—

139

—

107

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

108

#125

—

278

—

109

#125

—

139

—

110

C₄×Q₈

—

278

—

111

C₄×Q₈

—

139

—

112

C₂×C₄⋊C₄

—

278

—

113

C₂×C₄⋊C₄

—

278

—

114

C₂×C₄⋊C₄

—

278

—

115

C₂×C₄⋊C₄

—

139

—

116

C₂×C₄⋊C₄

—

139

—

117

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

118

C₂×C₄²

—

278

—

119

C₂×C₄²

—

139

—

120

#107

—

139

—

121

#106

—

139

—

122

C₄⋊C₈

—

139

—

123

#103

—

139

—

124

#103

C₂×D₁₃₉

yes

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

Dic₂₇₈

yes

—

128

C₂×Q₈

C₂²×D₁₃₉

yes

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

278

—

132

C₂²×C₄

—

278

—

133

C₂²×C₄

—

139

—

134

C₂²×C₄

—

139

—

135

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

278

—

138

C₂×C₈

—

139

—

139

C₂×C₈

C₂²×D₁₃₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

278

—

142

C₄⋊C₄

—

278

—

143

C₄⋊C₄

—

278

—

144

C₄⋊C₄

—

278

—

145

C₄⋊C₄

—

278

—

146

C₄⋊C₄

—

139

—

147

C₄⋊C₄

—

139

—

148

C₄⋊C₄

C₂²×D₁₃₉

yes

—

149

C₄²

—

278

—

150

C₄²

—

139

—

151

C₄²

—

139

—

152

C₂³

D₄×D₁₃₉

yes

characteristic

153

Q₈

—

154

Q₈

—

155

Q₈

C₂×Dic₂₇₈

yes

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

278

—

161

C₂×C₄

—

278

—

162

C₂×C₄

—

278

—

163

C₂×C₄

—

278

—

164

C₂×C₄

—

556

—

165

C₂×C₄

—

556

—

166

C₂×C₄

—

139

—

167

C₂×C₄

—

139

—

168

C₂×C₄

D₄×D₁₃₉

yes

characteristic

169

C₂×C₄

C₂×Dic₂₇₈

yes

—

170

C₂×C₄

#14555784

yes

—

171

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

172

C₈

—

173

C₈

—

278

—

174

C₂²

#17984819

yes

—

175

C₂²

#17984865

yes

—

176

C₂²

#17984871

yes

—

177

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

278

—

181

C₄

—

278

—

182

C₄

—

556

—

183

C₄

—

556

—

184

C₄

#17984819

yes

—

185

C₄

C₂²×Dic₂₇₈

yes

characteristic

186

C₄

C₂×#14555784

yes

characteristic

187

C₂

#22581262

yes

direct product factor

188

C₂

C₂×#17984819

yes

characteristic

189

C₂

C₂×#17984865

yes

characteristic

190

C₂

C₂×#17984871

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#103×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	C₂×#103	D₁₃₉
hypocenter	C₁₃₉	C₂×#479
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂×#103×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
3. #22581262	C₂	direct (N × H)
91. C₂×C₁₃₉	#479	semidirect
95. C₁₃₉	C₂×#479	semidirect
187. C₂	#22581262	direct (N × H)

H ≅ G/N

type

3. #22581262

C₂

direct (N × H)

91. C₂×C₁₃₉

#479

semidirect

95. C₁₃₉

C₂×#479

semidirect

187. C₂

#22581262

direct (N × H)

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

subgroup	index
C₂×#17984833	2
C₂×Q₈×Dic₁₃₉	2
#22581262	2
C₂×#17984776	2
C₂×#17984773	2
C₂×Dic₁₃₉⋊C₈	2
C₂×#17984766	2
C₂×#103×C₁₃₉	2
C₂×#479	139

subgroup

index

C₂×Q₈×Dic₁₃₉

C₂×Dic₁₃₉⋊C₈

C₂×#103×C₁₃₉

139

Subgroups — C₂×#22581262 #31816183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581262 #31816183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581262 #31816183