Group #31816309 — subgroups

Subgroups — C₂×#22581288 #31816309

Order 17792.

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

normal subgroups only

This group has 460 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×#17984821	C₂	1	1	yes	maximal, characteristic
2	#22581288	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984775	C₂	1	2	yes	maximal, complemented
4	C₂×#17984765	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
9	#17984821	—	2	2	no	—
10	#17984821	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₃₉	—	2	1	no	—
14	#17984775	C₂²	1	8	yes	complemented
15	#17984765	C₂²	1	8	yes	complemented
16	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₃₉	—	2	1	no	—
18	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
20	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₂×Dic₂₇₈	—	2	1	no	—
24	C₂×Dic₂₇₈	—	2	4	no	—
25	C₂×Dic₂₇₈	C₂³	1	2	yes	—
26	C₂²×Dic₁₃₉	—	2	1	no	—
27	C₂²×Dic₁₃₉	—	2	1	no	—
28	C₂×C₁₃₉⋊C₈	—	2	4	no	—
29	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
30	C₄⋊Dic₁₃₉	—	2	1	no	—
31	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
32	Dic₁₃₉⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₁₃₉	—	2	4	no	—
34	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₉	—	2	4	no	—
36	C₂²×C₄×C₁₃₉	—	2	4	no	—
37	C₂²×C₄×C₁₃₉	—	2	2	no	—
38	C₂²×C₄×C₁₃₉	—	4	4	no	—
39	C₂²×C₄×C₁₃₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
41	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₃₉	—	2	4	no	—
44	C₄⋊C₄×C₁₃₉	—	2	2	no	—
45	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
46	Dic₂₇₈	—	2	4	no	—
47	Dic₂₇₈	—	4	2	no	—
48	C₂×Dic₁₃₉	—	2	2	no	—
49	C₂×Dic₁₃₉	—	2	2	no	—
50	C₂×Dic₁₃₉	—	4	2	no	—
51	C₂×Dic₁₃₉	—	4	2	no	—
52	C₁₃₉⋊C₈	—	2	4	no	—
53	C₂³×C₁₃₉	—	2	2	no	—
54	C₂³×C₁₃₉	—	2	2	no	—
55	C₂³×C₁₃₉	Q₁₆	1	4	yes	—
56	C₂³×C₁₃₉	C₂×D₄	1	2	yes	—
57	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	—	2	2	no	—
59	C₂×C₄×C₁₃₉	—	2	4	no	—
60	C₂×C₄×C₁₃₉	—	2	4	no	—
61	C₂×C₄×C₁₃₉	—	2	4	no	—
62	C₂×C₄×C₁₃₉	—	4	8	no	—
63	C₂×C₄×C₁₃₉	—	4	4	no	—
64	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	#41	1	4	yes	—
67	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
68	C₂²×C₁₃₉	—	2	2	no	—
69	C₂²×C₁₃₉	—	2	2	no	—
70	C₂²×C₁₃₉	—	2	8	no	—
71	C₂²×C₁₃₉	#123	1	2	yes	—
72	C₂²×C₁₃₉	C₂×Q₁₆	1	4	yes	—
73	C₂²×C₁₃₉	C₂×Q₁₆	1	2	yes	—
74	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₁₃₉	—	2	2	no	—
77	C₄×C₁₃₉	—	4	4	no	—
78	C₄×C₁₃₉	#123	1	2	yes	—
79	C₄×C₁₃₉	C₂×#41	1	2	yes	—
80	Dic₁₃₉	—	4	2	no	—
81	Dic₁₃₉	—	4	2	no	—
82	C₂×C₁₃₉	—	2	4	no	—
83	C₂×C₁₃₉	#484	1	4	yes	complemented
84	C₂×C₁₃₉	C₂×#123	1	1	yes	characteristic
85	C₂×C₁₃₉	C₂²×Q₁₆	1	1	yes	characteristic
86	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₁₃₉	C₂×#484	1	1	yes	characteristic, complemented, Sylow
88	C₂×#484	—	139	1	no	maximal, Sylow
89	C₂×C₂²⋊Q₈	—	139	1	no	—
90	C₂²×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic
91	#484	—	139	8	no	—
92	C₂×#107	—	139	2	no	—
93	C₂×#103	—	139	2	no	—
94	C₂×C₂²⋊C₈	—	139	1	no	—
95	C₂²×Q₈	—	139	1	no	—
96	C₂³×C₄	—	2	1	no	—
97	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
98	C₂²×C₈	—	139	2	no	—
99	C₂²⋊Q₈	—	278	2	no	—
100	C₂²⋊Q₈	—	139	4	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	278	1	no	—
104	C₂×C₄⋊C₄	—	139	1	no	—
105	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
106	C₂×C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
107	C₂×C₂²⋊C₄	—	278	1	no	—
108	#107	—	139	8	no	—
109	#103	—	139	8	no	—
110	C₂²⋊C₈	—	139	4	no	—
111	C₂⁴	C₂²⋊D₁₃₉	1	1	yes	characteristic
112	C₂×Q₈	—	278	1	no	—
113	C₂×Q₈	—	278	4	no	—
114	C₂×Q₈	—	139	2	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	278	1	no	—
120	C₂²×C₄	—	278	1	no	—
121	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
122	C₂²×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
125	C₂×C₈	—	278	4	no	—
126	C₂×C₈	—	139	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	278	4	no	—
130	C₄⋊C₄	—	278	1	no	—
131	C₄⋊C₄	—	139	2	no	—
132	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
133	C₂²⋊C₄	—	278	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	#14555775	1	4	yes	—
137	C₂³	C₂×C₂²⋊D₁₃₉	1	2	yes	—
138	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
139	Q₈	—	278	4	no	—
140	Q₈	—	556	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	278	2	no	—
148	C₂×C₄	—	278	2	no	—
149	C₂×C₄	—	556	2	no	—
150	C₂×C₄	—	556	2	no	—
151	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
153	C₂×C₄	#14555785	1	4	yes	—
154	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
155	C₈	—	278	4	no	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#17984825	1	2	yes	—
160	C₂²	C₂×#14555775	1	4	yes	—
161	C₂²	C₂×#14555775	1	2	yes	—
162	C₂²	#17984868	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	4	4	no	—
166	C₄	—	556	2	no	—
167	C₄	—	556	2	no	—
168	C₄	#17984825	1	2	yes	—
169	C₄	C₂×#14555785	1	2	yes	—
170	C₂	—	2	4	no	—
171	C₂	#22581288	1	4	yes	direct product factor
172	C₂	C₂×#17984825	1	1	yes	characteristic
173	C₂	C₂²×#14555775	1	1	yes	characteristic
174	C₂	C₂×#17984868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984821

C₂

yes

maximal, characteristic

#22581288

C₂

yes

maximal, direct product factor

C₂×#17984775

C₂

yes

maximal, complemented

C₂×#17984765

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984821

—

#17984821

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×Dic₁₃₉⋊C₄

—

C₂×C₂²⋊Dic₁₃₉

—

#17984775

C₂²

yes

complemented

#17984765

C₂²

yes

complemented

C₂²⋊(C₁₃₉⋊C₈)

C₂²

yes

—

C₂³×C₄×C₁₃₉

—

C₂³×C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₂⁴×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

characteristic

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

Q₁₆

yes

—

C₂³×C₁₃₉

C₂×D₄

yes

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#123

yes

—

C₂²×C₁₃₉

C₂×Q₁₆

yes

—

C₂²×C₁₃₉

C₂×Q₁₆

yes

—

C₂²×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#123

yes

—

C₄×C₁₃₉

C₂×#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

C₂²×Q₁₆

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

C₂×#484

—

139

maximal, Sylow

C₂×C₂²⋊Q₈

—

139

—

C₂²×C₄⋊C₄

D₁₃₉

yes

characteristic

#484

—

139

—

C₂×#107

—

139

—

C₂×#103

—

139

—

C₂×C₂²⋊C₈

—

139

—

C₂²×Q₈

—

139

—

C₂³×C₄

—

C₂³×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

C₂²⋊Q₈

—

278

—

100

C₂²⋊Q₈

—

139

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

278

—

104

C₂×C₄⋊C₄

—

139

—

105

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

106

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

107

C₂×C₂²⋊C₄

—

278

—

108

#107

—

139

—

109

#103

—

139

—

110

C₂²⋊C₈

—

139

—

111

C₂⁴

C₂²⋊D₁₃₉

yes

characteristic

112

C₂×Q₈

—

278

—

113

C₂×Q₈

—

278

—

114

C₂×Q₈

—

139

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

278

—

120

C₂²×C₄

—

278

—

121

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

122

C₂²×C₄

C₂²⋊D₁₃₉

yes

characteristic

123

C₂²×C₄

C₂²×D₁₃₉

yes

—

124

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

125

C₂×C₈

—

278

—

126

C₂×C₈

—

139

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

278

—

130

C₄⋊C₄

—

278

—

131

C₄⋊C₄

—

139

—

132

C₄⋊C₄

C₂²×D₁₃₉

yes

—

133

C₂²⋊C₄

—

278

—

134

C₂³

—

135

C₂³

—

136

C₂³

#14555775

yes

—

137

C₂³

C₂×C₂²⋊D₁₃₉

yes

—

138

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

139

Q₈

—

278

—

140

Q₈

—

556

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

278

—

148

C₂×C₄

—

278

—

149

C₂×C₄

—

556

—

150

C₂×C₄

—

556

—

151

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

152

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

153

C₂×C₄

#14555785

yes

—

154

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

155

C₈

—

278

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#17984825

yes

—

160

C₂²

C₂×#14555775

yes

—

161

C₂²

C₂×#14555775

yes

—

162

C₂²

#17984868

yes

—

163

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

556

—

167

C₄

—

556

—

168

C₄

#17984825

yes

—

169

C₄

C₂×#14555785

yes

—

170

C₂

—

171

C₂

#22581288

yes

direct product factor

172

C₂

C₂×#17984825

yes

characteristic

173

C₂

C₂²×#14555775

yes

characteristic

174

C₂

C₂×#17984868

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂²×C₄⋊C₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂×D₄
prime radical	C₂⁴×C₁₃₉	D₄
hypercenter	C₂²×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₂×#484
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂²×C₄⋊C₄×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂×D₄

prime radical

C₂⁴×C₁₃₉

D₄

hypercenter

C₂²×C₄⋊C₄

D₁₃₉

hypocenter

C₁₃₉

C₂×#484

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #22581288	C₂	direct (N × H)
3. C₂×#17984775	C₂	semidirect
4. C₂×#17984765	C₂	semidirect
14. #17984775	C₂²	semidirect
15. #17984765	C₂²	semidirect
83. C₂×C₁₃₉	#484	semidirect
87. C₁₃₉	C₂×#484	semidirect
171. C₂	#22581288	direct (N × H)

H ≅ G/N

type

2. #22581288

C₂

direct (N × H)

3. C₂×#17984775

C₂

semidirect

4. C₂×#17984765

C₂

semidirect

14. #17984775

C₂²

semidirect

15. #17984765

C₂²

semidirect

83. C₂×C₁₃₉

#484

semidirect

87. C₁₃₉

C₂×#484

semidirect

171. C₂

#22581288

direct (N × H)

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

subgroup	index
C₂×#17984821	2
#22581288	2
C₂×#17984775	2
C₂×#17984765	2
C₂×C₂²⋊(C₁₃₉⋊C₈)	2
C₂²×C₄⋊C₄×C₁₃₉	2
C₂×#484	139

subgroup

index

C₂×C₂²⋊(C₁₃₉⋊C₈)

C₂²×C₄⋊C₄×C₁₃₉

C₂×#484

139

Subgroups — C₂×#22581288 #31816309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581288 #31816309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581288 #31816309