Group #30676289 — subgroups

Subgroups — C₂×#22553671 #30676289

Order 16768.

436 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862246	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553671	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16862170	C₂	1	2	yes	maximal, complemented
4	C₂×#16862167	C₂	1	2	yes	maximal
5	C₂×#16862157	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₂	—	2	1	no	—
8	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
9	#16862246	—	2	2	no	—
10	#16862246	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₁₃₁	—	2	1	no	—
12	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
13	#16862170	C₂²	1	8	yes	complemented
14	#16862167	C₂²	1	8	yes	—
15	#16862157	C₂²	1	4	yes	complemented
16	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
18	#124×C₁₃₁	—	2	2	no	—
19	#124×C₁₃₁	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₁₃₁	—	2	1	no	—
22	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₆₂	—	2	4	no	—
24	C₂×Dic₂₆₂	—	4	4	no	—
25	C₂²×Dic₁₃₁	—	2	2	no	—
26	C₂×C₁₃₁⋊C₈	—	2	4	no	—
27	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₁₃₁	—	2	2	no	—
29	C₄⋊Dic₁₃₁	—	2	4	no	—
30	C₄⋊Dic₁₃₁	C₂³	1	4	yes	—
31	C₂⁴×C₁₃₁	—	2	1	no	—
32	C₂×Q₈×C₁₃₁	—	2	1	no	—
33	C₂×Q₈×C₁₃₁	—	2	4	no	—
34	C₂×Q₈×C₁₃₁	C₂³	1	2	yes	—
35	C₂×D₄×C₁₃₁	—	2	1	no	—
36	C₂×D₄×C₁₃₁	—	2	4	no	—
37	C₂×D₄×C₁₃₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃₁	—	2	1	no	—
39	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
43	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
44	C₄²×C₁₃₁	D₄	1	2	yes	—
45	C₄²×C₁₃₁	C₂³	1	2	yes	—
46	Dic₂₆₂	—	4	8	no	—
47	C₂×Dic₁₃₁	—	2	4	no	—
48	C₂×Dic₁₃₁	—	4	4	no	—
49	C₁₃₁⋊C₈	—	2	4	no	—
50	C₂³×C₁₃₁	—	2	2	no	—
51	C₂³×C₁₃₁	—	4	2	no	—
52	C₂³×C₁₃₁	—	4	2	no	—
53	C₂³×C₁₃₁	—	4	2	no	—
54	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₃₁	—	2	4	no	—
56	Q₈×C₁₃₁	—	4	2	no	—
57	D₄×C₁₃₁	—	2	4	no	—
58	D₄×C₁₃₁	—	4	2	no	—
59	C₂×C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄×C₁₃₁	—	2	2	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	4	2	no	—
63	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	#41	1	4	yes	—
67	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
68	C₂²×C₁₃₁	—	4	2	no	—
69	C₂²×C₁₃₁	—	4	2	no	—
70	C₂²×C₁₃₁	—	4	2	no	—
71	C₂²×C₁₃₁	—	8	4	no	—
72	C₂²×C₁₃₁	#124	1	2	yes	—
73	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₂²×C₁₃₁	#137	1	2	yes	—
75	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₁₃₁	—	2	2	no	—
77	C₄×C₁₃₁	—	2	2	no	—
78	C₄×C₁₃₁	—	4	2	no	—
79	C₄×C₁₃₁	#124	1	2	yes	—
80	C₄×C₁₃₁	C₂×#41	1	2	yes	—
81	Dic₁₃₁	—	4	4	no	—
82	C₂×C₁₃₁	—	8	2	no	—
83	C₂×C₁₃₁	#489	1	4	yes	complemented
84	C₂×C₁₃₁	C₂×#124	1	1	yes	characteristic
85	C₂×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₁₃₁	C₂×#137	1	1	yes	characteristic
87	C₁₃₁	C₂×#489	1	1	yes	characteristic, complemented, Sylow
88	C₂×#489	—	131	1	no	maximal, Sylow
89	C₂×C₄⋊Q₈	—	131	1	no	—
90	C₂×#124	D₁₃₁	1	1	yes	characteristic
91	#489	—	131	8	no	—
92	C₂×#103	—	131	2	no	—
93	C₂×#102	—	131	2	no	—
94	C₂×#97	—	131	1	no	—
95	C₂²×Q₈	—	262	1	no	—
96	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
97	C₂²×D₄	C₂×D₁₃₁	1	1	yes	characteristic
98	C₂²×C₈	—	131	2	no	—
99	C₄⋊Q₈	—	262	2	no	—
100	C₄⋊Q₈	—	131	4	no	—
101	#124	—	2	2	no	—
102	#124	C₂×D₁₃₁	1	4	yes	—
103	C₂×C₄⋊C₄	—	262	1	no	—
104	C₂×C₄⋊C₄	—	131	2	no	—
105	C₂×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
108	#103	—	131	8	no	—
109	#102	—	131	8	no	—
110	#97	—	131	4	no	—
111	C₂⁴	—	2	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	—	262	4	no	—
115	C₂×Q₈	—	524	4	no	—
116	C₂×Q₈	C₂²×D₁₃₁	1	2	yes	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	C₂²×D₁₃₁	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	262	2	no	—
122	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
125	C₂×C₈	—	262	4	no	—
126	C₂×C₈	—	131	4	no	—
127	C₄⋊C₄	—	262	4	no	—
128	C₄⋊C₄	—	262	2	no	—
129	C₄⋊C₄	—	131	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
133	C₄²	C₂²×D₁₃₁	1	2	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	4	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
139	Q₈	—	2	4	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	524	8	no	—
142	D₄	—	2	4	no	—
143	D₄	—	4	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	262	4	no	—
149	C₂×C₄	—	524	4	no	—
150	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
153	C₂×C₄	#14523512	1	4	yes	—
154	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
155	C₈	—	262	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	4	no	—
160	C₂²	#16862233	1	2	yes	—
161	C₂²	#16862271	1	2	yes	—
162	C₂²	#16862276	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	524	4	no	—
168	C₄	#16862233	1	2	yes	—
169	C₄	C₂×#14523512	1	2	yes	—
170	C₂	—	8	2	no	—
171	C₂	#22553671	1	4	yes	direct product factor
172	C₂	C₂×#16862233	1	1	yes	characteristic
173	C₂	C₂×#16862271	1	1	yes	characteristic
174	C₂	C₂×#16862276	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862246

C₂

yes

maximal, characteristic, complemented

#22553671

C₂

yes

maximal, direct product factor

C₂×#16862170

C₂

yes

maximal, complemented

C₂×#16862167

C₂

yes

maximal

C₂×#16862157

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₂

—

C₂²×C₁₃₁⋊C₈

C₂²

yes

—

#16862246

—

#16862246

C₂²

yes

complemented

C₂×C₄⋊Dic₁₃₁

—

C₂×C₄⋊Dic₁₃₁

C₂²

yes

—

#16862170

C₂²

yes

complemented

#16862167

C₂²

yes

—

#16862157

C₂²

yes

complemented

C₂²×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂²×D₄×C₁₃₁

C₂²

yes

characteristic

#124×C₁₃₁

—

#124×C₁₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

C₂²×Dic₁₃₁

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂³

yes

—

C₂⁴×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

C₂³

yes

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#124

yes

—

C₂²×C₁₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃₁

#137

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#124

yes

—

C₄×C₁₃₁

C₂×#41

yes

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

C₂×#489

—

131

maximal, Sylow

C₂×C₄⋊Q₈

—

131

—

C₂×#124

D₁₃₁

yes

characteristic

#489

—

131

—

C₂×#103

—

131

—

C₂×#102

—

131

—

C₂×#97

—

131

—

C₂²×Q₈

—

262

—

C₂²×Q₈

C₂×D₁₃₁

yes

characteristic

C₂²×D₄

C₂×D₁₃₁

yes

characteristic

C₂²×C₈

—

131

—

C₄⋊Q₈

—

262

—

100

C₄⋊Q₈

—

131

—

101

#124

—

102

#124

C₂×D₁₃₁

yes

—

103

C₂×C₄⋊C₄

—

262

—

104

C₂×C₄⋊C₄

—

131

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

108

#103

—

131

—

109

#102

—

131

—

110

#97

—

131

—

111

C₂⁴

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

262

—

115

C₂×Q₈

—

524

—

116

C₂×Q₈

C₂²×D₁₃₁

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

C₂²×D₁₃₁

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

262

—

122

C₂²×C₄

C₂²⋊D₁₃₁

yes

characteristic

123

C₂²×C₄

C₂²⋊D₁₃₁

yes

characteristic

124

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

125

C₂×C₈

—

262

—

126

C₂×C₈

—

131

—

127

C₄⋊C₄

—

262

—

128

C₄⋊C₄

—

262

—

129

C₄⋊C₄

—

131

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

C₂²⋊D₁₃₁

yes

—

133

C₄²

C₂²×D₁₃₁

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×C₂²⋊D₁₃₁

yes

characteristic

139

Q₈

—

140

Q₈

—

141

Q₈

—

524

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

262

—

149

C₂×C₄

—

524

—

150

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

151

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

—

152

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

153

C₂×C₄

#14523512

yes

—

154

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

155

C₈

—

262

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#16862233

yes

—

161

C₂²

#16862271

yes

—

162

C₂²

#16862276

yes

—

163

C₂²

C₂²×C₂²⋊D₁₃₁

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

524

—

168

C₄

#16862233

yes

—

169

C₄

C₂×#14523512

yes

—

170

C₂

—

171

C₂

#22553671

yes

direct product factor

172

C₂

C₂×#16862233

yes

characteristic

173

C₂

C₂×#16862271

yes

characteristic

174

C₂

C₂×#16862276

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₂×#124×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂×D₄
prime radical	C₂²×D₄×C₁₃₁	C₂²
hypercenter	C₂×#124	D₁₃₁
hypocenter	C₁₃₁	C₂×#489
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₁

commutator, cosocle

C₂×C₄×C₁₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₁

Fitting

C₂×#124×C₁₃₁

C₂

socle

C₂³×C₁₃₁

C₂×D₄

prime radical

C₂²×D₄×C₁₃₁

C₂²

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#16862246	C₂	semidirect
2. #22553671	C₂	direct (N × H)
3. C₂×#16862170	C₂	semidirect
5. C₂×#16862157	C₂	semidirect
10. #16862246	C₂²	semidirect
13. #16862170	C₂²	semidirect
15. #16862157	C₂²	semidirect
83. C₂×C₁₃₁	#489	semidirect
87. C₁₃₁	C₂×#489	semidirect
171. C₂	#22553671	direct (N × H)

H ≅ G/N

type

1. C₂×#16862246

C₂

semidirect

2. #22553671

C₂

direct (N × H)

3. C₂×#16862170

C₂

semidirect

5. C₂×#16862157

C₂

semidirect

10. #16862246

C₂²

semidirect

13. #16862170

C₂²

semidirect

15. #16862157

C₂²

semidirect

83. C₂×C₁₃₁

#489

semidirect

87. C₁₃₁

C₂×#489

semidirect

171. C₂

#22553671

direct (N × H)

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

subgroup	index
C₂×#16862246	2
#22553671	2
C₂×#16862170	2
C₂×#16862167	2
C₂×#16862157	2
C₂×#124×C₁₃₁	2
C₂×#489	131

subgroup

index

C₂×#124×C₁₃₁

131

Subgroups — C₂×#22553671 #30676289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22553671 #30676289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553671 #30676289