Group #134628 — subgroups

Subgroups — #134628

Order 512.

287 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21723	C₂	1	1	yes	maximal, characteristic
2	#12901	C₂	1	1	yes	maximal, characteristic
3	#12881	C₂	1	1	yes	maximal, characteristic
4	C₂×#1366	C₂	1	2	yes	maximal
5	#7269	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	1	yes	characteristic
7	#2330	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	#1992	C₄	1	2	yes	—
13	#1944	—	2	1	no	—
14	#1942	—	2	1	no	—
15	#1845	C₂²	1	1	yes	characteristic
16	#1776	—	2	1	no	—
17	C₄×#107	—	2	1	no	—
18	#1726	C₂²	1	1	yes	characteristic
19	#1377	—	2	1	no	—
20	#1375	—	2	1	no	—
21	#1366	—	2	4	no	—
22	C₂×#125	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×M₃₂	—	2	2	no	—
25	#499	—	2	1	no	—
26	#499	—	2	1	no	—
27	#499	D₄	1	2	yes	—
28	#499	Q₈	1	2	yes	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×#106	—	2	1	no	—
33	C₂×#97	C₂×C₄	1	1	yes	characteristic
34	C₂×#97	D₄	1	1	yes	characteristic
35	C₂×#97	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#389	—	2	2	no	—
40	#389	—	4	2	no	—
41	#387	—	4	2	no	—
42	#384	C₂×C₄	1	2	yes	—
43	C₄×C₄⋊C₄	—	2	2	no	—
44	#376	C₂×C₄	1	1	yes	characteristic
45	#364	—	2	2	no	—
46	#364	—	2	2	no	—
47	#364	—	2	2	no	—
48	#364	—	2	2	no	—
49	#364	—	4	2	no	—
50	C₄⋊C₁₆	—	4	2	no	—
51	#340	—	2	2	no	—
52	#340	—	2	2	no	—
53	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	#125	—	2	2	no	—
58	#125	—	4	2	no	—
59	#125	C₄⋊C₄	1	4	yes	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₄⋊C₄	—	4	2	no	—
65	C₂×C₄²	—	4	2	no	—
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
69	M₃₂	—	4	2	no	—
70	M₃₂	—	4	2	no	—
71	M₃₂	—	4	2	no	—
72	M₃₂	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₂×C₁₆	—	4	2	no	—
75	#107	—	2	2	no	—
76	#107	—	2	2	no	—
77	#107	—	4	2	no	—
78	#107	—	4	1	no	—
79	#107	—	4	1	no	—
80	#106	—	2	2	no	—
81	#106	—	2	2	no	—
82	#97	—	2	2	no	—
83	#97	—	2	2	no	—
84	#97	—	4	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	SD₁₆	1	2	yes	—
88	C₄×C₈	Q₁₆	1	2	yes	—
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	#95	—	2	1	no	—
94	#95	—	2	1	no	—
95	#95	—	4	2	no	—
96	#95	—	4	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	#100	1	1	yes	characteristic
102	C₂²×C₄	#100	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	#103	1	2	yes	—
110	C₂×C₈	#121	1	1	yes	characteristic
111	C₂×C₈	#121	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₈	#135	1	2	yes	—
117	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₈	#137	1	1	yes	characteristic
119	C₂×C₈	#137	1	2	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	16	2	no	—
127	C₄⋊C₄	—	16	2	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	16	2	no	—
133	C₄²	#103	1	2	yes	—
134	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₄²	#117	1	1	yes	characteristic
136	C₁₆	—	8	2	no	—
137	C₁₆	—	8	2	no	—
138	C₂³	C₂×#100	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	#380	1	1	yes	characteristic
157	C₂×C₄	#413	1	1	yes	characteristic
158	C₂×C₄	C₂×#103	1	1	yes	characteristic
159	C₂×C₄	#416	1	1	yes	characteristic
160	C₂×C₄	#417	1	1	yes	characteristic
161	C₂×C₄	#419	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	4	1	no	—
164	C₈	#460	1	1	yes	characteristic
165	C₈	#462	1	1	yes	characteristic
166	C₈	#464	1	2	yes	—
167	C₈	#475	1	1	yes	characteristic
168	C₈	#477	1	1	yes	characteristic
169	C₈	#479	1	2	yes	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	#1548	1	1	yes	characteristic
173	C₂²	#1551	1	1	yes	characteristic
174	C₂²	#1800	1	1	yes	characteristic
175	C₄	—	4	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	4	no	—
181	C₄	—	16	2	no	—
182	C₄	#1542	1	1	yes	characteristic
183	C₄	#1559	1	1	yes	characteristic
184	C₄	#1803	1	1	yes	characteristic
185	C₄	#1805	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#8590	1	1	yes	characteristic
189	C₂	#11989	1	1	yes	characteristic
190	C₂	#11990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21723

C₂

yes

maximal, characteristic

#12901

C₂

yes

maximal, characteristic

#12881

C₂

yes

maximal, characteristic

C₂×#1366

C₂

yes

maximal

#7269

C₂

yes

maximal

C₂×#499

C₂²

yes

characteristic

#2330

C₄

yes

—

C₂×#364

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

#2150

—

#1992

C₄

yes

—

#1944

—

#1942

—

#1845

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂×C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

#499

—

#499

—

#499

D₄

yes

—

#499

Q₈

yes

—

—

—

—

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₄×C₄⋊C₄

—

#376

C₂×C₄

yes

—

—

—

—

—

C₄⋊C₁₆

—

#340

—

#340

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

C₄⋊C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

#100

yes

characteristic

102

C₂²×C₄

#100

yes

characteristic

103

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#103

yes

—

110

C₂×C₈

#121

yes

characteristic

111

C₂×C₈

#121

yes

characteristic

112

C₂×C₈

C₂²⋊Q₈

yes

characteristic

113

C₂×C₈

C₂²⋊Q₈

yes

characteristic

114

C₂×C₈

C₂×SD₁₆

yes

characteristic

115

C₂×C₈

C₂×Q₁₆

yes

characteristic

116

C₂×C₈

#135

yes

—

117

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₈

#137

yes

characteristic

119

C₂×C₈

#137

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

#103

yes

—

134

C₄²

C₂×C₄⋊C₄

yes

characteristic

135

C₄²

#117

yes

characteristic

136

C₁₆

—

137

C₁₆

—

138

C₂³

C₂×#100

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#380

yes

characteristic

157

C₂×C₄

#413

yes

characteristic

158

C₂×C₄

C₂×#103

yes

characteristic

159

C₂×C₄

#416

yes

characteristic

160

C₂×C₄

#417

yes

characteristic

161

C₂×C₄

#419

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

#460

yes

characteristic

165

C₈

#462

yes

characteristic

166

C₈

#464

yes

—

167

C₈

#475

yes

characteristic

168

C₈

#477

yes

characteristic

169

C₈

#479

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

#1548

yes

characteristic

173

C₂²

#1551

yes

characteristic

174

C₂²

#1800

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

#1542

yes

characteristic

183

C₄

#1559

yes

characteristic

184

C₄

#1803

yes

characteristic

185

C₄

#1805

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#8590

yes

characteristic

189

C₂

#11989

yes

characteristic

190

C₂

#11990

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1551

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#21723	2
#12901	2
#12881	2
C₂×#1366	2
#7269	2

subgroup

index

Subgroups — #134628

Conjugacy classes of subgroups

Maximal subgroups