Group #16927280 — subgroups

Subgroups — #16927280

Order 4352.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
2	#14527545	C₂	1	2	yes	maximal
3	#14527128	C₂	1	1	yes	maximal, characteristic
4	#14526653	C₂	1	2	yes	maximal
5	C₁₇×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3622	—	2	2	no	—
7	Q₈×#3622	—	2	2	no	—
8	#11762866	C₂²	1	2	yes	—
9	C₂×C₄⋊#3622	C₄	1	2	yes	—
10	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3622	C₄	1	2	yes	—
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762561	—	2	2	no	—
14	#11762559	—	2	2	no	—
15	#11762535	—	2	2	no	—
16	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#432	C₂²	1	2	yes	—
18	C₁₇×#324	—	2	2	no	—
19	C₂²×Q₈×C₁₇	Q₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	—	2	2	no	—
21	C₄×Q₈×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	2	yes	—
28	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
29	C₁₇×C₄⋊C₈	—	2	2	no	—
30	C₁₇×C₂²⋊C₈	—	2	2	no	—
31	C₁₇×#97	—	2	2	no	—
32	C₂²⋊#3622	—	2	2	no	—
33	C₄⋊#3622	—	2	4	no	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄⋊#3622	—	2	2	no	—
36	#10587474	—	2	2	no	—
37	C₄×#3622	—	2	2	no	—
38	C₄×#3622	—	2	2	no	—
39	C₄×#3622	D₄	1	2	yes	—
40	C₄×#3622	D₄	1	2	yes	—
41	C₂×Q₈×C₁₇	—	2	4	no	—
42	C₂×Q₈×C₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	—	4	2	no	—
49	C₄⋊C₄×C₁₇	—	2	4	no	—
50	C₄⋊C₄×C₁₇	—	2	2	no	—
51	C₄²×C₁₇	—	2	2	no	—
52	C₄²×C₁₇	—	2	2	no	—
53	C₄²×C₁₇	C₂×D₄	1	2	yes	—
54	C₄²×C₁₇	#41	1	2	yes	—
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	—	4	1	no	—
58	C₂×#3622	—	4	2	no	—
59	C₂×#3622	—	4	2	no	—
60	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×#3622	C₂²⋊C₄	1	2	yes	—
62	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
63	#8623	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	Q₈×C₁₇	—	2	8	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	#108	1	4	yes	—
75	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#121	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₁₇	—	8	2	no	—
80	#3622	—	2	1	no	—
81	#3622	—	2	2	no	—
82	#3622	—	4	2	no	—
83	#3622	—	8	2	no	—
84	#3622	#101	1	2	yes	—
85	C₂×C₈×Q₈	—	17	1	no	—
86	#1568	—	17	2	no	—
87	#1467	—	17	1	no	—
88	#1467	D₁₇	1	1	yes	characteristic
89	#1290	—	17	2	no	—
90	C₂²×C₁₇	—	2	1	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	#380	1	1	yes	characteristic
93	C₂²×C₁₇	C₂×#108	1	2	yes	—
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	—	2	4	no	—
96	C₄×C₁₇	—	2	2	no	—
97	C₄×C₁₇	—	2	2	no	—
98	C₄×C₁₇	—	4	2	no	—
99	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
100	C₄×C₁₇	#413	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
102	C₈×Q₈	—	34	2	no	—
103	C₈×Q₈	—	34	2	no	—
104	#432	—	17	2	no	—
105	#432	C₂×D₁₇	1	2	yes	—
106	C₂×C₄⋊C₈	—	17	2	no	—
107	C₂×C₄⋊C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	2	no	—
110	#335	—	34	2	no	—
111	#334	—	34	2	no	—
112	#324	—	2	2	no	—
113	#324	—	34	2	no	—
114	C₂×C₁₇	—	2	2	no	—
115	C₂×C₁₇	#1801	1	1	yes	characteristic
116	C₂×C₁₇	#1810	1	1	yes	characteristic
117	C₂×C₁₇	#1812	1	1	yes	characteristic
118	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
119	C₂²×C₈	—	17	1	no	—
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	17	2	no	—
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	Dic₃₄	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²⋊D₁₇	1	2	yes	—
126	C₂×C₄²	C₄×D₁₇	1	2	yes	—
127	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	34	4	no	—
130	C₄⋊C₈	—	34	2	no	—
131	C₄⋊C₈	—	34	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	34	2	no	—
134	#97	—	2	2	no	—
135	#97	—	34	2	no	—
136	C₄×C₈	—	17	2	no	—
137	C₄×C₈	—	17	2	no	—
138	C₄×C₈	—	34	2	no	—
139	C₄×C₈	—	34	2	no	—
140	C₁₇	#8623	1	1	yes	characteristic, complemented, Sylow
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	#63225	1	2	yes	—
144	C₂²×C₄	#63225	1	2	yes	—
145	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	34	2	no	—
153	C₂×C₈	—	34	2	no	—
154	C₂×C₈	—	68	1	no	—
155	C₂×C₈	—	68	2	no	—
156	C₂×C₈	—	68	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₁₇	1	2	yes	—
162	C₄²	#63254	1	2	yes	—
163	C₂³	C₂×#63225	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#10587493	1	4	yes	—
174	C₂×C₄	#10587563	1	1	yes	characteristic
175	C₂×C₄	#10587564	1	2	yes	—
176	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
177	C₂×C₄	#10587625	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	17	2	no	—
180	C₈	—	34	1	no	—
181	C₈	—	34	2	no	—
182	C₈	—	68	2	no	—
183	C₈	—	136	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#11762667	1	1	yes	characteristic
187	C₂²	C₂×#10587493	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₁₇	1	1	yes	characteristic
194	C₄	#11762792	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#14528778	1	1	yes	characteristic
197	C₂	#14528820	1	1	yes	characteristic
198	C₂	#14528828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×#3622

C₂

yes

maximal, characteristic

#14527545

C₂

yes

maximal

#14527128

C₂

yes

maximal, characteristic

#14526653

C₂

yes

maximal

C₁₇×#1467

C₂

yes

maximal, characteristic

Q₈×#3622

—

Q₈×#3622

—

#11762866

C₂²

yes

—

C₂×C₄⋊#3622

C₄

yes

—

C₂×C₄⋊#3622

C₂²

yes

characteristic

C₂×C₄×#3622

C₄

yes

—

C₂×C₄×#3622

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×#432

C₂²

yes

—

C₁₇×#324

—

C₂²×Q₈×C₁₇

Q₈

yes

characteristic

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

—

C₂²×#3622

C₂×C₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₁₇×#97

—

C₂²⋊#3622

—

—

—

—

—

—

—

D₄

yes

—

C₄×#3622

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

#8623

—

maximal, Sylow

C₂³×C₁₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#108

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

—

—

—

—

—

yes

—

C₂×C₈×Q₈

—

#1568

—

#1467

—

#1467

D₁₇

yes

characteristic

#1290

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#380

yes

characteristic

C₂²×C₁₇

C₂×#108

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#101

yes

characteristic

100

C₄×C₁₇

#413

yes

characteristic

101

C₂×C₄×Q₈

C₂×D₁₇

yes

characteristic

102

C₈×Q₈

—

103

C₈×Q₈

—

104

#432

—

105

#432

C₂×D₁₇

yes

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#335

—

111

#334

—

112

#324

—

113

#324

—

114

C₂×C₁₇

—

115

C₂×C₁₇

#1801

yes

characteristic

116

C₂×C₁₇

#1810

yes

characteristic

117

C₂×C₁₇

#1812

yes

characteristic

118

C₂²×Q₈

Dic₃₄

yes

characteristic

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₄×Q₈

—

123

C₄×Q₈

—

124

C₂×C₄⋊C₄

Dic₃₄

yes

characteristic

125

C₂×C₄⋊C₄

C₂²⋊D₁₇

yes

—

126

C₂×C₄²

C₄×D₁₇

yes

—

127

C₂×C₄²

C₂²×D₁₇

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₁₇

#8623

yes

characteristic, complemented, Sylow

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

#63225

yes

—

144

C₂²×C₄

#63225

yes

—

145

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

146

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₁₇

yes

—

162

C₄²

#63254

yes

—

163

C₂³

C₂×#63225

yes

characteristic

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#10587493

yes

—

174

C₂×C₄

#10587563

yes

characteristic

175

C₂×C₄

#10587564

yes

—

176

C₂×C₄

C₂²⋊C₄×D₁₇

yes

characteristic

177

C₂×C₄

#10587625

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

136

—

184

C₂²

—

185

C₂²

—

186

C₂²

#11762667

yes

characteristic

187

C₂²

C₂×#10587493

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₁₇

yes

characteristic

194

C₄

#11762792

yes

characteristic

195

C₂

—

196

C₂

#14528778

yes

characteristic

197

C₂

#14528820

yes

characteristic

198

C₂

#14528828

yes

characteristic

	subgroup	G/H
center	C₂²	#11762667
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1467	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1467	D₁₇
hypocenter	C₁₇	#8623
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#11762667

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1467

C₂

socle

C₂²×C₁₇

#380

prime radical

C₂³×C₁₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
140. C₁₇	#8623	semidirect

H ≅ G/N

type

140. C₁₇

#8623

semidirect

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

subgroup	index
C₂×Q₈×#3622	2
#14527545	2
#14527128	2
#14526653	2
C₁₇×#1467	2
#8623	17

subgroup

index

C₂×Q₈×#3622

C₁₇×#1467

Subgroups — #16927280

Conjugacy classes of subgroups

Complements

Maximal subgroups