Group #17189641 — subgroups

Subgroups — C₂×#14531029 #17189641

Order 4352.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#11763211	C₂	1	1	yes	maximal, characteristic
2	#14531029	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762613	C₂	1	4	yes	maximal
4	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587556	—	2	1	no	—
7	C₂²×#63214	C₂²	1	2	yes	—
8	#11763211	—	2	2	no	—
9	#11763211	C₂²	1	4	yes	—
10	C₂×#10587489	C₂²	1	2	yes	—
11	C₂×#10587488	—	2	1	no	—
12	#11762613	C₂²	1	16	yes	—
13	#11762585	C₂²	1	4	yes	—
14	C₂²×Q₁₆×C₁₇	—	2	1	no	—
15	C₂²×Q₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
17	C₁₇×#494	—	2	2	no	—
18	C₁₇×#494	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₇	—	2	4	no	—
23	C₂×Q₁₆×C₁₇	—	2	8	no	—
24	C₂×Q₁₆×C₁₇	—	4	4	no	—
25	C₂×Q₁₆×C₁₇	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×C₄⋊Q₈	—	2	4	no	—
29	C₁₇×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
31	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₇	—	2	2	no	—
33	C₂×#63225	—	4	2	no	—
34	#10587556	—	2	4	no	—
35	#10587556	—	4	2	no	—
36	C₄×C₈×C₁₇	D₄	1	2	yes	—
37	C₄×C₈×C₁₇	C₂³	1	2	yes	—
38	#10587489	—	2	2	no	—
39	#10587489	C₂³	1	4	yes	—
40	#10587488	—	2	4	no	—
41	C₂×#63214	—	2	4	no	—
42	C₂×#63214	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₇	—	2	4	no	—
44	C₂×Q₈×C₁₇	—	4	2	no	—
45	C₂×Q₈×C₁₇	—	4	4	no	—
46	C₂×Q₈×C₁₇	—	4	4	no	—
47	C₂²×C₄×C₁₇	—	4	2	no	—
48	C₂²×C₄×C₁₇	D₈	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₇	—	4	2	no	—
51	Q₁₆×C₁₇	—	2	8	no	—
52	Q₁₆×C₁₇	—	4	4	no	—
53	Q₁₆×C₁₇	—	4	8	no	—
54	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
55	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₇	#41	1	4	yes	—
58	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇	—	4	8	no	—
60	C₄²×C₁₇	D₈	1	2	yes	—
61	C₄²×C₁₇	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₁₇	—	2	4	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	#63225	—	4	8	no	—
65	#63214	—	2	4	no	—
66	C₂×#2245	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₇	—	4	4	no	—
69	Q₈×C₁₇	—	4	4	no	—
70	Q₈×C₁₇	—	8	4	no	—
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	—	8	4	no	—
74	C₂×C₄×C₁₇	#124	1	2	yes	—
75	C₂×C₄×C₁₇	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₁₇	—	4	4	no	—
80	C₂×Dic₁₇	—	8	4	no	—
81	C₈×C₁₇	#124	1	2	yes	—
82	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
83	C₈×C₁₇	C₂×#41	1	2	yes	—
84	C₂×#499	—	17	1	no	—
85	C₂×#494	D₁₇	1	1	yes	characteristic
86	#2245	—	17	8	no	—
87	C₂×#358	—	17	4	no	—
88	C₂×#346	—	17	1	no	—
89	C₂²×C₁₇	#486	1	2	yes	—
90	C₂²×C₁₇	#510	1	4	yes	—
91	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₁₇	—	2	4	no	—
93	C₄×C₁₇	—	8	4	no	—
94	C₄×C₁₇	#486	1	2	yes	—
95	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
96	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
97	Dic₁₇	—	8	4	no	—
98	C₂²×Q₁₆	—	2	1	no	—
99	C₂²×Q₁₆	C₂×D₁₇	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂×#125	—	34	1	no	—
102	C₂²×C₁₆	—	17	2	no	—
103	#499	—	17	4	no	—
104	#499	—	34	2	no	—
105	#494	—	2	2	no	—
106	#494	C₂×D₁₇	1	4	yes	—
107	C₂×#107	—	17	2	no	—
108	C₂×#106	—	34	1	no	—
109	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
110	#358	—	17	16	no	—
111	#346	—	17	4	no	—
112	C₂×C₁₇	#2245	1	4	yes	complemented
113	C₂×C₁₇	C₂×#486	1	1	yes	characteristic
114	C₂×C₁₇	C₂×#510	1	2	yes	—
115	C₂²×Q₈	—	2	2	no	—
116	C₂×Q₁₆	—	2	8	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂²×D₁₇	1	4	yes	—
121	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	2	no	—
125	#125	—	34	4	no	—
126	#125	—	68	2	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄⋊C₄	—	34	2	no	—
129	C₂×C₄⋊C₄	—	68	2	no	—
130	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
131	C₂×C₁₆	—	17	4	no	—
132	C₂×C₁₆	—	34	4	no	—
133	#107	—	17	4	no	—
134	#107	—	34	2	no	—
135	#106	—	34	4	no	—
136	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
137	C₄×C₈	C₂²×D₁₇	1	2	yes	—
138	C₁₇	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	68	2	no	—
145	C₂²×C₄	#63228	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
147	Q₁₆	—	2	8	no	—
148	Q₁₆	—	4	8	no	—
149	Q₁₆	—	4	4	no	—
150	C₂×C₈	#63229	1	4	yes	—
151	C₂×C₈	#63254	1	4	yes	—
152	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
154	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	34	4	no	—
157	C₄⋊C₄	—	68	2	no	—
158	C₄⋊C₄	—	68	8	no	—
159	C₄²	#63228	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
161	C₁₆	—	34	4	no	—
162	C₂³	C₂×#63228	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	Q₈	—	4	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	68	4	no	—
170	C₂×C₄	—	136	4	no	—
171	C₂×C₄	C₂×#63228	1	4	yes	—
172	C₂×C₄	C₂×#63228	1	1	yes	characteristic
173	C₂×C₄	C₂×#63229	1	2	yes	—
174	C₂×C₄	#10587622	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
176	C₈	C₂×#63229	1	4	yes	—
177	C₈	#10587622	1	2	yes	—
178	C₈	C₂×#63254	1	2	yes	—
179	C₂²	#11763155	1	2	yes	—
180	C₂²	#11763250	1	4	yes	—
181	C₂²	C₂²×#63228	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	136	4	no	—
185	C₄	#11763155	1	2	yes	—
186	C₄	C₂×#10587622	1	1	yes	characteristic
187	C₄	C₂²×#63229	1	1	yes	characteristic
188	C₂	#14531029	1	4	yes	direct product factor
189	C₂	C₂×#11763155	1	1	yes	characteristic
190	C₂	C₂×#11763250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763211

C₂

yes

maximal, characteristic

#14531029

C₂

yes

maximal, direct product factor

C₂×#11762613

C₂

yes

maximal

C₂×#11762585

C₂

yes

maximal, characteristic

C₂×C₁₇×#494

C₂

yes

maximal, characteristic

C₂×#10587556

—

C₂²×#63214

C₂²

yes

—

#11763211

—

#11763211

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

C₂×#10587488

—

#11762613

C₂²

yes

—

#11762585

C₂²

yes

—

C₂²×Q₁₆×C₁₇

—

C₂²×Q₁₆×C₁₇

C₂²

yes

—

C₂×C₁₇×C₄⋊Q₈

—

C₁₇×#494

—

C₁₇×#494

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

C₂³

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×#63225

—

#10587556

—

#10587556

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

#10587489

—

#10587489

C₂³

yes

—

#10587488

—

C₂×#63214

—

C₂×#63214

C₂³

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₈

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

C₂³×C₁₇

C₂×D₈

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#124

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#124

yes

—

C₈×C₁₇

C₂×SD₁₆

yes

—

C₈×C₁₇

C₂×#41

yes

—

C₂×#499

—

C₂×#494

D₁₇

yes

—

—

—

C₂²×C₁₇

#486

yes

—

C₂²×C₁₇

#510

yes

—

C₂²×C₁₇

C₂²×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#486

yes

—

C₄×C₁₇

C₂×#124

yes

characteristic

C₄×C₁₇

C₂²×SD₁₆

yes

characteristic

Dic₁₇

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×D₁₇

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×#125

—

102

C₂²×C₁₆

—

103

#499

—

104

#499

—

105

#494

—

106

#494

C₂×D₁₇

yes

—

107

C₂×#107

—

108

C₂×#106

—

109

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

110

#358

—

111

#346

—

112

C₂×C₁₇

#2245

yes

complemented

113

C₂×C₁₇

C₂×#486

yes

characteristic

114

C₂×C₁₇

C₂×#510

yes

—

115

C₂²×Q₈

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂²×D₁₇

yes

—

121

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

122

C₂²×C₈

C₂²×D₁₇

yes

characteristic

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

#125

—

126

#125

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

#107

—

134

#107

—

135

#106

—

136

C₄×C₈

C₂²⋊D₁₇

yes

—

137

C₄×C₈

C₂²×D₁₇

yes

—

138

C₁₇

C₂×#2245

yes

characteristic, complemented, Sylow

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

#63228

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

149

Q₁₆

—

150

C₂×C₈

#63229

yes

—

151

C₂×C₈

#63254

yes

—

152

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

153

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

154

C₂×C₈

C₂³×D₁₇

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

#63228

yes

—

160

C₄²

C₂×C₂²⋊D₁₇

yes

—

161

C₁₆

—

162

C₂³

C₂×#63228

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

136

—

171

C₂×C₄

C₂×#63228

yes

—

172

C₂×C₄

C₂×#63228

yes

characteristic

173

C₂×C₄

C₂×#63229

yes

—

174

C₂×C₄

#10587622

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₁₇

yes

characteristic

176

C₈

C₂×#63229

yes

—

177

C₈

#10587622

yes

—

178

C₈

C₂×#63254

yes

—

179

C₂²

#11763155

yes

—

180

C₂²

#11763250

yes

—

181

C₂²

C₂²×#63228

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

136

—

185

C₄

#11763155

yes

—

186

C₄

C₂×#10587622

yes

characteristic

187

C₄

C₂²×#63229

yes

characteristic

188

C₂

#14531029

yes

direct product factor

189

C₂

C₂×#11763155

yes

characteristic

190

C₂

C₂×#11763250

yes

—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	C₂×#2245
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#63228

commutator, cosocle

C₂×C₈×C₁₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₇

Fitting

C₂×C₁₇×#494

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
2. #14531029	C₂	direct (N × H)
112. C₂×C₁₇	#2245	semidirect
138. C₁₇	C₂×#2245	semidirect
188. C₂	#14531029	direct (N × H)

H ≅ G/N

type

2. #14531029

C₂

direct (N × H)

112. C₂×C₁₇

#2245

semidirect

138. C₁₇

C₂×#2245

semidirect

188. C₂

#14531029

direct (N × H)

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

subgroup	index
C₂×#11763211	2
#14531029	2
C₂×#11762613	2
C₂×#11762585	2
C₂×C₁₇×#494	2
C₂×#2245	17

subgroup

index

C₂×C₁₇×#494

Subgroups — C₂×#14531029 #17189641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14531029 #17189641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531029 #17189641