Group #16868110 — subgroups

Subgroups — C₁₇×#10366 #16868110

Order 4352.

402 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₁₇×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#1962	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1517	C₂	1	2	yes	maximal
4	C₁₇×#1286	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#532	C₂²	1	1	yes	characteristic
7	C₁₇×#431	C₂²	1	2	yes	—
8	C₁₇×#423	C₂²	1	1	yes	characteristic
9	C₂×C₁₇×#104	—	2	2	no	—
10	C₁₇×#389	—	2	2	no	—
11	C₁₇×#387	—	2	4	no	—
12	C₁₇×#382	C₂²	1	2	yes	—
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#327	—	2	4	no	—
15	C₂×#41×C₁₇	—	2	1	no	—
16	C₂×M₁₆×C₁₇	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
18	C₁₇×#127	Q₈	1	1	yes	characteristic
19	C₁₇×#124	—	2	1	no	—
20	C₁₇×#121	—	2	2	no	—
21	C₄×D₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂×C₄²×C₁₇	—	2	2	no	—
25	C₂×C₄²×C₁₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₁₇×C₄⋊C₈	—	2	2	no	—
28	C₁₇×C₄⋊C₈	—	2	2	no	—
29	C₁₇×#104	—	4	4	no	—
30	C₁₇×C₂²⋊C₈	—	2	2	no	—
31	C₄×C₈×C₁₇	—	2	2	no	—
32	C₁₇×#95	—	2	4	no	—
33	C₁₇×#95	—	2	2	no	—
34	C₁₇×#95	D₄	1	4	yes	—
35	#41×C₁₇	—	4	2	no	—
36	#41×C₁₇	—	4	2	no	—
37	C₂×Q₈×C₁₇	—	2	1	no	—
38	C₂×D₄×C₁₇	—	2	1	no	—
39	C₂×D₄×C₁₇	—	2	2	no	—
40	C₂×D₄×C₁₇	—	4	1	no	—
41	C₂²×C₄×C₁₇	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	4	no	—
43	C₂²×C₄×C₁₇	—	4	1	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₁₇	—	4	2	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	—	4	2	no	—
49	C₄⋊C₄×C₁₇	—	2	2	no	—
50	C₄⋊C₄×C₁₇	—	4	8	no	—
51	C₄⋊C₄×C₁₇	—	4	2	no	—
52	C₄⋊C₄×C₁₇	—	4	2	no	—
53	C₂²⋊C₄×C₁₇	—	4	2	no	—
54	C₄²×C₁₇	—	2	2	no	—
55	C₄²×C₁₇	—	2	2	no	—
56	C₄²×C₁₇	—	4	4	no	—
57	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₇	#41	1	1	yes	characteristic
59	C₄²×C₁₇	#41	1	2	yes	—
60	#10366	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₇	—	4	1	no	—
62	C₂³×C₁₇	#120	1	1	yes	characteristic
63	Q₈×C₁₇	—	4	2	no	—
64	D₄×C₁₇	—	4	2	no	—
65	D₄×C₁₇	—	8	2	no	—
66	D₄×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	—	4	2	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	8	no	—
72	C₂×C₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	—	4	4	no	—
74	C₂×C₄×C₁₇	—	8	1	no	—
75	C₂×C₄×C₁₇	—	8	1	no	—
76	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#124	1	2	yes	—
79	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#135	1	2	yes	—
81	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
82	C₈×C₁₇	—	8	2	no	—
83	C₈×C₁₇	—	8	1	no	—
84	#2864	C₂×C₁₇	1	1	yes	characteristic, complemented
85	#1962	C₂×C₁₇	1	1	yes	characteristic
86	#1517	C₂×C₁₇	1	2	yes	—
87	#1286	C₂×C₁₇	1	2	yes	complemented
88	#1276	C₂×C₁₇	1	1	yes	characteristic, complemented
89	C₂²×C₁₇	—	2	2	no	—
90	C₂²×C₁₇	—	8	1	no	—
91	C₂²×C₁₇	—	8	2	no	—
92	C₂²×C₁₇	#393	1	1	yes	characteristic
93	C₂²×C₁₇	#449	1	2	yes	—
94	C₄×C₁₇	—	2	2	no	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	4	4	no	—
97	C₄×C₁₇	—	8	4	no	—
98	C₄×C₁₇	—	8	1	no	—
99	C₄×C₁₇	#478	1	2	yes	—
100	C₄×C₁₇	#488	1	1	yes	characteristic
101	C₄×C₁₇	#490	1	1	yes	characteristic
102	#532	C₂²×C₁₇	1	1	yes	characteristic
103	#431	C₂²×C₁₇	1	2	yes	—
104	#423	C₂²×C₁₇	1	1	yes	characteristic
105	C₂×#104	—	2	2	no	—
106	#389	—	2	2	no	—
107	#387	—	2	4	no	—
108	#382	C₂²×C₁₇	1	2	yes	—
109	C₄³	C₂²×C₁₇	1	1	yes	characteristic
110	#327	—	2	4	no	—
111	C₂×C₁₇	—	2	2	no	—
112	C₂×C₁₇	—	16	1	no	—
113	C₂×C₁₇	#2025	1	1	yes	characteristic
114	C₂×C₁₇	#2031	1	2	yes	—
115	C₂×#41	—	2	1	no	—
116	C₂×M₁₆	—	2	1	no	—
117	C₄⋊Q₈	Q₈×C₁₇	1	1	yes	characteristic
118	#127	Q₈×C₁₇	1	1	yes	characteristic
119	#124	—	2	1	no	—
120	#121	—	2	2	no	—
121	C₄×D₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	D₄×C₁₇	1	2	yes	—
126	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	#104	—	4	4	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	#95	—	2	4	no	—
133	#95	—	2	2	no	—
134	#95	D₄×C₁₇	1	4	yes	—
135	C₁₇	#10366	1	1	yes	characteristic, direct product factor, Sylow
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×D₄	—	2	1	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	4	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₂×Q₈×C₁₇	1	1	yes	characteristic
159	C₄²	#41×C₁₇	1	1	yes	characteristic
160	C₄²	#41×C₁₇	1	2	yes	—
161	C₂³	—	4	1	no	—
162	C₂³	C₁₇×#120	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	C₁₇×C₂²⋊Q₈	1	2	yes	—
177	C₂×C₄	C₁₇×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	C₁₇×#124	1	2	yes	—
179	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
180	C₂×C₄	C₁₇×#135	1	2	yes	—
181	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	8	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	C₁₇×#393	1	1	yes	characteristic
188	C₂²	C₁₇×#449	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	4	no	—
193	C₄	—	8	1	no	—
194	C₄	C₁₇×#478	1	2	yes	—
195	C₄	C₁₇×#488	1	1	yes	characteristic
196	C₄	C₁₇×#490	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	16	1	no	—
199	C₂	C₁₇×#2025	1	1	yes	characteristic
200	C₂	C₁₇×#2031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#2864

C₂

yes

maximal, characteristic, complemented

C₁₇×#1962

C₂

yes

maximal, characteristic

C₁₇×#1517

C₂

yes

maximal

C₁₇×#1286

C₂

yes

maximal, complemented

C₁₇×#1276

C₂

yes

maximal, characteristic, complemented

C₁₇×#532

C₂²

yes

characteristic

C₁₇×#431

C₂²

yes

—

C₁₇×#423

C₂²

yes

characteristic

C₂×C₁₇×#104

—

C₁₇×#389

—

C₁₇×#387

—

C₁₇×#382

C₂²

yes

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#327

—

C₂×#41×C₁₇

—

C₂×M₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

Q₈

yes

characteristic

C₁₇×#127

Q₈

yes

characteristic

C₁₇×#124

—

C₁₇×#121

—

C₄×D₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#104

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

D₄

yes

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

#10366

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

#120

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×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

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#124

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

#2864

C₂×C₁₇

yes

characteristic, complemented

#1962

C₂×C₁₇

yes

characteristic

#1517

C₂×C₁₇

yes

—

#1286

C₂×C₁₇

yes

complemented

#1276

C₂×C₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#393

yes

characteristic

C₂²×C₁₇

#449

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#478

yes

—

100

C₄×C₁₇

#488

yes

characteristic

101

C₄×C₁₇

#490

yes

characteristic

102

#532

C₂²×C₁₇

yes

characteristic

103

#431

C₂²×C₁₇

yes

—

104

#423

C₂²×C₁₇

yes

characteristic

105

C₂×#104

—

106

#389

—

107

#387

—

108

#382

C₂²×C₁₇

yes

—

109

C₄³

C₂²×C₁₇

yes

characteristic

110

#327

—

111

C₂×C₁₇

—

112

C₂×C₁₇

—

113

C₂×C₁₇

#2025

yes

characteristic

114

C₂×C₁₇

#2031

yes

—

115

C₂×#41

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

Q₈×C₁₇

yes

characteristic

118

#127

Q₈×C₁₇

yes

characteristic

119

#124

—

120

#121

—

121

C₄×D₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

—

125

C₂×C₄²

D₄×C₁₇

yes

—

126

C₂×C₄²

C₂³×C₁₇

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

#104

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

#95

—

133

#95

—

134

#95

D₄×C₁₇

yes

—

135

C₁₇

#10366

yes

characteristic, direct product factor, Sylow

136

#41

—

137

#41

—

138

C₂×Q₈

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₁₇

yes

—

146

C₂²×C₄

C₂×D₄×C₁₇

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×Q₈×C₁₇

yes

characteristic

159

C₄²

#41×C₁₇

yes

characteristic

160

C₄²

#41×C₁₇

yes

—

161

C₂³

—

162

C₂³

C₁₇×#120

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₁₇×C₂²⋊Q₈

yes

—

177

C₂×C₄

C₁₇×C₂²⋊Q₈

yes

characteristic

178

C₂×C₄

C₁₇×#124

yes

—

179

C₂×C₄

C₁₇×#124

yes

characteristic

180

C₂×C₄

C₁₇×#135

yes

—

181

C₂×C₄

C₁₇×AGL(1,ℤ/8)

yes

—

182

C₈

—

183

C₈

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₁₇×#393

yes

characteristic

188

C₂²

C₁₇×#449

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₁₇×#478

yes

—

195

C₄

C₁₇×#488

yes

characteristic

196

C₄

C₁₇×#490

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

C₁₇×#2025

yes

characteristic

200

C₂

C₁₇×#2031

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₁₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#393

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

prime radical

C₁₇×#532

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
1. C₁₇×#2864	C₂	semidirect
4. C₁₇×#1286	C₂	semidirect
5. C₁₇×#1276	C₂	semidirect
60. #10366	C₁₇	direct (N × H)
84. #2864	C₂×C₁₇	semidirect
87. #1286	C₂×C₁₇	semidirect
88. #1276	C₂×C₁₇	semidirect
135. C₁₇	#10366	direct (N × H)

H ≅ G/N

type

1. C₁₇×#2864

C₂

semidirect

4. C₁₇×#1286

C₂

semidirect

5. C₁₇×#1276

C₂

semidirect

60. #10366

C₁₇

direct (N × H)

84. #2864

C₂×C₁₇

semidirect

87. #1286

C₂×C₁₇

semidirect

88. #1276

C₂×C₁₇

semidirect

135. C₁₇

#10366

direct (N × H)

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

subgroup	index
C₁₇×#2864	2
C₁₇×#1962	2
C₁₇×#1517	2
C₁₇×#1286	2
C₁₇×#1276	2
#10366	17

subgroup

index

C₁₇×#2864

C₁₇×#1962

C₁₇×#1517

C₁₇×#1286

C₁₇×#1276

#10366

Subgroups — C₁₇×#10366 #16868110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#10366 #16868110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10366 #16868110