Group #16868376 — subgroups

Subgroups — C₁₇×#10632 #16868376

Order 4352.

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

normal subgroups only

This group has 310 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×#1960	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1849	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1546	C₂	1	2	yes	maximal
4	C₁₇×#1286	C₂	1	2	yes	maximal
5	C₁₇×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₁₇×#431	C₂²	1	2	yes	—
8	C₂×C₁₇×#108	—	2	1	no	—
9	C₂×C₁₇×#108	—	2	1	no	—
10	C₁₇×#423	C₂²	1	2	yes	—
11	C₂×C₁₇×#103	—	2	1	no	—
12	C₂×C₁₇×#103	—	2	1	no	—
13	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
14	C₁₇×#382	C₂²	1	1	yes	characteristic
15	C₁₇×#332	—	2	4	no	—
16	C₂²×Q₈×C₁₇	—	2	1	no	—
17	C₂×M₁₆×C₁₇	—	2	2	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	1	no	—
22	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
25	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	C₁₇×#108	—	4	2	no	—
27	C₁₇×#108	—	4	2	no	—
28	C₁₇×C₄⋊C₈	—	2	2	no	—
29	C₁₇×C₄⋊C₈	—	2	2	no	—
30	C₁₇×C₄⋊C₈	—	2	2	no	—
31	C₁₇×#103	—	4	2	no	—
32	C₁₇×#103	—	4	2	no	—
33	C₁₇×C₂²⋊C₈	—	2	2	no	—
34	C₁₇×#97	—	2	2	no	—
35	C₄×C₈×C₁₇	—	2	2	no	—
36	C₁₇×#95	—	2	2	no	—
37	C₂×Q₈×C₁₇	—	2	2	no	—
38	C₂×Q₈×C₁₇	—	4	1	no	—
39	C₂×Q₈×C₁₇	—	4	1	no	—
40	C₂×Q₈×C₁₇	—	4	1	no	—
41	C₂×Q₈×C₁₇	—	4	1	no	—
42	C₂×Q₈×C₁₇	—	4	1	no	—
43	C₂²×C₄×C₁₇	—	4	1	no	—
44	C₂²×C₄×C₁₇	—	4	1	no	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	#41	1	2	yes	—
47	M₁₆×C₁₇	—	4	2	no	—
48	M₁₆×C₁₇	—	4	2	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	—	2	1	no	—
52	C₂×C₈×C₁₇	—	2	1	no	—
53	C₂×C₈×C₁₇	—	4	2	no	—
54	C₂×C₈×C₁₇	—	4	2	no	—
55	C₂×C₈×C₁₇	D₈	1	2	yes	—
56	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
57	C₄⋊C₄×C₁₇	—	2	4	no	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄⋊C₄×C₁₇	—	4	1	no	—
60	C₄⋊C₄×C₁₇	—	4	1	no	—
61	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₇	#41	1	1	yes	characteristic
63	C₄²×C₁₇	#41	1	2	yes	—
64	#10632	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₇	#124	1	1	yes	characteristic
66	Q₈×C₁₇	—	4	2	no	—
67	Q₈×C₁₇	—	4	2	no	—
68	Q₈×C₁₇	—	8	1	no	—
69	Q₈×C₁₇	—	8	1	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	8	1	no	—
73	C₂×C₄×C₁₇	—	8	1	no	—
74	C₂×C₄×C₁₇	—	8	2	no	—
75	C₂×C₄×C₁₇	—	8	1	no	—
76	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#123	1	2	yes	—
78	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#126	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
84	C₈×C₁₇	—	2	2	no	—
85	C₈×C₁₇	—	2	2	no	—
86	C₈×C₁₇	—	8	2	no	—
87	C₈×C₁₇	—	8	2	no	—
88	#1960	C₂×C₁₇	1	1	yes	characteristic
89	#1849	C₂×C₁₇	1	1	yes	characteristic
90	#1546	C₂×C₁₇	1	2	yes	—
91	#1286	C₂×C₁₇	1	2	yes	—
92	#1281	C₂×C₁₇	1	1	yes	characteristic
93	C₂²×C₁₇	—	2	2	no	—
94	C₂²×C₁₇	#397	1	1	yes	characteristic
95	C₂²×C₁₇	#486	1	1	yes	characteristic
96	C₂²×C₁₇	#489	1	1	yes	characteristic
97	C₄×C₁₇	—	2	2	no	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	8	2	no	—
100	C₄×C₁₇	—	16	1	no	—
101	C₄×C₁₇	#459	1	1	yes	characteristic
102	C₄×C₁₇	#461	1	1	yes	characteristic
103	C₄×C₁₇	#480	1	1	yes	characteristic
104	C₄×C₁₇	#481	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
106	#431	C₂²×C₁₇	1	2	yes	—
107	C₂×#108	—	2	1	no	—
108	C₂×#108	—	2	1	no	—
109	#423	C₂²×C₁₇	1	2	yes	—
110	C₂×#103	—	2	1	no	—
111	C₂×#103	—	2	1	no	—
112	C₂×#97	C₂²×C₁₇	1	1	yes	characteristic
113	#382	C₂²×C₁₇	1	1	yes	characteristic
114	#332	—	2	4	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	—	2	1	no	—
117	C₂×C₁₇	#2071	1	1	yes	characteristic
118	C₂×C₁₇	#2082	1	2	yes	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	1	no	—
125	C₄⋊Q₈	D₄×C₁₇	1	2	yes	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
129	#108	—	4	2	no	—
130	#108	—	4	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	#103	—	4	2	no	—
135	#103	—	4	2	no	—
136	C₂²⋊C₈	—	2	2	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	#95	—	2	2	no	—
140	C₁₇	#10632	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
150	C₂²×C₄	#41×C₁₇	1	2	yes	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	D₈×C₁₇	1	2	yes	—
160	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
166	C₄²	#41×C₁₇	1	2	yes	—
167	C₄²	#41×C₁₇	1	1	yes	characteristic
168	C₂³	C₁₇×#124	1	1	yes	characteristic
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
180	C₂×C₄	C₁₇×#123	1	2	yes	—
181	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
182	C₂×C₄	C₁₇×#126	1	2	yes	—
183	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
185	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×C₄	C₁₇×#137	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	8	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	C₁₇×#397	1	1	yes	characteristic
193	C₂²	C₁₇×#486	1	1	yes	characteristic
194	C₂²	C₁₇×#489	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	8	2	no	—
198	C₄	—	16	1	no	—
199	C₄	C₁₇×#459	1	1	yes	characteristic
200	C₄	C₁₇×#461	1	1	yes	characteristic
201	C₄	C₁₇×#480	1	1	yes	characteristic
202	C₄	C₁₇×#481	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	C₁₇×#2071	1	1	yes	characteristic
206	C₂	C₁₇×#2082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#1960

C₂

yes

maximal, characteristic

C₁₇×#1849

C₂

yes

maximal, characteristic

C₁₇×#1546

C₂

yes

maximal

C₁₇×#1286

C₂

yes

maximal

C₁₇×#1281

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₁₇×#431

C₂²

yes

—

C₂×C₁₇×#108

—

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

—

C₂×C₁₇×#103

—

C₂×C₁₇×#103

—

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

characteristic

C₁₇×#332

—

C₂²×Q₈×C₁₇

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×#108

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#103

—

C₁₇×#103

—

C₁₇×C₂²⋊C₈

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

C₁₇×#95

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

#10632

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

#124

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#123

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#1960

C₂×C₁₇

yes

characteristic

#1849

C₂×C₁₇

yes

characteristic

#1546

C₂×C₁₇

yes

—

#1286

C₂×C₁₇

yes

—

#1281

C₂×C₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

100

C₄×C₁₇

—

101

C₄×C₁₇

#459

yes

characteristic

102

C₄×C₁₇

#461

yes

characteristic

103

C₄×C₁₇

#480

yes

characteristic

104

C₄×C₁₇

#481

yes

characteristic

105

C₂×C₄⋊Q₈

C₂²×C₁₇

yes

characteristic

106

#431

C₂²×C₁₇

yes

—

107

C₂×#108

—

108

C₂×#108

—

109

#423

C₂²×C₁₇

yes

—

110

C₂×#103

—

111

C₂×#103

—

112

C₂×#97

C₂²×C₁₇

yes

characteristic

113

#382

C₂²×C₁₇

yes

characteristic

114

#332

—

115

C₂×C₁₇

—

116

C₂×C₁₇

—

117

C₂×C₁₇

#2071

yes

characteristic

118

C₂×C₁₇

#2082

yes

—

119

C₂²×Q₈

—

120

C₂×M₁₆

—

121

C₂²×C₈

D₄×C₁₇

yes

characteristic

122

C₂²×C₈

D₄×C₁₇

yes

characteristic

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

D₄×C₁₇

yes

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂³×C₁₇

yes

characteristic

129

#108

—

130

#108

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

#103

—

135

#103

—

136

C₂²⋊C₈

—

137

#97

—

138

C₄×C₈

—

139

#95

—

140

C₁₇

#10632

yes

characteristic, direct product factor, Sylow

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂×D₄×C₁₇

yes

characteristic

150

C₂²×C₄

#41×C₁₇

yes

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

D₈×C₁₇

yes

—

160

C₂×C₈

SD₁₆×C₁₇

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

C₂×D₄×C₁₇

yes

characteristic

166

C₄²

#41×C₁₇

yes

—

167

C₄²

#41×C₁₇

yes

characteristic

168

C₂³

C₁₇×#124

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₁₇×#121

yes

characteristic

180

C₂×C₄

C₁₇×#123

yes

—

181

C₂×C₄

C₁₇×#123

yes

characteristic

182

C₂×C₄

C₁₇×#126

yes

—

183

C₂×C₄

C₂×D₈×C₁₇

yes

characteristic

184

C₂×C₄

C₂×SD₁₆×C₁₇

yes

characteristic

185

C₂×C₄

C₁₇×AGL(1,ℤ/8)

yes

characteristic

186

C₂×C₄

C₁₇×#137

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

C₁₇×#397

yes

characteristic

193

C₂²

C₁₇×#486

yes

characteristic

194

C₂²

C₁₇×#489

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

C₁₇×#459

yes

characteristic

200

C₄

C₁₇×#461

yes

characteristic

201

C₄

C₁₇×#480

yes

characteristic

202

C₄

C₁₇×#481

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

C₁₇×#2071

yes

characteristic

206

C₂

C₁₇×#2082

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂³×C₁₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

prime radical

C₂³×C₁₇

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

N	H ≅ G/N	type
64. #10632	C₁₇	direct (N × H)
140. C₁₇	#10632	direct (N × H)

H ≅ G/N

type

64. #10632

C₁₇

direct (N × H)

140. C₁₇

#10632

direct (N × H)

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

subgroup	index
C₁₇×#1960	2
C₁₇×#1849	2
C₁₇×#1546	2
C₁₇×#1286	2
C₁₇×#1281	2
#10632	17

subgroup

index

C₁₇×#1960

C₁₇×#1849

C₁₇×#1546

C₁₇×#1286

C₁₇×#1281

#10632

Subgroups — C₁₇×#10632 #16868376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#10632 #16868376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10632 #16868376