Group #16871185 — subgroups

Subgroups — C₁₇×#13441 #16871185

Order 4352.

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×C₁₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#362	C₂	1	2	yes	maximal
4	C₁₇×#1979	C₂	1	2	yes	maximal
5	C₁₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂×C₁₇×Q₃₂	—	2	2	no	—
8	C₂×C₁₇×Q₃₂	—	2	4	no	—
9	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×#108	C₂²	1	2	yes	—
12	C₁₇×#419	—	2	2	no	—
13	C₂×C₁₇×#101	—	2	2	no	—
14	C₁₇×#405	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊C₁₆	—	2	2	no	—
16	C₁₇×#362	—	2	4	no	—
17	C₂²×Q₈×C₁₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₇	—	2	2	no	—
19	C₂×Q₁₆×C₁₇	—	2	2	no	—
20	C₂×Q₁₆×C₁₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₇	D₄	1	4	yes	—
24	C₂×M₁₆×C₁₇	—	2	2	no	—
25	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₁₇×#117	D₄	1	1	yes	characteristic
28	C₁₇×Q₃₂	—	2	8	no	—
29	C₁₇×M₃₂	—	2	2	no	—
30	C₂×C₁₆×C₁₇	—	2	2	no	—
31	C₂×C₁₆×C₁₇	—	2	1	no	—
32	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
33	C₁₇×#108	—	2	2	no	—
34	C₁₇×#108	—	2	2	no	—
35	C₁₇×#103	—	4	4	no	—
36	C₁₇×#101	—	4	4	no	—
37	C₁₇×#97	—	2	1	no	—
38	C₄×C₈×C₁₇	—	2	1	no	—
39	C₂×Q₈×C₁₇	—	2	4	no	—
40	C₂×Q₈×C₁₇	—	4	2	no	—
41	C₂×Q₈×C₁₇	—	4	2	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂×Q₈×C₁₇	—	4	2	no	—
44	C₂×Q₈×C₁₇	—	4	2	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₁₇	—	2	4	no	—
48	Q₁₆×C₁₇	—	2	4	no	—
49	Q₁₆×C₁₇	—	4	2	no	—
50	Q₁₆×C₁₇	—	4	2	no	—
51	M₁₆×C₁₇	—	2	2	no	—
52	M₁₆×C₁₇	—	4	2	no	—
53	M₁₆×C₁₇	—	4	2	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	—	4	2	no	—
56	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
61	C₂²⋊C₄×C₁₇	—	2	1	no	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₁₆×C₁₇	—	2	2	no	—
64	C₁₆×C₁₇	—	4	1	no	—
65	#13441	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₇	#135	1	1	yes	characteristic
67	Q₈×C₁₇	—	4	4	no	—
68	Q₈×C₁₇	—	4	4	no	—
69	Q₈×C₁₇	—	8	2	no	—
70	Q₈×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	—	8	2	no	—
74	C₂×C₄×C₁₇	—	8	2	no	—
75	C₂×C₄×C₁₇	#121	1	2	yes	—
76	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₁₇	—	4	1	no	—
80	C₈×C₁₇	—	8	2	no	—
81	C₈×C₁₇	#120	1	1	yes	characteristic
82	C₈×C₁₇	#123	1	2	yes	—
83	C₈×C₁₇	#124	1	1	yes	characteristic
84	C₂²×Q₃₂	C₂×C₁₇	1	1	yes	characteristic
85	#2151	C₂×C₁₇	1	1	yes	characteristic
86	C₂×#362	C₂×C₁₇	1	2	yes	—
87	#1979	C₂×C₁₇	1	2	yes	—
88	#1381	C₂×C₁₇	1	1	yes	characteristic
89	C₂²×C₁₇	—	2	2	no	—
90	C₂²×C₁₇	#463	1	2	yes	—
91	C₂²×C₁₇	#466	1	1	yes	characteristic
92	C₄×C₁₇	—	4	2	no	—
93	C₄×C₁₇	—	8	4	no	—
94	C₄×C₁₇	#394	1	1	yes	characteristic
95	C₄×C₁₇	#459	1	2	yes	—
96	C₄×C₁₇	#469	1	1	yes	characteristic
97	C₂²×Q₁₆	C₂²×C₁₇	1	2	yes	—
98	C₂×Q₃₂	—	2	4	no	—
99	C₂×Q₃₂	—	2	2	no	—
100	C₂×M₃₂	C₂²×C₁₇	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
102	C₂×#108	C₂²×C₁₇	1	2	yes	—
103	#419	—	2	2	no	—
104	C₂×#101	—	2	2	no	—
105	#405	C₂²×C₁₇	1	1	yes	characteristic
106	C₄⋊C₁₆	—	2	2	no	—
107	#362	—	2	4	no	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	#2054	1	1	yes	characteristic
110	C₂×C₁₇	#2222	1	2	yes	—
111	C₂²×Q₈	—	2	2	no	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	D₄×C₁₇	1	4	yes	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	D₄×C₁₇	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
121	#117	D₄×C₁₇	1	1	yes	characteristic
122	Q₃₂	—	2	8	no	—
123	M₃₂	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	D₄×C₁₇	1	2	yes	—
127	#108	—	2	2	no	—
128	#108	—	2	2	no	—
129	#103	—	4	4	no	—
130	#101	—	4	4	no	—
131	#97	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₁₇	#13441	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
152	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
154	C₂×C₈	#41×C₁₇	1	2	yes	—
155	C₄⋊C₄	D₈×C₁₇	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	4	1	no	—
160	C₂³	C₁₇×#135	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₁₇×#121	1	2	yes	—
170	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
171	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
172	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
173	C₈	—	4	1	no	—
174	C₈	—	8	2	no	—
175	C₈	C₁₇×#120	1	1	yes	characteristic
176	C₈	C₁₇×#123	1	2	yes	—
177	C₈	C₁₇×#124	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	C₁₇×#463	1	2	yes	—
180	C₂²	C₁₇×#466	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₄	C₁₇×#394	1	1	yes	characteristic
184	C₄	C₁₇×#459	1	2	yes	—
185	C₄	C₁₇×#469	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	C₁₇×#2054	1	1	yes	characteristic
188	C₂	C₁₇×#2222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₇×Q₃₂

C₂

yes

maximal, characteristic

C₁₇×#2151

C₂

yes

maximal, characteristic

C₂×C₁₇×#362

C₂

yes

maximal

C₁₇×#1979

C₂

yes

maximal

C₁₇×#1381

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₇

C₂²

yes

—

C₂×C₁₇×Q₃₂

—

C₂×C₁₇×Q₃₂

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₁₇×#419

—

C₂×C₁₇×#101

—

C₁₇×#405

C₂²

yes

characteristic

C₁₇×C₄⋊C₁₆

—

C₁₇×#362

—

C₂²×Q₈×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

D₄

yes

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

D₄

yes

characteristic

C₁₇×Q₃₂

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

C₁₇×#103

—

C₁₇×#101

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

C₂×C₈×C₁₇

yes

C₂×C₈×C₁₇

yes

—

C₄⋊C₄×C₁₇

D₈

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

#13441

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

#135

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

#121

yes

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#120

yes

characteristic

C₈×C₁₇

#123

yes

—

C₈×C₁₇

#124

yes

characteristic

C₂²×Q₃₂

C₂×C₁₇

yes

characteristic

#2151

C₂×C₁₇

yes

characteristic

C₂×#362

C₂×C₁₇

yes

—

#1979

C₂×C₁₇

yes

—

#1381

C₂×C₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

#463

yes

—

C₂²×C₁₇

#466

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#394

yes

characteristic

C₄×C₁₇

#459

yes

—

C₄×C₁₇

#469

yes

characteristic

C₂²×Q₁₆

C₂²×C₁₇

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

100

C₂×M₃₂

C₂²×C₁₇

yes

characteristic

101

C₂²×C₁₆

C₂²×C₁₇

yes

characteristic

102

C₂×#108

C₂²×C₁₇

yes

—

103

#419

—

104

C₂×#101

—

105

#405

C₂²×C₁₇

yes

characteristic

106

C₄⋊C₁₆

—

107

#362

—

108

C₂×C₁₇

—

109

C₂×C₁₇

#2054

yes

characteristic

110

C₂×C₁₇

#2222

yes

—

111

C₂²×Q₈

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

D₄×C₁₇

yes

—

118

C₂×M₁₆

—

119

C₂×M₁₆

D₄×C₁₇

yes

characteristic

120

C₂²×C₈

C₂³×C₁₇

yes

characteristic

121

#117

D₄×C₁₇

yes

characteristic

122

Q₃₂

—

123

M₃₂

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

D₄×C₁₇

yes

—

127

#108

—

128

#108

—

129

#103

—

130

#101

—

131

#97

—

132

C₄×C₈

—

133

C₁₇

#13441

yes

characteristic, direct product factor, Sylow

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×D₄×C₁₇

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂×D₄×C₁₇

yes

—

152

C₂×C₈

C₂×D₄×C₁₇

yes

characteristic

153

C₂×C₈

#41×C₁₇

yes

characteristic

154

C₂×C₈

#41×C₁₇

yes

—

155

C₄⋊C₄

D₈×C₁₇

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

C₁₇×#135

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₁₇×#121

yes

—

170

C₂×C₄

C₁₇×#121

yes

characteristic

171

C₂×C₄

C₂×D₈×C₁₇

yes

characteristic

172

C₂×C₄

C₁₇×AGL(1,ℤ/8)

yes

—

173

C₈

—

174

C₈

—

175

C₈

C₁₇×#120

yes

characteristic

176

C₈

C₁₇×#123

yes

—

177

C₈

C₁₇×#124

yes

characteristic

178

C₂²

—

179

C₂²

C₁₇×#463

yes

—

180

C₂²

C₁₇×#466

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

C₁₇×#394

yes

characteristic

184

C₄

C₁₇×#459

yes

—

185

C₄

C₁₇×#469

yes

characteristic

186

C₂

—

187

C₂

C₁₇×#2054

yes

characteristic

188

C₂

C₁₇×#2222

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂³×C₁₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₇

prime radical

C₂³×C₁₇

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#405 ◁ G

N	H ≅ G/N	type
65. #13441	C₁₇	direct (N × H)
133. C₁₇	#13441	direct (N × H)

H ≅ G/N

type

65. #13441

C₁₇

direct (N × H)

133. C₁₇

#13441

direct (N × H)

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

subgroup	index
C₂²×C₁₇×Q₃₂	2
C₁₇×#2151	2
C₂×C₁₇×#362	2
C₁₇×#1979	2
C₁₇×#1381	2
#13441	17

subgroup

index

C₂²×C₁₇×Q₃₂

C₁₇×#2151

C₂×C₁₇×#362

C₁₇×#1979

C₁₇×#1381

#13441

Subgroups — C₁₇×#13441 #16871185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#13441 #16871185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13441 #16871185