Group #16865349 — subgroups

Subgroups — C₁₇×#7605 #16865349

Order 4352.

440 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1839	C₂	1	2	yes	maximal
3	C₁₇×#1272	C₂	1	4	yes	maximal
4	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Q₈×C₁₇	C₄	1	2	yes	—
6	C₁₇×#391	C₂²	1	2	yes	—
7	C₁₇×#389	—	2	2	no	—
8	C₁₇×#384	C₄	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
10	C₄³×C₁₇	C₂²	1	1	yes	characteristic
11	C₁₇×#336	—	2	2	no	—
12	C₁₇×#336	—	2	2	no	—
13	C₁₇×#336	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₁₇	—	2	4	no	—
16	C₁₇×C₄⋊Q₈	—	2	2	no	—
17	C₁₇×C₄⋊Q₈	C₈	1	4	yes	complemented
18	C₄×Q₈×C₁₇	—	4	4	no	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	—	2	2	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	2	yes	—
27	C₂×C₄²×C₁₇	D₄	1	2	yes	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₁₇×#95	—	2	4	no	—
30	C₂×Q₈×C₁₇	—	2	2	no	—
31	C₂×Q₈×C₁₇	—	4	2	no	—
32	C₂×Q₈×C₁₇	—	4	2	no	—
33	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₁₇	—	2	2	no	—
35	C₂²×C₄×C₁₇	—	2	2	no	—
36	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₁₇	—	4	4	no	—
42	C₂×C₈×C₁₇	—	4	4	no	—
43	C₂×C₈×C₁₇	—	4	4	no	—
44	C₄⋊C₄×C₁₇	—	2	8	no	—
45	C₄⋊C₄×C₁₇	—	4	2	no	—
46	C₄⋊C₄×C₁₇	—	4	4	no	—
47	C₄⋊C₄×C₁₇	—	4	4	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	4	2	no	—
51	C₄²×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	—	4	4	no	—
53	C₄²×C₁₇	C₂×C₈	1	2	yes	—
54	C₄²×C₁₇	M₁₆	1	2	yes	—
55	#7605	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	Q₈×C₁₇	—	4	4	no	—
58	Q₈×C₁₇	—	4	4	no	—
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	4	2	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₁₇	#100	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#137	1	4	yes	—
78	C₈×C₁₇	—	8	4	no	—
79	C₄×C₄⋊Q₈	C₂×C₁₇	1	1	yes	characteristic
80	#1839	C₂×C₁₇	1	2	yes	—
81	#1272	C₂×C₁₇	1	4	yes	—
82	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₁₇	#410	1	1	yes	characteristic
84	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
85	C₂²×C₁₇	#419	1	2	yes	—
86	C₂²×C₁₇	#421	1	2	yes	—
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	4	no	—
91	C₄×C₁₇	—	2	4	no	—
92	C₄×C₁₇	—	4	1	no	—
93	C₄×C₁₇	—	4	4	no	—
94	C₄×C₁₇	—	8	2	no	—
95	C₂×C₄⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₄×C₁₇	1	2	yes	—
97	#391	C₂²×C₁₇	1	2	yes	—
98	#389	—	2	2	no	—
99	#384	C₄×C₁₇	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	2	no	—
101	C₄³	C₂²×C₁₇	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#336	—	2	2	no	—
104	#336	C₂²×C₁₇	1	4	yes	—
105	C₂×C₁₇	#1460	1	1	yes	characteristic
106	C₂×C₁₇	#1479	1	2	yes	—
107	C₂×C₁₇	#1515	1	2	yes	—
108	C₂×C₁₇	#1536	1	1	yes	characteristic
109	C₂×C₁₇	#1558	1	1	yes	characteristic
110	C₂²×Q₈	C₂×C₄×C₁₇	1	2	yes	—
111	C₂²×C₈	—	2	4	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	C₈×C₁₇	1	4	yes	complemented
114	C₄×Q₈	—	4	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
122	C₂×C₄²	D₄×C₁₇	1	2	yes	—
123	C₂×C₄²	D₄×C₁₇	1	2	yes	—
124	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
125	#95	—	2	4	no	—
126	C₁₇	#7605	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	C₂×C₈×C₁₇	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
135	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×C₈×C₁₇	1	2	yes	—
151	C₄²	M₁₆×C₁₇	1	2	yes	—
152	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₁₇×C₂²⋊C₈	1	4	yes	—
170	C₂×C₄	C₁₇×#100	1	2	yes	—
171	C₂×C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
172	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
173	C₂×C₄	C₁₇×#137	1	4	yes	—
174	C₈	—	8	4	no	—
175	C₂²	C₂×C₁₇×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²	C₁₇×#410	1	1	yes	characteristic
177	C₂²	C₂×C₁₇×#100	1	1	yes	characteristic
178	C₂²	C₁₇×#419	1	2	yes	—
179	C₂²	C₁₇×#421	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₂	C₁₇×#1460	1	1	yes	characteristic
189	C₂	C₁₇×#1479	1	2	yes	—
190	C₂	C₁₇×#1515	1	2	yes	—
191	C₂	C₁₇×#1536	1	1	yes	characteristic
192	C₂	C₁₇×#1558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₁₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₁₇×#1839

C₂

yes

maximal

C₁₇×#1272

C₂

yes

maximal

C₂×C₁₇×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈×C₁₇

C₄

yes

—

C₁₇×#391

C₂²

yes

—

C₁₇×#389

—

C₁₇×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₇

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#336

—

C₁₇×#336

—

C₁₇×#336

C₂²

yes

—

C₂²×Q₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₈

yes

complemented

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₁₇×#95

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

M₁₆

yes

—

#7605

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

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

—

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⋊C₈

yes

—

C₂×C₄×C₁₇

#100

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₈×C₁₇

—

C₄×C₄⋊Q₈

C₂×C₁₇

yes

characteristic

#1839

C₂×C₁₇

yes

—

#1272

C₂×C₁₇

yes

—

C₂²×C₁₇

C₂×C₂²⋊C₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#421

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₂×C₄⋊Q₈

C₄×C₁₇

yes

characteristic

C₂×C₄×Q₈

C₄×C₁₇

yes

—

#391

C₂²×C₁₇

yes

—

#389

—

#384

C₄×C₁₇

yes

characteristic

100

C₄×C₄⋊C₄

—

101

C₄³

C₂²×C₁₇

yes

characteristic

102

#336

—

103

#336

—

104

#336

C₂²×C₁₇

yes

—

105

C₂×C₁₇

#1460

yes

characteristic

106

C₂×C₁₇

#1479

yes

—

107

C₂×C₁₇

#1515

yes

—

108

C₂×C₁₇

#1536

yes

characteristic

109

C₂×C₁₇

#1558

yes

characteristic

110

C₂²×Q₈

C₂×C₄×C₁₇

yes

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

C₈×C₁₇

yes

complemented

114

C₄×Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×C₄×C₁₇

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×C₄×C₁₇

yes

characteristic

121

C₂×C₄²

C₂×C₄×C₁₇

yes

characteristic

122

C₂×C₄²

D₄×C₁₇

yes

—

123

C₂×C₄²

D₄×C₁₇

yes

—

124

C₂×C₄²

C₂³×C₁₇

yes

characteristic

125

#95

—

126

C₁₇

#7605

yes

characteristic, direct product factor, Sylow

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₂×C₈×C₁₇

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊C₄×C₁₇

yes

characteristic

134

C₂²×C₄

C₂²⋊C₄×C₁₇

yes

—

135

C₂²×C₄

C₂²⋊C₄×C₁₇

yes

characteristic

136

C₂²×C₄

C₂²×C₄×C₁₇

yes

characteristic

137

C₂²×C₄

C₂×D₄×C₁₇

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂×C₈×C₁₇

yes

—

151

C₄²

M₁₆×C₁₇

yes

—

152

C₂³

C₂×C₂²⋊C₄×C₁₇

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₁₇×C₂²⋊C₈

yes

—

170

C₂×C₄

C₁₇×#100

yes

—

171

C₂×C₄

C₂²×C₈×C₁₇

yes

characteristic

172

C₂×C₄

C₂×M₁₆×C₁₇

yes

characteristic

173

C₂×C₄

C₁₇×#137

yes

—

174

C₈

—

175

C₂²

C₂×C₁₇×C₂²⋊C₈

yes

characteristic

176

C₂²

C₁₇×#410

yes

characteristic

177

C₂²

C₂×C₁₇×#100

yes

characteristic

178

C₂²

C₁₇×#419

yes

—

179

C₂²

C₁₇×#421

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

C₁₇×#1460

yes

characteristic

189

C₂

C₁₇×#1479

yes

—

190

C₂

C₁₇×#1515

yes

—

191

C₂

C₁₇×#1536

yes

characteristic

192

C₂

C₁₇×#1558

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₇

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₁₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

N	H ≅ G/N	type
17. C₁₇×C₄⋊Q₈	C₈	semidirect
55. #7605	C₁₇	direct (N × H)
113. C₄⋊Q₈	C₈×C₁₇	semidirect
126. C₁₇	#7605	direct (N × H)

H ≅ G/N

type

17. C₁₇×C₄⋊Q₈

C₈

semidirect

55. #7605

C₁₇

direct (N × H)

113. C₄⋊Q₈

C₈×C₁₇

semidirect

126. C₁₇

#7605

direct (N × H)

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

subgroup	index
C₄×C₁₇×C₄⋊Q₈	2
C₁₇×#1839	2
C₁₇×#1272	2
#7605	17

subgroup

index

C₄×C₁₇×C₄⋊Q₈

C₁₇×#1839

C₁₇×#1272

#7605

Subgroups — C₁₇×#7605 #16865349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#7605 #16865349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7605 #16865349