Group #16865397 — subgroups

Subgroups — C₁₇×#7653 #16865397

Order 4352.

456 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 170 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×#1831	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	C₂×C₁₇×C₄⋊C₈	—	2	2	no	—
7	C₁₇×#391	C₂²	1	2	yes	—
8	C₁₇×#389	—	2	2	no	—
9	C₁₇×#384	C₄	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
11	C₄³×C₁₇	C₂²	1	1	yes	characteristic
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	4	no	—
17	C₄×Q₈×C₁₇	—	4	4	no	—
18	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₇	—	2	2	no	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₇	D₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×C₄⋊C₈	—	4	4	no	—
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	M₁₆	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	2	no	—
50	C₄²×C₁₇	—	4	2	no	—
51	C₄²×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	—	4	4	no	—
53	C₄²×C₁₇	SD₁₆	1	4	yes	—
54	C₄²×C₁₇	Q₁₆	1	4	yes	—
55	#7653	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	2	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	1	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	4	no	—
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	2	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	4	no	—
72	C₂×C₄×C₁₇	#100	1	2	yes	—
73	C₂×C₄×C₁₇	#103	1	4	yes	—
74	C₂×C₄×C₁₇	#103	1	4	yes	—
75	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
78	C₈×C₁₇	—	8	4	no	—
79	C₄×C₄⋊Q₈	C₂×C₁₇	1	1	yes	characteristic
80	#1831	C₂×C₁₇	1	2	yes	—
81	#1272	C₂×C₁₇	1	4	yes	—
82	C₂²×C₁₇	#407	1	1	yes	characteristic
83	C₂²×C₁₇	#410	1	1	yes	characteristic
84	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
85	C₂²×C₁₇	C₂×#103	1	2	yes	—
86	C₂²×C₁₇	#421	1	2	yes	—
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	—	2	4	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	4	1	no	—
92	C₄×C₁₇	—	4	4	no	—
93	C₄×C₁₇	—	8	2	no	—
94	C₄×C₁₇	#333	1	4	yes	—
95	C₂×C₄⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
96	C₂×C₄×Q₈	C₄×C₁₇	1	2	yes	—
97	C₂×C₄⋊C₈	—	2	2	no	—
98	#391	C₂²×C₁₇	1	2	yes	—
99	#389	—	2	2	no	—
100	#384	C₄×C₁₇	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	2	2	no	—
102	C₄³	C₂²×C₁₇	1	1	yes	characteristic
103	#336	—	2	2	no	—
104	#336	C₂²×C₁₇	1	4	yes	—
105	C₂×C₁₇	#1466	1	1	yes	characteristic
106	C₂×C₁₇	#1487	1	2	yes	—
107	C₂×C₁₇	#1518	1	2	yes	—
108	C₂×C₁₇	#1536	1	1	yes	characteristic
109	C₂×C₁₇	C₂×#333	1	1	yes	characteristic
110	C₂²×Q₈	C₂×C₄×C₁₇	1	2	yes	—
111	C₂²×C₈	—	2	4	no	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄×Q₈	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
120	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
121	C₂×C₄²	D₄×C₁₇	1	2	yes	—
122	C₂×C₄²	D₄×C₁₇	1	2	yes	—
123	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
124	C₄⋊C₈	—	4	4	no	—
125	#95	—	2	4	no	—
126	C₁₇	#7653	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	M₁₆×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	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	SD₁₆×C₁₇	1	4	yes	—
151	C₄²	Q₁₆×C₁₇	1	4	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	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₁₇×#100	1	2	yes	—
169	C₂×C₄	C₁₇×#103	1	4	yes	—
170	C₂×C₄	C₁₇×#103	1	4	yes	—
171	C₂×C₄	C₂×M₁₆×C₁₇	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₁₇	1	2	yes	—
173	C₂×C₄	C₂×Q₁₆×C₁₇	1	2	yes	—
174	C₈	—	8	4	no	—
175	C₂²	C₁₇×#407	1	1	yes	characteristic
176	C₂²	C₁₇×#410	1	1	yes	characteristic
177	C₂²	C₂×C₁₇×#100	1	1	yes	characteristic
178	C₂²	C₂×C₁₇×#103	1	2	yes	—
179	C₂²	C₁₇×#421	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	C₁₇×#333	1	4	yes	—
188	C₂	C₁₇×#1466	1	1	yes	characteristic
189	C₂	C₁₇×#1487	1	2	yes	—
190	C₂	C₁₇×#1518	1	2	yes	—
191	C₂	C₁₇×#1536	1	1	yes	characteristic
192	C₂	C₂×C₁₇×#333	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₁₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₁₇×#1831

C₂

yes

maximal

C₁₇×#1272

C₂

yes

maximal

C₂×C₁₇×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈×C₁₇

C₄

yes

—

C₂×C₁₇×C₄⋊C₈

—

C₁₇×#391

C₂²

yes

—

C₁₇×#389

—

C₁₇×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₇

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#336

—

C₁₇×#336

C₂²

yes

—

C₂²×Q₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

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×C₄⋊C₈

—

C₁₇×#95

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

M₁₆

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

SD₁₆

yes

—

C₄²×C₁₇

Q₁₆

yes

—

#7653

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

#100

yes

—

C₂×C₄×C₁₇

#103

yes

—

C₂×C₄×C₁₇

#103

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

—

C₄×C₄⋊Q₈

C₂×C₁₇

yes

characteristic

#1831

C₂×C₁₇

yes

—

#1272

C₂×C₁₇

yes

—

C₂²×C₁₇

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

#333

yes

—

C₂×C₄⋊Q₈

C₄×C₁₇

yes

characteristic

C₂×C₄×Q₈

C₄×C₁₇

yes

—

C₂×C₄⋊C₈

—

#391

C₂²×C₁₇

yes

—

#389

—

100

#384

C₄×C₁₇

yes

characteristic

101

C₄×C₄⋊C₄

—

102

C₄³

C₂²×C₁₇

yes

characteristic

103

#336

—

104

#336

C₂²×C₁₇

yes

—

105

C₂×C₁₇

#1466

yes

characteristic

106

C₂×C₁₇

#1487

yes

—

107

C₂×C₁₇

#1518

yes

—

108

C₂×C₁₇

#1536

yes

characteristic

109

C₂×C₁₇

C₂×#333

yes

characteristic

110

C₂²×Q₈

C₂×C₄×C₁₇

yes

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×C₄×C₁₇

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×C₄×C₁₇

yes

characteristic

120

C₂×C₄²

C₂×C₄×C₁₇

yes

characteristic

121

C₂×C₄²

D₄×C₁₇

yes

—

122

C₂×C₄²

D₄×C₁₇

yes

—

123

C₂×C₄²

C₂³×C₁₇

yes

characteristic

124

C₄⋊C₈

—

125

#95

—

126

C₁₇

#7653

yes

characteristic, direct product factor, Sylow

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

M₁₆×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

SD₁₆×C₁₇

yes

—

151

C₄²

Q₁₆×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

C₁₇×#100

yes

—

169

C₂×C₄

C₁₇×#103

yes

—

170

C₂×C₄

C₁₇×#103

yes

—

171

C₂×C₄

C₂×M₁₆×C₁₇

yes

—

172

C₂×C₄

C₂×SD₁₆×C₁₇

yes

—

173

C₂×C₄

C₂×Q₁₆×C₁₇

yes

—

174

C₈

—

175

C₂²

C₁₇×#407

yes

characteristic

176

C₂²

C₁₇×#410

yes

characteristic

177

C₂²

C₂×C₁₇×#100

yes

characteristic

178

C₂²

C₂×C₁₇×#103

yes

—

179

C₂²

C₁₇×#421

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₁₇×#333

yes

—

188

C₂

C₁₇×#1466

yes

characteristic

189

C₂

C₁₇×#1487

yes

—

190

C₂

C₁₇×#1518

yes

—

191

C₂

C₁₇×#1536

yes

characteristic

192

C₂

C₂×C₁₇×#333

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₇

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₁₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₇

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

N	H ≅ G/N	type
55. #7653	C₁₇	direct (N × H)
126. C₁₇	#7653	direct (N × H)

H ≅ G/N

type

55. #7653

C₁₇

direct (N × H)

126. C₁₇

#7653

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×#1831	2
C₁₇×#1272	2
#7653	17

subgroup

index

C₄×C₁₇×C₄⋊Q₈

C₁₇×#1831

C₁₇×#1272

#7653

Subgroups — C₁₇×#7653 #16865397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#7653 #16865397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7653 #16865397