Group #17952298 — subgroups

Subgroups — #17952298

Order 4352.

390 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 140 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	#14544331	C₂	1	2	yes	maximal
2	#14543997	C₂	1	2	yes	maximal
3	#14543612	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	#11763825	C₂²	1	4	yes	—
6	#11763672	C₄	1	4	yes	complemented
7	#11763670	C₄	1	4	yes	complemented
8	#11763651	—	2	4	no	—
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	—	2	2	no	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²⋊#3624	—	2	4	no	—
25	C₄⋊#3624	C₂×C₄	1	8	yes	—
26	C₄×#3624	—	2	4	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₄×#3622	—	2	2	no	—
35	C₂²×C₄×C₁₇	—	2	1	no	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	2	1	no	—
38	C₂²×Dic₁₇	#41	1	2	yes	—
39	C₂×#3624	—	2	8	no	—
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
47	C₄²×C₁₇	—	2	1	no	—
48	C₄²×C₁₇	M₁₆	1	2	yes	—
49	C₄×Dic₁₇	—	2	8	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	M₁₆	1	2	yes	—
52	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
53	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×#3622	—	2	1	no	—
56	C₂×#3622	—	2	1	no	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	C₄²	1	2	yes	—
60	C₂×#3622	C₄⋊C₄	1	2	yes	—
61	#8267	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₁₇	—	2	4	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₁₇	—	2	4	no	—
72	C₂×Dic₁₇	—	2	2	no	—
73	C₂×Dic₁₇	—	2	4	no	—
74	C₂×Dic₁₇	C₄×D₄	1	2	yes	—
75	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
76	#3624	—	4	8	no	—
77	C₈×C₁₇	—	2	2	no	—
78	C₈×C₁₇	#97	1	2	yes	—
79	C₈×C₁₇	#97	1	2	yes	—
80	#3622	—	2	2	no	—
81	#3622	—	2	2	no	—
82	C₄²×C₈	—	17	1	no	—
83	#1564	—	17	2	no	—
84	#1452	—	17	2	no	—
85	#1276	—	17	2	no	—
86	C₂²×C₁₇	—	2	2	no	—
87	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₇	#432	1	1	yes	characteristic
89	C₂²×C₁₇	#432	1	1	yes	characteristic
90	C₄×C₁₇	—	2	4	no	—
91	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
92	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
93	C₄×C₁₇	#423	1	1	yes	characteristic
94	C₄×C₁₇	#423	1	1	yes	characteristic
95	Dic₁₇	—	2	4	no	—
96	Dic₁₇	—	2	2	no	—
97	Dic₁₇	#443	1	4	yes	—
98	#431	—	17	4	no	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	—	17	1	no	—
102	C₂×C₄×C₈	—	17	1	no	—
103	C₂×C₄×C₈	—	17	1	no	—
104	C₂×C₄×C₈	#595	1	1	yes	characteristic
105	C₄³	—	17	1	no	—
106	#335	—	17	4	no	—
107	#334	—	17	4	no	—
108	#322	—	34	4	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	#1767	1	1	yes	characteristic
112	C₂×C₁₇	#1772	1	2	yes	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	C₂×#595	1	1	yes	characteristic
116	C₂²×C₈	C₂×#595	1	1	yes	characteristic
117	C₂×C₄²	—	17	1	no	—
118	C₂×C₄²	—	17	1	no	—
119	C₂×C₄²	—	34	2	no	—
120	C₂×C₄²	C₂×#595	1	1	yes	characteristic
121	C₄⋊C₈	—	17	8	no	—
122	C₂²⋊C₈	—	34	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	17	2	no	—
125	C₄×C₈	—	17	2	no	—
126	C₄×C₈	—	17	2	no	—
127	C₄×C₈	—	17	2	no	—
128	C₄×C₈	—	34	2	no	—
129	C₄×C₈	—	34	4	no	—
130	C₄×C₈	—	34	2	no	—
131	C₄×C₈	—	34	2	no	—
132	C₁₇	#8267	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	17	2	no	—
135	C₂²×C₄	—	34	1	no	—
136	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	17	2	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	34	1	no	—
142	C₂×C₈	—	34	1	no	—
143	C₂×C₈	—	34	1	no	—
144	C₂×C₈	—	34	1	no	—
145	C₂×C₈	—	34	8	no	—
146	C₂×C₈	#63243	1	2	yes	—
147	C₂×C₈	C₄×#595	1	1	yes	characteristic
148	C₂×C₈	C₄×#595	1	1	yes	characteristic
149	C₂×C₈	C₄⋊#595	1	2	yes	—
150	C₂×C₈	#63247	1	2	yes	—
151	C₄²	—	2	1	no	—
152	C₄²	—	17	2	no	—
153	C₄²	—	17	1	no	—
154	C₄²	—	17	1	no	—
155	C₄²	—	17	2	no	—
156	C₄²	—	34	8	no	—
157	C₄²	—	34	1	no	—
158	C₄²	#63243	1	2	yes	—
159	C₂³	#10587677	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	2	no	—
164	C₂×C₄	—	34	4	no	—
165	C₂×C₄	—	34	4	no	—
166	C₂×C₄	—	34	2	no	—
167	C₂×C₄	C₂×#63243	1	1	yes	characteristic
168	C₂×C₄	C₂×#63243	1	1	yes	characteristic
169	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
172	C₂×C₄	C₂×#63247	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	34	2	no	—
175	C₈	—	34	2	no	—
176	C₈	—	68	8	no	—
177	C₈	#10587542	1	2	yes	—
178	C₈	#10587542	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
181	C₂²	#11763830	1	1	yes	characteristic
182	C₂²	#11763831	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	17	4	no	—
185	C₄	—	34	4	no	—
186	C₄	—	34	2	no	—
187	C₄	C₂×#10587542	1	1	yes	characteristic
188	C₄	C₄×#63243	1	1	yes	characteristic
189	C₄	#11763812	1	1	yes	characteristic
190	C₄	#11763813	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#14544570	1	1	yes	characteristic
194	C₂	#14544586	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544331

C₂

yes

maximal

#14543997

C₂

yes

maximal

#14543612

C₂

yes

maximal

C₄×C₈×Dic₁₇

C₂

yes

maximal, characteristic

#11763825

C₂²

yes

—

#11763672

C₄

yes

complemented

#11763670

C₄

yes

complemented

#11763651

—

C₂×C₄×#3622

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

C₄²×Dic₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂³

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²⋊#3624

—

C₄⋊#3624

C₂×C₄

yes

—

C₄×#3624

—

C₄×C₈×C₁₇

—

C₈×Dic₁₇

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

#41

yes

—

C₂×#3624

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

M₁₆

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

M₁₆

yes

—

C₄×Dic₁₇

C₂²×C₄

yes

—

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

—

—

—

—

C₄²

yes

—

C₂×#3622

C₄⋊C₄

yes

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄×D₄

yes

—

C₂×Dic₁₇

C₄×Q₈

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

#97

yes

—

C₈×C₁₇

#97

yes

—

#3622

—

#3622

—

C₄²×C₈

—

#1564

—

#1452

—

#1276

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄×C₄⋊C₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

C₂×#97

yes

characteristic

C₄×C₁₇

C₄×M₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#443

yes

—

#431

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

#595

yes

characteristic

105

C₄³

—

106

#335

—

107

#334

—

108

#322

—

109

C₂×C₁₇

—

110

C₂×C₁₇

—

111

C₂×C₁₇

#1767

yes

characteristic

112

C₂×C₁₇

#1772

yes

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂×#595

yes

characteristic

116

C₂²×C₈

C₂×#595

yes

characteristic

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×#595

yes

characteristic

121

C₄⋊C₈

—

122

C₂²⋊C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₁₇

#8267

yes

characteristic, complemented, Sylow

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂²×#595

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#63243

yes

—

147

C₂×C₈

C₄×#595

yes

characteristic

148

C₂×C₈

C₄×#595

yes

characteristic

149

C₂×C₈

C₄⋊#595

yes

—

150

C₂×C₈

#63247

yes

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#63243

yes

—

159

C₂³

#10587677

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×#63243

yes

characteristic

168

C₂×C₄

C₂×#63243

yes

characteristic

169

C₂×C₄

D₁₇⋊M₁₆

yes

characteristic

170

C₂×C₄

C₂×C₄×#595

yes

characteristic

171

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

172

C₂×C₄

C₂×#63247

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

#10587542

yes

—

178

C₈

#10587542

yes

—

179

C₂²

—

180

C₂²

C₄×C₄⋊#595

yes

characteristic

181

C₂²

#11763830

yes

characteristic

182

C₂²

#11763831

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×#10587542

yes

characteristic

188

C₄

C₄×#63243

yes

characteristic

189

C₄

#11763812

yes

characteristic

190

C₄

#11763813

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

#14544570

yes

characteristic

194

C₂

#14544586

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	#117
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8267
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₂×C₄×C₈×C₁₇

C₄

socle

C₂²×C₁₇

C₄×C₄⋊C₄

prime radical

C₂³×C₁₇

#117

cosocle

C₂×C₄×Dic₁₇

C₂³

C₂×C₄×C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
6. #11763672	C₄	semidirect
7. #11763670	C₄	semidirect
132. C₁₇	#8267	semidirect

H ≅ G/N

type

6. #11763672

C₄

semidirect

7. #11763670

C₄

semidirect

132. C₁₇

#8267

semidirect

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

subgroup	index
#14544331	2
#14543997	2
#14543612	2
C₄×C₈×Dic₁₇	2
#8267	17

subgroup

index

C₄×C₈×Dic₁₇

Subgroups — #17952298

Conjugacy classes of subgroups

Complements

Maximal subgroups