Group #16954131 — subgroups

Subgroups — #16954131

Order 4352.

326 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#11762614	C₂	1	2	yes	maximal
2	#14529645	C₂	1	1	yes	maximal, characteristic
3	#14528948	C₂	1	1	yes	maximal, characteristic
4	#14526876	C₂	1	2	yes	maximal
5	C₁₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	C₂²	1	1	yes	characteristic
7	C₂×#10587489	C₂²	1	1	yes	characteristic
8	C₂×#10587489	C₂²	1	2	yes	—
9	C₂×#10587488	—	2	1	no	—
10	C₂×#10587487	—	2	1	no	—
11	#11762715	—	2	1	no	—
12	#11762675	—	2	1	no	—
13	#11762614	—	2	4	no	—
14	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₃₄	—	2	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₇	—	2	1	no	—
21	C₂×C₄⋊Dic₁₇	—	2	2	no	—
22	C₂×Dic₆₈	—	2	1	no	—
23	C₂×Dic₆₈	—	2	2	no	—
24	C₂×Dic₆₈	—	2	2	no	—
25	C₂×Dic₆₈	D₄	1	2	yes	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	—	2	2	no	—
28	C₂×C₁₆×C₁₇	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	#10587502	—	4	2	no	—
31	#10587502	—	4	2	no	—
32	#10587489	—	2	2	no	—
33	#10587489	—	2	2	no	—
34	#10587489	—	2	2	no	—
35	#10587488	—	4	2	no	—
36	#10587487	—	4	2	no	—
37	C₂²×C₄×C₁₇	D₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	4	1	no	—
40	C₂²×Dic₁₇	—	4	2	no	—
41	C₂²×Dic₁₇	—	4	1	no	—
42	C₂×Dic₃₄	—	2	2	no	—
43	C₂×Dic₃₄	—	4	1	no	—
44	C₂×Dic₃₄	—	4	1	no	—
45	C₂×Dic₃₄	—	4	1	no	—
46	C₂×Dic₃₄	—	4	1	no	—
47	C₂×Dic₃₄	—	4	1	no	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	#41	1	2	yes	—
53	C₂×C₈×C₁₇	#41	1	2	yes	—
54	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
55	C₄²×C₁₇	—	2	1	no	—
56	C₄²×C₁₇	—	2	1	no	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	4	1	no	—
59	C₄⋊Dic₁₇	—	4	1	no	—
60	C₄⋊Dic₁₇	—	4	2	no	—
61	Dic₆₈	—	2	4	no	—
62	Dic₆₈	—	4	2	no	—
63	C₁₆×C₁₇	—	4	2	no	—
64	#13483	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
75	C₂×Dic₁₇	—	4	2	no	—
76	C₂×Dic₁₇	—	8	1	no	—
77	C₂×Dic₁₇	—	8	1	no	—
78	C₂×Dic₁₇	—	8	2	no	—
79	C₂×Dic₁₇	—	8	1	no	—
80	C₂×Dic₁₇	—	8	2	no	—
81	C₂×Dic₁₇	—	8	2	no	—
82	C₂×Dic₁₇	—	8	2	no	—
83	Dic₃₄	—	4	2	no	—
84	Dic₃₄	—	4	2	no	—
85	Dic₃₄	—	8	1	no	—
86	Dic₃₄	—	8	1	no	—
87	C₈×C₁₇	—	2	2	no	—
88	C₈×C₁₇	#123	1	2	yes	—
89	C₈×C₁₇	#126	1	2	yes	—
90	C₂×#358	—	17	2	no	—
91	#1969	—	17	1	no	—
92	#1846	—	17	1	no	—
93	#1380	—	17	2	no	—
94	#1374	D₁₇	1	1	yes	characteristic
95	C₂²×C₁₇	#459	1	1	yes	characteristic
96	C₂²×C₁₇	#480	1	1	yes	characteristic
97	C₂²×C₁₇	#486	1	1	yes	characteristic
98	C₂²×C₁₇	#508	1	2	yes	—
99	C₂²×C₁₇	#510	1	2	yes	—
100	C₄×C₁₇	—	4	2	no	—
101	C₄×C₁₇	#397	1	1	yes	characteristic
102	C₄×C₁₇	#461	1	1	yes	characteristic
103	C₄×C₁₇	#481	1	1	yes	characteristic
104	C₄×C₁₇	#489	1	1	yes	characteristic
105	Dic₁₇	—	8	2	no	—
106	Dic₁₇	—	16	2	no	—
107	Dic₁₇	—	16	1	no	—
108	C₂²×Q₁₆	—	17	1	no	—
109	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
110	C₂×#107	—	17	2	no	—
111	C₂×#107	—	17	1	no	—
112	C₂×#106	—	34	1	no	—
113	C₂×#103	—	34	1	no	—
114	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
115	#391	—	34	1	no	—
116	#382	—	34	1	no	—
117	#358	—	34	4	no	—
118	C₂×C₁₇	#2071	1	1	yes	characteristic
119	C₂×C₁₇	#2211	1	2	yes	—
120	C₂×C₁₇	#2237	1	2	yes	—
121	C₂×C₁₇	#2242	1	1	yes	characteristic
122	C₂×C₁₇	#2245	1	1	yes	characteristic
123	C₂²×Q₈	—	34	1	no	—
124	C₂×Q₁₆	—	17	2	no	—
125	C₂×Q₁₆	—	34	2	no	—
126	C₂×Q₁₆	—	34	2	no	—
127	C₂×Q₁₆	—	34	1	no	—
128	C₂²×C₈	D₆₈	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	34	2	no	—
131	C₂×C₄⋊C₄	—	34	1	no	—
132	C₂×C₄²	D₆₈	1	1	yes	characteristic
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	#107	—	34	2	no	—
137	#107	—	34	2	no	—
138	#107	—	34	2	no	—
139	#106	—	68	2	no	—
140	#103	—	68	2	no	—
141	C₄×C₈	—	2	2	no	—
142	#95	—	68	2	no	—
143	#95	—	68	2	no	—
144	C₁₇	#13483	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	34	2	no	—
146	C₂×Q₈	—	68	1	no	—
147	C₂×Q₈	—	68	1	no	—
148	C₂×Q₈	—	68	1	no	—
149	C₂×Q₈	—	68	1	no	—
150	C₂×Q₈	—	68	1	no	—
151	C₂²×C₄	—	68	1	no	—
152	C₂²×C₄	—	68	1	no	—
153	C₂²×C₄	—	68	2	no	—
154	C₂²×C₄	D₁₃₆	1	2	yes	—
155	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
156	Q₁₆	—	34	4	no	—
157	Q₁₆	—	68	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	#63219	1	2	yes	—
161	C₂×C₈	#63252	1	2	yes	—
162	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
163	C₂×C₈	#63254	1	2	yes	—
164	C₂×C₈	#63256	1	1	yes	characteristic
165	C₄⋊C₄	—	68	1	no	—
166	C₄⋊C₄	—	68	2	no	—
167	C₄⋊C₄	—	68	1	no	—
168	C₄⋊C₄	—	68	2	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₁₆	—	4	2	no	—
172	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
173	Q₈	—	68	2	no	—
174	Q₈	—	68	2	no	—
175	Q₈	—	136	1	no	—
176	Q₈	—	136	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	68	2	no	—
181	C₂×C₄	—	136	2	no	—
182	C₂×C₄	—	136	2	no	—
183	C₂×C₄	—	136	2	no	—
184	C₂×C₄	—	136	2	no	—
185	C₂×C₄	—	136	1	no	—
186	C₂×C₄	—	136	1	no	—
187	C₂×C₄	—	136	1	no	—
188	C₂×C₄	#10587561	1	1	yes	characteristic
189	C₂×C₄	#10587572	1	1	yes	characteristic
190	C₂×C₄	#10587581	1	1	yes	characteristic
191	C₂×C₄	C₂×#63219	1	1	yes	characteristic
192	C₂×C₄	#10587594	1	1	yes	characteristic
193	C₂×C₄	#10587595	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	#10587569	1	2	yes	—
196	C₈	#10587584	1	2	yes	—
197	C₂²	#11763005	1	1	yes	characteristic
198	C₂²	#11763123	1	1	yes	characteristic
199	C₂²	#11763156	1	1	yes	characteristic
200	C₂²	#11763241	1	2	yes	—
201	C₂²	#11763252	1	2	yes	—
202	C₄	—	4	2	no	—
203	C₄	—	272	2	no	—
204	C₄	—	272	1	no	—
205	C₄	—	136	2	no	—
206	C₄	#11762736	1	1	yes	characteristic
207	C₄	#11763015	1	1	yes	characteristic
208	C₄	#11763128	1	1	yes	characteristic
209	C₄	#11763169	1	1	yes	characteristic
210	C₂	#14530189	1	1	yes	characteristic
211	C₂	#14530847	1	2	yes	—
212	C₂	#14530993	1	2	yes	—
213	C₂	#14531016	1	1	yes	characteristic
214	C₂	#14531030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762614

C₂

yes

maximal

#14529645

C₂

yes

maximal, characteristic

#14528948

C₂

yes

maximal, characteristic

#14526876

C₂

yes

maximal

C₁₇×#1374

C₂

yes

maximal, characteristic

C₂²×Dic₆₈

C₂²

yes

characteristic

C₂×#10587489

C₂²

yes

characteristic

C₂×#10587489

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Dic₃₄

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

D₄

yes

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

—

—

—

—

—

—

—

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

#13483

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#123

yes

—

C₈×C₁₇

#126

yes

—

—

—

—

—

D₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#510

yes

—

100

C₄×C₁₇

—

101

C₄×C₁₇

#397

yes

characteristic

102

C₄×C₁₇

#461

yes

characteristic

103

C₄×C₁₇

#481

yes

characteristic

104

C₄×C₁₇

#489

yes

characteristic

105

Dic₁₇

—

106

Dic₁₇

—

107

Dic₁₇

—

108

C₂²×Q₁₆

—

109

C₂²×C₁₆

C₂×D₁₇

yes

—

110

C₂×#107

—

111

C₂×#107

—

112

C₂×#106

—

113

C₂×#103

—

114

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

115

#391

—

116

#382

—

117

#358

—

118

C₂×C₁₇

#2071

yes

characteristic

119

C₂×C₁₇

#2211

yes

—

120

C₂×C₁₇

#2237

yes

—

121

C₂×C₁₇

#2242

yes

characteristic

122

C₂×C₁₇

#2245

yes

characteristic

123

C₂²×Q₈

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂²×C₈

D₆₈

yes

characteristic

129

C₂²×C₈

C₂²×D₁₇

yes

characteristic

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

D₆₈

yes

characteristic

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

#107

—

137

#107

—

138

#107

—

139

#106

—

140

#103

—

141

C₄×C₈

—

142

#95

—

143

#95

—

144

C₁₇

#13483

yes

characteristic, complemented, Sylow

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

D₁₃₆

yes

—

155

C₂²×C₄

C₂×D₆₈

yes

characteristic

156

Q₁₆

—

157

Q₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

#63219

yes

—

161

C₂×C₈

#63252

yes

—

162

C₂×C₈

D₄×D₁₇

yes

characteristic

163

C₂×C₈

#63254

yes

—

164

C₂×C₈

#63256

yes

characteristic

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₁₆

—

172

C₂³

C₂×D₁₃₆

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

136

—

176

Q₈

—

136

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

136

—

182

C₂×C₄

—

136

—

183

C₂×C₄

—

136

—

184

C₂×C₄

—

136

—

185

C₂×C₄

—

136

—

186

C₂×C₄

—

136

—

187

C₂×C₄

—

136

—

188

C₂×C₄

#10587561

yes

characteristic

189

C₂×C₄

#10587572

yes

characteristic

190

C₂×C₄

#10587581

yes

characteristic

191

C₂×C₄

C₂×#63219

yes

characteristic

192

C₂×C₄

#10587594

yes

characteristic

193

C₂×C₄

#10587595

yes

characteristic

194

C₈

—

195

C₈

#10587569

yes

—

196

C₈

#10587584

yes

—

197

C₂²

#11763005

yes

characteristic

198

C₂²

#11763123

yes

characteristic

199

C₂²

#11763156

yes

characteristic

200

C₂²

#11763241

yes

—

201

C₂²

#11763252

yes

—

202

C₄

—

203

C₄

—

272

—

204

C₄

—

272

—

205

C₄

—

136

—

206

C₄

#11762736

yes

characteristic

207

C₄

#11763015

yes

characteristic

208

C₄

#11763128

yes

characteristic

209

C₄

#11763169

yes

characteristic

210

C₂

#14530189

yes

characteristic

211

C₂

#14530847

yes

—

212

C₂

#14530993

yes

—

213

C₂

#14531016

yes

characteristic

214

C₂

#14531030

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1374	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	#1374	D₁₇
hypocenter	C₁₇	#13483
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

subgroup

G/H

center

C₂³

C₂×D₁₃₆

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1374

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

N	H ≅ G/N	type
144. C₁₇	#13483	semidirect

H ≅ G/N

type

144. C₁₇

#13483

semidirect

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

subgroup	index
C₂×#11762614	2
#14529645	2
#14528948	2
#14526876	2
C₁₇×#1374	2
#13483	17

subgroup

index

C₁₇×#1374

Subgroups — #16954131

Conjugacy classes of subgroups

Complements

Maximal subgroups