Group #16934234 — subgroups

Subgroups — #16934234

Order 4352.

454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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	#14539370	C₂	1	1	yes	maximal, characteristic
2	#14529761	C₂	1	2	yes	maximal
3	#14527540	C₂	1	1	yes	maximal, characteristic
4	#14527240	C₂	1	2	yes	maximal
5	C₄×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
6	#11763366	C₂²	1	2	yes	—
7	#11763214	C₄	1	1	yes	characteristic, complemented
8	#11763189	C₄	1	1	yes	characteristic, complemented
9	#11763160	C₄	1	2	yes	complemented
10	#11763050	—	2	2	no	—
11	C₄×Dic₆₈	—	2	2	no	—
12	#11762833	C₂²	1	2	yes	—
13	#11762569	—	2	4	no	—
14	#11762560	—	2	2	no	—
15	#11762541	—	2	4	no	—
16	C₂×C₁₇×#97	—	2	2	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₄³×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	2	yes	—
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	—	2	1	no	—
24	C₂×C₄²×C₁₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	#10587612	—	2	2	no	—
27	#10587610	—	2	2	no	—
28	#10587610	—	2	2	no	—
29	C₂×#63223	—	2	2	no	—
30	C₂×Dic₆₈	—	2	2	no	—
31	#10587556	—	2	2	no	—
32	#10587555	C₂×C₄	1	2	yes	—
33	#10587555	C₂×C₄	1	2	yes	—
34	C₄×Dic₃₄	—	2	2	no	—
35	C₄×Dic₃₄	—	2	2	no	—
36	C₁₇×#97	—	2	8	no	—
37	C₄×C₈×C₁₇	—	2	1	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	2	yes	—
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	#10587489	—	2	2	no	—
43	#10587487	—	2	4	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	—	2	2	no	—
46	C₂²×C₄×C₁₇	—	2	2	no	—
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
50	C₂×Dic₃₄	—	2	2	no	—
51	C₂×Dic₃₄	—	2	2	no	—
52	C₂×C₈×C₁₇	—	2	4	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	4	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₇	—	2	4	no	—
58	C₄²×C₁₇	—	2	4	no	—
59	C₄²×C₁₇	—	4	2	no	—
60	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₇	#41	1	1	yes	characteristic
64	C₂²⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	2	2	no	—
66	C₄⋊Dic₁₇	—	2	2	no	—
67	C₄⋊Dic₁₇	—	2	2	no	—
68	C₄⋊Dic₁₇	—	2	2	no	—
69	#63225	—	4	2	no	—
70	#63225	—	4	2	no	—
71	C₄×Dic₁₇	—	4	2	no	—
72	#63223	—	4	4	no	—
73	C₂×#3622	—	4	2	no	—
74	Dic₆₈	—	4	4	no	—
75	#9850	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	4	no	—
80	C₂×C₄×C₁₇	—	4	2	no	—
81	C₂×C₄×C₁₇	—	4	2	no	—
82	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#137	1	4	yes	—
89	C₂×Dic₁₇	—	4	2	no	—
90	C₂×Dic₁₇	—	4	2	no	—
91	Dic₃₄	—	4	4	no	—
92	Dic₃₄	—	4	4	no	—
93	C₈×C₁₇	—	2	4	no	—
94	C₈×C₁₇	—	2	2	no	—
95	C₈×C₁₇	#104	1	4	yes	—
96	#3622	—	8	2	no	—
97	#3170	—	17	1	no	—
98	#1994	—	17	2	no	—
99	C₄×#97	D₁₇	1	1	yes	characteristic
100	#1566	—	17	1	no	—
101	#1493	—	17	2	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	—	2	1	no	—
104	C₂²×C₁₇	#390	1	1	yes	characteristic
105	C₂²×C₁₇	#470	1	2	yes	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	1	no	—
108	C₄×C₁₇	—	4	4	no	—
109	C₄×C₁₇	C₂×#104	1	2	yes	—
110	C₄×C₁₇	#441	1	2	yes	—
111	C₄×C₁₇	#444	1	2	yes	—
112	C₄×C₁₇	#497	1	2	yes	—
113	Dic₁₇	—	8	2	no	—
114	Dic₁₇	—	8	2	no	—
115	#533	—	17	2	no	—
116	#500	—	17	1	no	—
117	#494	—	17	1	no	—
118	#487	—	17	2	no	—
119	#467	—	34	2	no	—
120	C₄×Q₁₆	—	34	2	no	—
121	#423	—	17	2	no	—
122	C₂×#97	—	2	2	no	—
123	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	C₄³	C₂×D₁₇	1	1	yes	characteristic
126	#339	—	34	4	no	—
127	#334	—	34	2	no	—
128	Q₈⋊C₈	—	34	4	no	—
129	C₂×C₁₇	—	2	2	no	—
130	C₂×C₁₇	#1962	1	1	yes	characteristic
131	C₂×C₁₇	#1971	1	1	yes	characteristic
132	C₂×C₁₇	#1975	1	1	yes	characteristic
133	C₂×Q₁₆	—	34	2	no	—
134	C₂×M₁₆	—	34	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
137	C₄⋊Q₈	—	17	2	no	—
138	C₄⋊Q₈	—	17	2	no	—
139	#125	—	34	2	no	—
140	C₂²⋊Q₈	—	34	2	no	—
141	C₂²⋊Q₈	—	34	2	no	—
142	C₄×Q₈	—	34	2	no	—
143	C₄×Q₈	—	34	2	no	—
144	#117	—	34	2	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	D₆₈	1	2	yes	—
148	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
149	#107	—	34	2	no	—
150	C₄⋊C₈	—	34	2	no	—
151	C₄⋊C₈	—	34	2	no	—
152	#103	—	34	4	no	—
153	#97	—	2	8	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
156	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
157	C₄×C₈	D₆₈	1	2	yes	—
158	C₄×C₈	D₆₈	1	2	yes	—
159	C₁₇	#9850	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₈	—	34	2	no	—
161	C₂×Q₈	—	34	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
164	C₂²×C₄	#63252	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	Q₁₆	—	68	4	no	—
167	M₁₆	—	68	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	68	2	no	—
172	C₂×C₈	#63227	1	2	yes	—
173	C₂×C₈	#63227	1	2	yes	—
174	C₄⋊C₄	—	34	2	no	—
175	C₄⋊C₄	—	34	2	no	—
176	C₄⋊C₄	—	34	2	no	—
177	C₄⋊C₄	—	34	2	no	—
178	C₄⋊C₄	—	68	2	no	—
179	C₄⋊C₄	—	68	2	no	—
180	C₂²⋊C₄	—	68	2	no	—
181	C₄²	—	2	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	68	2	no	—
185	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
186	C₄²	C₂×D₆₈	1	1	yes	characteristic
187	C₄²	C₂×D₆₈	1	1	yes	characteristic
188	C₄²	#63252	1	1	yes	characteristic
189	C₂³	#10587616	1	1	yes	characteristic
190	Q₈	—	68	4	no	—
191	Q₈	—	68	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	68	2	no	—
198	C₂×C₄	—	68	2	no	—
199	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
200	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
201	C₂×C₄	#10587560	1	1	yes	characteristic
202	C₂×C₄	#10587561	1	1	yes	characteristic
203	C₂×C₄	#10587595	1	4	yes	—
204	C₂×C₄	C₂×#63227	1	1	yes	characteristic
205	C₂×C₄	#10587616	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	136	2	no	—
209	C₈	#10587476	1	4	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#11762711	1	1	yes	characteristic
213	C₂²	#11763067	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	136	2	no	—
218	C₄	—	136	2	no	—
219	C₄	C₂×#10587476	1	2	yes	—
220	C₄	#11762917	1	2	yes	—
221	C₄	#11762932	1	2	yes	—
222	C₄	#11763203	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#14529612	1	1	yes	characteristic
225	C₂	#14529655	1	1	yes	characteristic
226	C₂	#14529677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14539370

C₂

yes

maximal, characteristic

#14529761

C₂

yes

maximal

#14527540

C₂

yes

maximal, characteristic

#14527240

C₂

yes

maximal

C₄×C₁₇×#97

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

—

#11763214

C₄

yes

characteristic, complemented

#11763189

C₄

yes

characteristic, complemented

#11763160

C₄

yes

complemented

#11763050

—

C₄×Dic₆₈

—

#11762833

C₂²

yes

—

#11762569

—

#11762560

—

#11762541

—

C₂×C₁₇×#97

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄³×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

—

C₂×Dic₆₈

—

#10587556

—

#10587555

C₂×C₄

yes

—

#10587555

C₂×C₄

yes

—

C₄×Dic₃₄

—

C₄×Dic₃₄

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₄×Dic₁₇

—

#63223

—

C₂×#3622

—

Dic₆₈

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#104

yes

—

#3622

—

#3170

—

#1994

—

C₄×#97

D₁₇

yes

characteristic

100

#1566

—

101

#1493

—

102

C₂²×C₁₇

—

103

C₂²×C₁₇

—

104

C₂²×C₁₇

#390

yes

characteristic

105

C₂²×C₁₇

#470

yes

—

106

C₄×C₁₇

—

107

C₄×C₁₇

—

108

C₄×C₁₇

—

109

C₄×C₁₇

C₂×#104

yes

—

110

C₄×C₁₇

#441

yes

—

111

C₄×C₁₇

#444

yes

—

112

C₄×C₁₇

#497

yes

—

113

Dic₁₇

—

114

Dic₁₇

—

115

#533

—

116

#500

—

117

#494

—

118

#487

—

119

#467

—

120

C₄×Q₁₆

—

121

#423

—

122

C₂×#97

—

123

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

124

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

125

C₄³

C₂×D₁₇

yes

characteristic

126

#339

—

127

#334

—

128

Q₈⋊C₈

—

129

C₂×C₁₇

—

130

C₂×C₁₇

#1962

yes

characteristic

131

C₂×C₁₇

#1971

yes

characteristic

132

C₂×C₁₇

#1975

yes

characteristic

133

C₂×Q₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

C₂²⋊D₁₇

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#125

—

140

C₂²⋊Q₈

—

141

C₂²⋊Q₈

—

142

C₄×Q₈

—

143

C₄×Q₈

—

144

#117

—

145

C₂×C₄²

—

146

C₂×C₄²

—

147

C₂×C₄²

D₆₈

yes

—

148

C₂×C₄²

C₂²×D₁₇

yes

characteristic

149

#107

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

153

#97

—

154

C₄×C₈

—

155

C₄×C₈

C₄×D₁₇

yes

characteristic

156

C₄×C₈

C₄×D₁₇

yes

characteristic

157

C₄×C₈

D₆₈

yes

—

158

C₄×C₈

D₆₈

yes

—

159

C₁₇

#9850

yes

characteristic, complemented, Sylow

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×D₆₈

yes

characteristic

164

C₂²×C₄

#63252

yes

characteristic

165

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

166

Q₁₆

—

167

M₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

#63227

yes

—

173

C₂×C₈

#63227

yes

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₂²⋊C₄

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

C₂×C₄×D₁₇

yes

characteristic

186

C₄²

C₂×D₆₈

yes

characteristic

187

C₄²

C₂×D₆₈

yes

characteristic

188

C₄²

#63252

yes

characteristic

189

C₂³

#10587616

yes

characteristic

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₄×D₆₈

yes

characteristic

200

C₂×C₄

C₄×D₆₈

yes

characteristic

201

C₂×C₄

#10587560

yes

characteristic

202

C₂×C₄

#10587561

yes

characteristic

203

C₂×C₄

#10587595

yes

—

204

C₂×C₄

C₂×#63227

yes

characteristic

205

C₂×C₄

#10587616

yes

characteristic

206

C₈

—

207

C₈

—

208

C₈

—

136

—

209

C₈

#10587476

yes

—

210

C₂²

—

211

C₂²

—

212

C₂²

#11762711

yes

characteristic

213

C₂²

#11763067

yes

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

136

—

218

C₄

—

136

—

219

C₄

C₂×#10587476

yes

—

220

C₄

#11762917

yes

—

221

C₄

#11762932

yes

—

222

C₄

#11763203

yes

—

223

C₂

—

224

C₂

#14529612

yes

characteristic

225

C₂

#14529655

yes

characteristic

226

C₂

#14529677

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587560
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#97	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#97	D₁₇
hypocenter	C₁₇	#9850
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97

subgroup

G/H

center

C₂×C₄

#10587560

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₄×C₁₇×#97

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

hypercenter

C₄×#97

D₁₇

hypocenter

C₁₇

#9850

LCS

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97

N	H ≅ G/N	type
7. #11763214	C₄	semidirect
8. #11763189	C₄	semidirect
9. #11763160	C₄	semidirect
159. C₁₇	#9850	semidirect

H ≅ G/N

type

7. #11763214

C₄

semidirect

8. #11763189

C₄

semidirect

9. #11763160

C₄

semidirect

159. C₁₇

#9850

semidirect

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

subgroup	index
#14539370	2
#14529761	2
#14527540	2
#14527240	2
C₄×C₁₇×#97	2
#9850	17

subgroup

index

C₄×C₁₇×#97

Subgroups — #16934234

Conjugacy classes of subgroups

Complements

Maximal subgroups