Group #16939314 — subgroups

Subgroups — #16939314

Order 4352.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14529723	C₂	1	1	yes	maximal, characteristic
2	#14528958	C₂	1	1	yes	maximal, characteristic
3	#14527554	C₂	1	1	yes	maximal, characteristic
4	#14527214	C₂	1	1	yes	maximal, characteristic
5	#14527212	C₂	1	1	yes	maximal, characteristic
6	#14526614	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1570	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₃₄	C₂²	1	1	yes	characteristic
9	#11762862	C₂²	1	1	yes	characteristic
10	C₂×#10587493	—	2	2	no	—
11	#11762833	C₂²	1	1	yes	characteristic
12	C₂×#10587474	C₂²	1	1	yes	characteristic
13	#11762563	—	2	2	no	—
14	#11762559	—	2	2	no	—
15	#11762541	—	2	2	no	—
16	#11762539	—	2	2	no	—
17	C₁₇×#431	C₂²	1	1	yes	characteristic
18	C₁₇×#423	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₁₇×#334	—	2	2	no	—
21	C₂²×Dic₃₄	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	—	2	1	no	—
24	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	Q₈	1	1	yes	characteristic
26	C₂×#63225	—	2	1	no	—
27	C₂×C₄×Dic₁₇	—	2	1	no	—
28	C₂×#63223	—	2	1	no	—
29	C₂²×#3622	D₄	1	2	yes	—
30	C₄×Dic₃₄	—	2	1	no	—
31	C₄×Dic₃₄	—	2	1	no	—
32	C₄×Dic₃₄	—	2	1	no	—
33	C₄×Dic₃₄	—	2	1	no	—
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	—	2	1	no	—
36	C₁₇×C₄⋊C₈	—	2	1	no	—
37	C₁₇×C₄⋊C₈	—	2	1	no	—
38	C₁₇×C₄⋊C₈	—	2	1	no	—
39	C₁₇×C₂²⋊C₈	—	2	1	no	—
40	C₄×C₈×C₁₇	D₄	1	2	yes	—
41	C₂²⋊#3622	—	2	1	no	—
42	#10587493	—	4	4	no	—
43	C₄⋊#3622	—	2	1	no	—
44	C₄⋊#3622	—	2	1	no	—
45	C₄⋊#3622	—	2	1	no	—
46	#10587474	—	2	1	no	—
47	#10587474	—	2	1	no	—
48	C₄×#3622	D₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	2	1	no	—
53	C₂²×Dic₁₇	—	2	1	no	—
54	C₂×Dic₃₄	—	2	1	no	—
55	C₂×Dic₃₄	—	2	1	no	—
56	C₂×Dic₃₄	—	4	2	no	—
57	M₁₆×C₁₇	—	4	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	4	1	no	—
60	C₂×C₈×C₁₇	—	4	1	no	—
61	C₂×C₈×C₁₇	—	4	2	no	—
62	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₇	#41	1	1	yes	characteristic
65	C₄²×C₁₇	#41	1	1	yes	characteristic
66	C₄⋊Dic₁₇	—	2	1	no	—
67	C₄⋊Dic₁₇	—	2	1	no	—
68	#63225	—	4	1	no	—
69	#63225	—	4	1	no	—
70	C₄×Dic₁₇	—	4	1	no	—
71	C₄×Dic₁₇	—	4	1	no	—
72	#63223	—	4	2	no	—
73	C₂×#3622	—	2	2	no	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	2	2	no	—
76	C₂×#3622	—	4	1	no	—
77	C₂×#3622	SD₁₆	1	4	yes	—
78	#10663	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	—	4	1	no	—
81	C₂×C₄×C₁₇	—	4	1	no	—
82	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₁₇	#137	1	2	yes	—
90	C₂×Dic₁₇	—	2	2	no	—
91	C₂×Dic₁₇	—	4	1	no	—
92	C₂×Dic₁₇	—	4	1	no	—
93	C₂×Dic₁₇	—	4	2	no	—
94	C₂×Dic₁₇	—	4	1	no	—
95	Dic₃₄	—	2	2	no	—
96	Dic₃₄	—	2	2	no	—
97	Dic₃₄	—	4	1	no	—
98	Dic₃₄	—	4	1	no	—
99	C₈×C₁₇	—	4	2	no	—
100	C₈×C₁₇	—	8	1	no	—
101	C₈×C₁₇	—	8	1	no	—
102	#3622	—	2	4	no	—
103	#3622	—	4	2	no	—
104	#3622	—	8	1	no	—
105	#1985	—	17	1	no	—
106	#1849	—	17	1	no	—
107	#1570	—	17	1	no	—
108	#1570	D₁₇	1	1	yes	characteristic
109	#1487	—	17	1	no	—
110	#1487	—	17	1	no	—
111	#1275	—	17	1	no	—
112	C₂²×C₁₇	—	2	1	no	—
113	C₂²×C₁₇	—	2	1	no	—
114	C₂²×C₁₇	#399	1	1	yes	characteristic
115	C₂²×C₁₇	#475	1	2	yes	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	2	1	no	—
118	C₄×C₁₇	—	2	1	no	—
119	C₄×C₁₇	—	2	1	no	—
120	C₄×C₁₇	#483	1	2	yes	—
121	C₄×C₁₇	#492	1	1	yes	characteristic
122	C₄×C₁₇	#497	1	1	yes	characteristic
123	Dic₁₇	—	4	2	no	—
124	Dic₁₇	—	8	1	no	—
125	C₂×C₄×Q₈	—	17	1	no	—
126	#431	—	17	1	no	—
127	#431	C₂×D₁₇	1	1	yes	characteristic
128	C₂×#108	—	34	2	no	—
129	#423	—	17	1	no	—
130	#423	C₂×D₁₇	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
132	C₂×#97	—	17	1	no	—
133	#336	—	34	2	no	—
134	#334	—	2	2	no	—
135	#334	—	34	2	no	—
136	Q₈⋊C₈	—	34	2	no	—
137	Q₈⋊C₈	—	34	2	no	—
138	C₂×C₁₇	—	2	2	no	—
139	C₂×C₁₇	#2091	1	1	yes	characteristic
140	C₂×C₁₇	#2096	1	1	yes	characteristic
141	C₂×C₁₇	#2098	1	1	yes	characteristic
142	C₂²×Q₈	—	17	1	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	34	1	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	17	2	no	—
147	C₄×Q₈	—	34	1	no	—
148	C₄×Q₈	—	34	1	no	—
149	C₄×Q₈	—	34	1	no	—
150	C₄×Q₈	—	34	1	no	—
151	C₂×C₄⋊C₄	—	17	1	no	—
152	C₂×C₄⋊C₄	—	34	1	no	—
153	C₂×C₄²	—	34	1	no	—
154	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
155	#108	—	68	4	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	34	1	no	—
162	C₄⋊C₈	—	34	1	no	—
163	C₄⋊C₈	—	34	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	34	1	no	—
166	#97	—	34	1	no	—
167	#97	—	34	1	no	—
168	C₄×C₈	—	17	2	no	—
169	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
170	C₁₇	#10663	1	1	yes	characteristic, complemented, Sylow
171	C₂×Q₈	—	34	1	no	—
172	C₂×Q₈	—	34	1	no	—
173	C₂×Q₈	—	68	2	no	—
174	C₂²×C₄	—	34	1	no	—
175	C₂²×C₄	—	34	1	no	—
176	C₂²×C₄	#63252	1	1	yes	characteristic
177	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
178	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
179	M₁₆	—	4	2	no	—
180	M₁₆	—	68	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	17	4	no	—
186	C₂×C₈	—	34	2	no	—
187	C₂×C₈	—	34	2	no	—
188	C₂×C₈	—	34	2	no	—
189	C₂×C₈	—	68	1	no	—
190	C₄⋊C₄	—	34	1	no	—
191	C₄⋊C₄	—	34	1	no	—
192	C₄⋊C₄	—	68	1	no	—
193	C₄⋊C₄	—	68	1	no	—
194	C₄²	—	68	1	no	—
195	C₄²	—	68	1	no	—
196	C₄²	#63252	1	1	yes	characteristic
197	C₄²	D₄×D₁₇	1	1	yes	characteristic
198	C₄²	#63256	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
200	C₂³	#10587582	1	1	yes	characteristic
201	Q₈	—	34	2	no	—
202	Q₈	—	34	2	no	—
203	Q₈	—	68	1	no	—
204	Q₈	—	68	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	34	2	no	—
208	C₂×C₄	—	68	1	no	—
209	C₂×C₄	—	68	1	no	—
210	C₂×C₄	—	68	2	no	—
211	C₂×C₄	—	68	1	no	—
212	C₂×C₄	#10587580	1	1	yes	characteristic
213	C₂×C₄	#10587580	1	1	yes	characteristic
214	C₂×C₄	#10587582	1	1	yes	characteristic
215	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
216	C₂×C₄	#10587601	1	2	yes	—
217	C₂×C₄	#10587615	1	1	yes	characteristic
218	C₂×C₄	#10587626	1	1	yes	characteristic
219	C₂×C₄	#10587632	1	1	yes	characteristic
220	C₈	—	4	2	no	—
221	C₈	—	8	1	no	—
222	C₈	—	8	1	no	—
223	C₈	—	34	4	no	—
224	C₈	—	68	2	no	—
225	C₈	—	136	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	#11762744	1	1	yes	characteristic
229	C₂²	#11763093	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	68	2	no	—
235	C₄	—	136	1	no	—
236	C₄	#11763141	1	2	yes	—
237	C₄	#11763180	1	1	yes	characteristic
238	C₄	#11763200	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	#14530299	1	1	yes	characteristic
241	C₂	#14530324	1	1	yes	characteristic
242	C₂	#14530340	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14529723

C₂

yes

maximal, characteristic

#14528958

C₂

yes

maximal, characteristic

#14527554

C₂

yes

maximal, characteristic

#14527214

C₂

yes

maximal, characteristic

#14527212

C₂

yes

maximal, characteristic

#14526614

C₂

yes

maximal, characteristic

C₁₇×#1570

C₂

yes

maximal, characteristic

C₂×C₄×Dic₃₄

C₂²

yes

characteristic

#11762862

C₂²

yes

characteristic

C₂×#10587493

—

#11762833

C₂²

yes

characteristic

C₂×#10587474

C₂²

yes

—

—

—

—

C₁₇×#431

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×#334

—

C₂²×Dic₃₄

Q₈

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

Q₈

yes

characteristic

C₂×#63225

—

C₂×C₄×Dic₁₇

—

C₂×#63223

—

C₂²×#3622

D₄

yes

—

C₄×Dic₃₄

—

C₄×Dic₃₄

—

C₄×Dic₃₄

—

C₄×Dic₃₄

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

D₄

yes

—

C₂²⋊#3622

—

—

—

—

—

—

—

D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

—

—

—

—

—

SD₁₆

yes

—

#10663

—

maximal, Sylow

C₂³×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

100

C₈×C₁₇

—

101

C₈×C₁₇

—

102

#3622

—

103

#3622

—

104

#3622

—

105

#1985

—

106

#1849

—

107

#1570

—

108

#1570

D₁₇

yes

characteristic

109

#1487

—

110

#1487

—

111

#1275

—

112

C₂²×C₁₇

—

113

C₂²×C₁₇

—

114

C₂²×C₁₇

#399

yes

characteristic

115

C₂²×C₁₇

#475

yes

—

116

C₄×C₁₇

—

117

C₄×C₁₇

—

118

C₄×C₁₇

—

119

C₄×C₁₇

—

120

C₄×C₁₇

#483

yes

—

121

C₄×C₁₇

#492

yes

characteristic

122

C₄×C₁₇

#497

yes

characteristic

123

Dic₁₇

—

124

Dic₁₇

—

125

C₂×C₄×Q₈

—

126

#431

—

127

#431

C₂×D₁₇

yes

characteristic

128

C₂×#108

—

129

#423

—

130

#423

C₂×D₁₇

yes

characteristic

131

C₂×C₄⋊C₈

C₂×D₁₇

yes

characteristic

132

C₂×#97

—

133

#336

—

134

#334

—

135

#334

—

136

Q₈⋊C₈

—

137

Q₈⋊C₈

—

138

C₂×C₁₇

—

139

C₂×C₁₇

#2091

yes

characteristic

140

C₂×C₁₇

#2096

yes

characteristic

141

C₂×C₁₇

#2098

yes

characteristic

142

C₂²×Q₈

—

143

C₂×M₁₆

—

144

C₂×M₁₆

—

145

C₂²×C₈

—

146

C₂²×C₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

C₄×Q₈

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

—

153

C₂×C₄²

—

154

C₂×C₄²

C₂²×D₁₇

yes

characteristic

155

#108

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

C₄⋊C₈

—

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

C₂²⋊C₈

—

165

C₂²⋊C₈

—

166

#97

—

167

#97

—

168

C₄×C₈

—

169

C₄×C₈

C₂²⋊D₁₇

yes

—

170

C₁₇

#10663

yes

characteristic, complemented, Sylow

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

#63252

yes

characteristic

177

C₂²×C₄

D₄×D₁₇

yes

characteristic

178

C₂²×C₄

Q₈×D₁₇

yes

characteristic

179

M₁₆

—

180

M₁₆

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄²

—

195

C₄²

—

196

C₄²

#63252

yes

characteristic

197

C₄²

D₄×D₁₇

yes

characteristic

198

C₄²

#63256

yes

characteristic

199

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

200

C₂³

#10587582

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

#10587580

yes

characteristic

213

C₂×C₄

#10587580

yes

characteristic

214

C₂×C₄

#10587582

yes

characteristic

215

C₂×C₄

SD₁₆×D₁₇

yes

—

216

C₂×C₄

#10587601

yes

—

217

C₂×C₄

#10587615

yes

characteristic

218

C₂×C₄

#10587626

yes

characteristic

219

C₂×C₄

#10587632

yes

characteristic

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

—

136

—

226

C₂²

—

227

C₂²

—

228

C₂²

#11762744

yes

characteristic

229

C₂²

#11763093

yes

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

136

—

236

C₄

#11763141

yes

—

237

C₄

#11763180

yes

characteristic

238

C₄

#11763200

yes

characteristic

239

C₂

—

240

C₂

#14530299

yes

characteristic

241

C₂

#14530324

yes

characteristic

242

C₂

#14530340

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1570	C₂
socle	C₂²×C₁₇	#399
prime radical	C₂³×C₁₇	C₂²⋊Q₈
hypercenter	#1570	D₁₇
hypocenter	C₁₇	#10663
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

subgroup

G/H

center

C₂×C₄

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1570

C₂

socle

C₂²×C₁₇

#399

prime radical

C₂³×C₁₇

C₂²⋊Q₈

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

N	H ≅ G/N	type
170. C₁₇	#10663	semidirect

H ≅ G/N

type

170. C₁₇

#10663

semidirect

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

subgroup	index
#14529723	2
#14528958	2
#14527554	2
#14527214	2
#14527212	2
#14526614	2
C₁₇×#1570	2
#10663	17

subgroup

index

C₁₇×#1570

Subgroups — #16939314

Conjugacy classes of subgroups

Complements

Maximal subgroups