Group #16954155 — subgroups

Subgroups — #16954155

Order 4352.

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

normal subgroups only

This group has 302 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	C₂×#11762624	C₂	1	2	yes	maximal
2	#14529655	C₂	1	1	yes	maximal, characteristic
3	#14526884	C₂	1	2	yes	maximal
4	#14526868	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	C₂²	1	1	yes	characteristic
7	C₂×#10587483	C₂²	1	2	yes	—
8	C₂×#10587488	C₂²	1	1	yes	characteristic
9	C₂×#10587487	—	2	1	no	—
10	#11762715	—	2	1	no	—
11	#11762624	—	2	4	no	—
12	C₂×C₁₇×#108	—	2	1	no	—
13	C₂×C₁₇×#108	C₂²	1	2	yes	—
14	C₁₇×#423	—	2	1	no	—
15	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
16	C₂²×Dic₃₄	—	2	1	no	—
17	C₂×M₁₆×C₁₇	—	2	2	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₇	—	2	1	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	C₁₇×#108	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	C₁₇×#108	—	4	2	no	—
29	C₁₇×C₄⋊C₈	—	4	2	no	—
30	C₁₇×#97	—	2	2	no	—
31	#10587502	—	4	2	no	—
32	#10587488	—	2	2	no	—
33	#10587487	—	4	2	no	—
34	#10587483	—	2	2	no	—
35	#10587483	—	2	2	no	—
36	C₂×#63214	—	2	2	no	—
37	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	4	1	no	—
40	C₂²×Dic₁₇	—	4	1	no	—
41	C₂×Dic₃₄	—	2	2	no	—
42	C₂×Dic₃₄	—	4	1	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	M₁₆×C₁₇	—	4	2	no	—
48	M₁₆×C₁₇	—	4	2	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	—	4	2	no	—
52	C₂×C₈×C₁₇	D₈	1	2	yes	—
53	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	#41	1	2	yes	—
55	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₇	#41	1	2	yes	—
57	C₄²×C₁₇	—	2	1	no	—
58	C₄²×C₁₇	—	2	1	no	—
59	C₄⋊Dic₁₇	—	4	1	no	—
60	C₄⋊Dic₁₇	—	4	1	no	—
61	Dic₆₈	—	2	4	no	—
62	Dic₆₈	—	4	2	no	—
63	#63214	—	4	2	no	—
64	#13488	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
74	C₂×Dic₁₇	—	4	2	no	—
75	C₂×Dic₁₇	—	8	1	no	—
76	C₂×Dic₁₇	—	8	1	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	C₂×Dic₁₇	—	8	1	no	—
79	Dic₃₄	—	4	2	no	—
80	Dic₃₄	—	4	2	no	—
81	Dic₃₄	—	8	1	no	—
82	Dic₃₄	—	8	1	no	—
83	C₈×C₁₇	—	2	2	no	—
84	C₈×C₁₇	—	8	2	no	—
85	C₈×C₁₇	#123	1	2	yes	—
86	C₈×C₁₇	#126	1	2	yes	—
87	C₂×#362	—	17	2	no	—
88	#1971	—	17	1	no	—
89	#1849	D₁₇	1	1	yes	characteristic
90	#1383	—	17	2	no	—
91	#1377	—	17	1	no	—
92	C₂²×C₁₇	—	2	2	no	—
93	C₂²×C₁₇	#461	1	1	yes	characteristic
94	C₂²×C₁₇	#481	1	1	yes	characteristic
95	C₂²×C₁₇	#486	1	1	yes	characteristic
96	C₄×C₁₇	—	4	2	no	—
97	C₄×C₁₇	#397	1	1	yes	characteristic
98	C₄×C₁₇	#459	1	1	yes	characteristic
99	C₄×C₁₇	#480	1	1	yes	characteristic
100	C₄×C₁₇	#489	1	1	yes	characteristic
101	Dic₁₇	—	8	2	no	—
102	Dic₁₇	—	16	1	no	—
103	C₂²×Q₁₆	—	17	1	no	—
104	C₂×M₃₂	—	17	2	no	—
105	C₂×#108	—	2	1	no	—
106	C₂×#108	C₂×D₁₇	1	2	yes	—
107	C₂×#106	—	17	1	no	—
108	#423	—	2	1	no	—
109	C₂×#103	—	34	1	no	—
110	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
111	#391	—	34	1	no	—
112	#362	—	34	4	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	#2071	1	1	yes	characteristic
116	C₂×C₁₇	#2239	1	2	yes	—
117	C₂²×Q₈	—	34	1	no	—
118	C₂×Q₁₆	—	17	2	no	—
119	C₂×Q₁₆	—	34	2	no	—
120	C₂×Q₁₆	—	34	1	no	—
121	C₂×Q₁₆	—	34	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	34	1	no	—
126	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
127	M₃₂	—	34	2	no	—
128	M₃₂	—	34	2	no	—
129	C₂×C₁₆	—	34	2	no	—
130	#108	—	2	2	no	—
131	#108	—	2	2	no	—
132	#108	—	4	2	no	—
133	#106	—	34	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	#103	—	68	2	no	—
136	#97	—	2	2	no	—
137	#95	—	68	2	no	—
138	C₁₇	#13488	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₈	—	34	2	no	—
140	C₂×Q₈	—	68	1	no	—
141	C₂×Q₈	—	68	1	no	—
142	C₂×Q₈	—	68	1	no	—
143	C₂×Q₈	—	68	1	no	—
144	C₂×Q₈	—	68	1	no	—
145	C₂²×C₄	—	68	1	no	—
146	C₂²×C₄	—	68	1	no	—
147	C₂²×C₄	#63230	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	Q₁₆	—	34	4	no	—
150	Q₁₆	—	68	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#63228	1	2	yes	—
157	C₂×C₈	#63252	1	2	yes	—
158	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
159	C₂×C₈	#63254	1	1	yes	characteristic
160	C₂×C₈	#63256	1	2	yes	—
161	C₄⋊C₄	—	68	1	no	—
162	C₄⋊C₄	—	68	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	68	2	no	—
166	C₂³	C₂×#63230	1	1	yes	characteristic
167	Q₈	—	68	2	no	—
168	Q₈	—	68	2	no	—
169	Q₈	—	136	1	no	—
170	Q₈	—	136	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	68	2	no	—
174	C₂×C₄	—	136	2	no	—
175	C₂×C₄	—	136	1	no	—
176	C₂×C₄	—	136	1	no	—
177	C₂×C₄	—	136	1	no	—
178	C₂×C₄	#10587615	1	1	yes	characteristic
179	C₂×C₄	C₂×#63228	1	1	yes	characteristic
180	C₂×C₄	#10587618	1	1	yes	characteristic
181	C₂×C₄	#10587624	1	1	yes	characteristic
182	C₂×C₄	#10587628	1	1	yes	characteristic
183	C₂×C₄	#10587633	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#10587580	1	2	yes	—
187	C₈	#10587584	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#11763011	1	1	yes	characteristic
190	C₂²	#11763127	1	1	yes	characteristic
191	C₂²	#11763157	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	272	1	no	—
194	C₄	—	136	2	no	—
195	C₄	#11762735	1	1	yes	characteristic
196	C₄	#11763001	1	1	yes	characteristic
197	C₄	#11763122	1	1	yes	characteristic
198	C₄	#11763171	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#14530188	1	1	yes	characteristic
202	C₂	#14531003	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762624

C₂

yes

maximal

#14529655

C₂

yes

maximal, characteristic

#14526884

C₂

yes

maximal

#14526868

C₂

yes

maximal, characteristic

C₁₇×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₆₈

C₂²

yes

characteristic

C₂×#10587483

C₂²

yes

—

C₂×#10587488

C₂²

yes

—

—

—

C₂×C₁₇×#108

—

C₂×C₁₇×#108

C₂²

yes

—

C₁₇×#423

—

C₂×C₁₇×#97

C₂²

yes

characteristic

C₂²×Dic₃₄

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

D₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₁₇×#97

—

—

—

—

—

—

—

C₂²×C₄×C₁₇

SD₁₆

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

#63214

—

#13488

—

maximal, Sylow

C₂³×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#123

yes

—

C₈×C₁₇

#126

yes

—

C₂×#362

—

#1971

—

#1849

D₁₇

yes

characteristic

#1383

—

#1377

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

100

C₄×C₁₇

#489

yes

characteristic

101

Dic₁₇

—

102

Dic₁₇

—

103

C₂²×Q₁₆

—

104

C₂×M₃₂

—

105

C₂×#108

—

106

C₂×#108

C₂×D₁₇

yes

—

107

C₂×#106

—

108

#423

—

109

C₂×#103

—

110

C₂×#97

C₂×D₁₇

yes

characteristic

111

#391

—

112

#362

—

113

C₂×C₁₇

—

114

C₂×C₁₇

—

115

C₂×C₁₇

#2071

yes

characteristic

116

C₂×C₁₇

#2239

yes

—

117

C₂²×Q₈

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×M₁₆

—

123

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

124

C₂²×C₈

C₂²×D₁₇

yes

characteristic

125

C₂×C₄⋊C₄

—

126

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

127

M₃₂

—

128

M₃₂

—

129

C₂×C₁₆

—

130

#108

—

131

#108

—

132

#108

—

133

#106

—

134

C₄⋊C₈

—

135

#103

—

136

#97

—

137

#95

—

138

C₁₇

#13488

yes

characteristic, complemented, Sylow

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

#63230

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#63228

yes

—

157

C₂×C₈

#63252

yes

—

158

C₂×C₈

D₄×D₁₇

yes

characteristic

159

C₂×C₈

#63254

yes

characteristic

160

C₂×C₈

#63256

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

166

C₂³

C₂×#63230

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

—

136

—

170

Q₈

—

136

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

136

—

175

C₂×C₄

—

136

—

176

C₂×C₄

—

136

—

177

C₂×C₄

—

136

—

178

C₂×C₄

#10587615

yes

characteristic

179

C₂×C₄

C₂×#63228

yes

characteristic

180

C₂×C₄

#10587618

yes

characteristic

181

C₂×C₄

#10587624

yes

characteristic

182

C₂×C₄

#10587628

yes

characteristic

183

C₂×C₄

#10587633

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#10587580

yes

—

187

C₈

#10587584

yes

—

188

C₂²

—

189

C₂²

#11763011

yes

characteristic

190

C₂²

#11763127

yes

characteristic

191

C₂²

#11763157

yes

characteristic

192

C₄

—

193

C₄

—

272

—

194

C₄

—

136

—

195

C₄

#11762735

yes

characteristic

196

C₄

#11763001

yes

characteristic

197

C₄

#11763122

yes

characteristic

198

C₄

#11763171

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#14530188

yes

characteristic

202

C₂

#14531003

yes

—

	subgroup	G/H
center	C₂²	#11763127
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1849	C₂
socle	C₂²×C₁₇	#481
prime radical	C₂³×C₁₇	C₂×SD₁₆
hypercenter	#1849	D₁₇
hypocenter	C₁₇	#13488
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#11763127

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1849

C₂

socle

C₂²×C₁₇

#481

prime radical

C₂³×C₁₇

C₂×SD₁₆

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
138. C₁₇	#13488	semidirect

H ≅ G/N

type

138. C₁₇

#13488

semidirect

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

subgroup	index
C₂×#11762624	2
#14529655	2
#14526884	2
#14526868	2
C₁₇×#1849	2
#13488	17

subgroup

index

C₁₇×#1849

Subgroups — #16954155

Conjugacy classes of subgroups

Complements

Maximal subgroups