Group #17961071 — subgroups

Subgroups — #17961071

Order 4352.

306 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 116 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×#11763738	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763737	C₂	1	1	yes	maximal, characteristic
3	C₄×#10587536	C₂	1	2	yes	maximal
4	#14543844	C₂	1	2	yes	maximal
5	D₁₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587536	C₂²	1	2	yes	—
7	C₂×#10587536	C₂²	1	2	yes	—
8	C₄×#63243	—	2	2	no	—
9	#11763738	C₄	1	4	yes	complemented
10	#11763737	C₄	1	4	yes	complemented
11	#11763685	—	2	2	no	—
12	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
15	#11762585	C₄	1	1	yes	characteristic
16	#11762584	C₄	1	1	yes	characteristic
17	#11762584	C₄	1	1	yes	characteristic
18	C₁₇×#346	C₄	1	1	yes	characteristic
19	C₂×#63243	—	2	2	no	—
20	C₂×#63243	—	2	2	no	—
21	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
23	C₄²×D₁₇	D₄	1	1	yes	characteristic
24	#10587542	—	4	2	no	—
25	C₄×#3624	—	4	2	no	—
26	#10587536	—	2	2	no	—
27	#10587536	—	2	2	no	—
28	#10587536	C₂×C₄	1	4	yes	—
29	#10587536	C₂×C₄	1	4	yes	—
30	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
35	C₂×#63214	C₂×C₄	1	1	yes	characteristic
36	C₂×#63214	C₂×C₄	1	1	yes	characteristic
37	C₄×#3622	Q₈	1	1	yes	characteristic
38	C₁₆×D₁₇	—	2	1	no	—
39	C₁₆×D₁₇	—	2	1	no	—
40	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
41	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
43	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×#3624	—	4	2	no	—
45	C₂×#3624	—	4	2	no	—
46	#63243	—	2	4	no	—
47	#63243	—	2	4	no	—
48	#63243	—	4	2	no	—
49	#63243	—	4	2	no	—
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₄×Dic₁₇	D₈	1	2	yes	—
55	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
56	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
57	C₈×D₁₇	—	2	1	no	—
58	C₈×D₁₇	—	2	1	no	—
59	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₈×D₁₇	#41	1	1	yes	characteristic
62	C₈×D₁₇	#41	1	1	yes	characteristic
63	C₁₆×C₁₇	—	2	1	no	—
64	C₁₆×C₁₇	C₄²	1	2	yes	—
65	#63214	—	2	1	no	—
66	#63214	C₄²	1	2	yes	—
67	#12676	—	17	1	no	maximal, Sylow
68	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#107	1	2	yes	—
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×Dic₁₇	#107	1	2	yes	—
72	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
73	C₄×D₁₇	—	4	2	no	—
74	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
75	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
76	C₄×D₁₇	#135	1	1	yes	characteristic
77	C₄×D₁₇	#135	1	1	yes	characteristic
78	#3624	—	4	4	no	—
79	#3624	—	4	4	no	—
80	C₈×C₁₇	—	2	1	no	—
81	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
82	C₈×C₁₇	#117	1	1	yes	characteristic
83	#3622	—	2	1	no	—
84	#3622	C₄×Q₈	1	1	yes	characteristic
85	#3622	C₄×Q₈	1	1	yes	characteristic
86	C₂×#368	—	17	1	no	—
87	C₂×#368	—	17	1	no	—
88	C₂×#346	—	17	1	no	—
89	C₄×#108	—	17	2	no	—
90	#1381	—	17	2	no	—
91	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
92	C₂×D₁₇	—	2	2	no	—
93	C₂×D₁₇	C₄×Q₁₆	1	1	yes	characteristic
94	C₂×D₁₇	C₄×Q₁₆	1	1	yes	characteristic
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₁₇	#427	1	1	yes	characteristic
98	Dic₁₇	—	2	2	no	—
99	Dic₁₇	C₄×D₈	1	1	yes	characteristic
100	Dic₁₇	C₄×D₈	1	1	yes	characteristic
101	C₂²×C₁₆	—	17	1	no	—
102	C₂²×C₁₆	—	17	1	no	—
103	C₂×#108	—	17	2	no	—
104	C₂×#108	—	17	2	no	—
105	C₄×M₁₆	—	34	2	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	#368	—	17	4	no	—
108	#368	—	17	4	no	—
109	#346	—	17	1	no	—
110	#346	—	17	1	no	—
111	#346	—	17	1	no	—
112	#346	#595	1	1	yes	characteristic
113	#341	—	34	2	no	—
114	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
115	C₂×C₁₇	#2161	1	2	yes	—
116	D₁₇	—	2	1	no	—
117	D₁₇	—	2	1	no	—
118	C₂×M₁₆	—	34	2	no	—
119	C₂×M₁₆	—	34	2	no	—
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	17	1	no	—
122	C₂×C₄²	—	17	1	no	—
123	C₂×C₁₆	—	17	1	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	2	no	—
126	C₂×C₁₆	—	17	1	no	—
127	C₂×C₁₆	—	34	1	no	—
128	C₂×C₁₆	—	34	1	no	—
129	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
130	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
131	#108	—	17	4	no	—
132	#108	—	17	4	no	—
133	#108	—	34	2	no	—
134	#108	—	34	2	no	—
135	#97	—	68	2	no	—
136	C₄×C₈	—	17	1	no	—
137	C₄×C₈	—	17	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	68	2	no	—
140	C₄×C₈	C₂×#595	1	1	yes	characteristic
141	C₁₇	#12676	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₄	—	17	1	no	—
143	C₂²×C₄	—	17	2	no	—
144	M₁₆	—	34	4	no	—
145	M₁₆	—	34	4	no	—
146	M₁₆	—	68	2	no	—
147	M₁₆	—	68	2	no	—
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	34	1	no	—
155	C₂×C₈	—	34	1	no	—
156	C₂×C₈	—	68	2	no	—
157	C₂×C₈	—	68	2	no	—
158	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
159	C₂×C₈	C₂²×#595	1	1	yes	characteristic
160	C₄²	—	17	1	no	—
161	C₄²	—	17	2	no	—
162	C₄²	C₄⋊#595	1	1	yes	characteristic
163	C₁₆	—	2	1	no	—
164	C₁₆	—	17	2	no	—
165	C₁₆	—	34	1	no	—
166	C₁₆	C₄×#595	1	2	yes	—
167	C₂³	—	17	1	no	—
168	C₂×C₄	—	17	1	no	—
169	C₂×C₄	—	17	1	no	—
170	C₂×C₄	—	17	1	no	—
171	C₂×C₄	—	17	1	no	—
172	C₂×C₄	—	17	1	no	—
173	C₂×C₄	—	17	2	no	—
174	C₂×C₄	—	68	2	no	—
175	C₂×C₄	#10587534	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
177	C₈	—	2	1	no	—
178	C₈	—	17	1	no	—
179	C₈	—	17	1	no	—
180	C₈	—	34	1	no	—
181	C₈	—	68	4	no	—
182	C₈	—	68	4	no	—
183	C₈	C₂×C₄×#595	1	1	yes	characteristic
184	C₈	#10587677	1	1	yes	characteristic
185	C₂²	—	17	1	no	—
186	C₂²	—	17	1	no	—
187	C₂²	—	34	2	no	—
188	C₂²	C₂×#10587534	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	17	1	no	—
191	C₄	—	17	1	no	—
192	C₄	—	34	2	no	—
193	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
194	C₄	#11763819	1	1	yes	characteristic
195	C₂	—	34	1	no	—
196	C₂	—	34	1	no	—
197	C₂	C₄×#10587534	1	1	yes	characteristic
198	C₂	#14545140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763738

C₂

yes

maximal, characteristic

C₂×#11763737

C₂

yes

maximal, characteristic

C₄×#10587536

C₂

yes

maximal

#14543844

C₂

yes

maximal

D₁₇×#346

C₂

yes

maximal, characteristic

C₂×#10587536

C₂²

yes

—

C₂×#10587536

C₂²

yes

—

C₄×#63243

—

#11763738

C₄

yes

complemented

#11763737

C₄

yes

complemented

#11763685

—

C₂×C₁₆×D₁₇

C₂²

yes

characteristic

C₂×C₁₆×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic

#11762584

C₄

yes

characteristic

#11762584

C₄

yes

characteristic

C₁₇×#346

C₄

yes

characteristic

C₂×#63243

—

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#10587536

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₄×#3622

Q₈

yes

characteristic

C₁₆×D₁₇

—

C₁₆×D₁₇

—

C₁₆×D₁₇

C₂×C₄

yes

—

C₁₆×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

Q₁₆

yes

—

C₂×C₄×D₁₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

—

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

C₂²×C₄

yes

characteristic

C₈×D₁₇

C₂²×C₄

yes

C₈×D₁₇

yes

C₈×D₁₇

yes

C₁₆×C₁₇

—

C₁₆×C₁₇

C₄²

yes

—

#63214

—

#63214

C₄²

yes

—

#12676

—

maximal, Sylow

C₂²×D₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#107

yes

—

C₂×Dic₁₇

C₂×D₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

C₄×D₁₇

C₄×D₄

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₄²

yes

C₈×C₁₇

yes

—

C₄×Q₈

yes

characteristic

#3622

C₄×Q₈

yes

—

—

—

—

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₄×Q₁₆

yes

characteristic

C₂×D₁₇

C₄×Q₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₁₇

#427

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄×D₈

yes

characteristic

100

Dic₁₇

C₄×D₈

yes

characteristic

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

103

C₂×#108

—

104

C₂×#108

—

105

C₄×M₁₆

—

106

C₂×C₄×C₈

—

107

#368

—

108

#368

—

109

#346

—

110

#346

—

111

#346

—

112

yes

113

—

114

C₂×C₁₇

C₄×#107

yes

characteristic

115

C₂×C₁₇

#2161

yes

—

116

D₁₇

—

117

D₁₇

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂×C₄²

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

C₂×C₁₆

C₂×#595

yes

characteristic

130

C₂×C₁₆

C₂×#595

yes

characteristic

131

#108

—

132

#108

—

133

#108

—

134

#108

—

135

#97

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₂×#595

yes

characteristic

141

C₁₇

#12676

yes

characteristic, complemented, Sylow

142

C₂²×C₄

—

143

C₂²×C₄

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

C₄⋊#595

yes

characteristic

159

C₂×C₈

C₂²×#595

yes

characteristic

160

C₄²

—

161

C₄²

—

162

C₄²

C₄⋊#595

yes

characteristic

163

C₁₆

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

C₄×#595

yes

—

167

C₂³

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#10587534

yes

—

176

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

C₂×C₄×#595

yes

characteristic

184

C₈

#10587677

yes

characteristic

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#10587534

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

C₄×C₄⋊#595

yes

characteristic

194

C₄

#11763819

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₄×#10587534

yes

characteristic

198

C₂

#14545140

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587534
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂²×D₁₇	C₂×Q₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12676
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#10587534

commutator

C₈×C₁₇

C₂×C₄²

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂²×D₁₇

C₂×Q₁₆

cosocle

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
9. #11763738	C₄	semidirect
10. #11763737	C₄	semidirect
141. C₁₇	#12676	semidirect

H ≅ G/N

type

9. #11763738

C₄

semidirect

10. #11763737

C₄

semidirect

141. C₁₇

#12676

semidirect

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

subgroup	index
C₂×#11763738	2
C₂×#11763737	2
C₄×#10587536	2
#14543844	2
D₁₇×#346	2
#12676	17

subgroup

index

D₁₇×#346

Subgroups — #17961071

Conjugacy classes of subgroups

Complements

Maximal subgroups