Group #17982485 — subgroups

Subgroups — #17982485

Order 4352.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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×#11763914	C₂	1	1	yes	maximal, characteristic
2	#14545946	C₂	1	2	yes	maximal
3	C₂×#11763904	C₂	1	1	yes	maximal, characteristic
4	#14545864	C₂	1	2	yes	maximal
5	#14544587	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587666	C₂²	1	2	yes	—
7	C₂×#10587665	C₂²	1	1	yes	characteristic
8	C₂×#10587665	C₂²	1	1	yes	characteristic
9	#11763914	C₄	1	4	yes	complemented
10	#11763904	C₄	1	4	yes	complemented
11	#11763903	—	2	2	no	—
12	C₄×#63241	—	2	2	no	—
13	C₂×#10587535	C₄	1	1	yes	characteristic
14	#11763774	C₂²	1	2	yes	—
15	C₂×C₈×#595	C₂²	1	1	yes	characteristic
16	#11763679	C₄	1	2	yes	—
17	#11762844	C₄	1	1	yes	characteristic
18	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
19	C₂×#63243	C₂×C₄	1	2	yes	—
20	C₂×#63242	C₂³	1	1	yes	characteristic
21	C₂×#63241	—	2	2	no	—
22	C₂×#63241	—	2	1	no	—
23	C₂×#63241	D₄	1	2	yes	—
24	#10587666	—	2	2	no	—
25	#10587666	C₂×C₄	1	4	yes	—
26	#10587665	—	2	1	no	—
27	#10587665	C₂×C₄	1	2	yes	—
28	#10587665	C₂×C₄	1	2	yes	—
29	#10587665	Q₈	1	2	yes	—
30	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
31	#10587578	C₂×C₄	1	2	yes	—
32	#10587542	—	2	2	no	—
33	C₄×#3624	—	2	2	no	—
34	#10587535	C₈	1	4	yes	—
35	C₈×#595	C₂×C₄	1	2	yes	—
36	C₈×#595	C₂×C₄	1	2	yes	—
37	C₁₇×#106	C₈	1	1	yes	characteristic
38	#10587488	C₈	1	1	yes	characteristic
39	#10587478	C₈	1	2	yes	—
40	C₂²×#595	—	2	1	no	—
41	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×#3624	—	2	2	no	—
43	C₂×#3624	C₂×D₄	1	1	yes	characteristic
44	C₂×#3624	#41	1	1	yes	characteristic
45	C₄×#595	—	2	1	no	—
46	C₄×#595	C₄²	1	2	yes	—
47	#63243	—	2	2	no	—
48	#63243	C₂×C₈	1	4	yes	—
49	#63242	C₂²×C₄	1	1	yes	characteristic
50	#63242	C₂²×C₄	1	1	yes	characteristic
51	#63242	C₂×Q₈	1	1	yes	characteristic
52	#63242	#41	1	1	yes	characteristic
53	#63241	—	2	2	no	—
54	#63241	—	2	4	no	—
55	#63241	C₄⋊C₄	1	4	yes	—
56	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
57	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
58	#63227	—	2	2	no	—
59	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
60	C₄×Dic₁₇	—	2	2	no	—
61	C₂×#3622	C₂×C₈	1	1	yes	characteristic
62	C₈×D₁₇	C₄²	1	2	yes	—
63	C₈×D₁₇	C₂×C₈	1	2	yes	—
64	#12665	—	17	1	no	maximal, Sylow
65	C₂²×D₁₇	#131	1	1	yes	characteristic
66	C₂×#595	—	2	2	no	—
67	C₂×#595	—	4	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×Dic₁₇	—	2	2	no	—
71	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
72	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
73	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₄×D₁₇	#131	1	1	yes	characteristic
76	#3624	—	2	4	no	—
77	#3624	C₂×C₄⋊C₄	1	1	yes	characteristic
78	#3624	#117	1	1	yes	characteristic
79	#3624	C₄×D₄	1	1	yes	characteristic
80	#3624	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₁₇	C₄×C₈	1	2	yes	—
82	#3622	C₄×C₈	1	2	yes	—
83	C₂×#364	—	17	1	no	—
84	#2108	—	17	2	no	—
85	C₂×#346	—	17	1	no	—
86	#1772	—	17	1	no	—
87	#1379	—	17	2	no	—
88	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
89	C₂×D₁₇	—	2	2	no	—
90	C₂×D₁₇	#405	1	1	yes	characteristic
91	C₂×D₁₇	C₈×Q₈	1	1	yes	characteristic
92	#595	—	4	2	no	—
93	C₄×C₁₇	—	4	2	no	—
94	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
96	Dic₁₇	—	4	2	no	—
97	Dic₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
98	Dic₁₇	C₈×D₄	1	1	yes	characteristic
99	C₂×M₃₂	—	17	2	no	—
100	C₂²×C₁₆	—	17	1	no	—
101	C₂²×C₁₆	—	17	1	no	—
102	C₂×#108	—	17	1	no	—
103	#427	—	17	1	no	—
104	#405	—	17	2	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	#364	—	17	4	no	—
107	#346	—	17	4	no	—
108	#346	—	34	2	no	—
109	C₄×C₁₆	—	34	2	no	—
110	#339	—	17	2	no	—
111	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₁₇	#2171	1	2	yes	—
113	D₁₇	—	2	1	no	—
114	D₁₇	—	2	1	no	—
115	C₂×M₁₆	—	17	2	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	#117	—	17	2	no	—
119	C₂×C₄²	—	17	1	no	—
120	M₃₂	—	17	4	no	—
121	M₃₂	—	34	2	no	—
122	C₂×C₁₆	—	17	2	no	—
123	C₂×C₁₆	—	17	2	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	2	no	—
126	C₂×C₁₆	—	34	1	no	—
127	C₂×C₁₆	—	34	2	no	—
128	C₂×C₁₆	—	34	1	no	—
129	#108	—	17	4	no	—
130	#107	—	17	2	no	—
131	#106	—	17	1	no	—
132	#106	#3633	1	1	yes	characteristic
133	#97	—	34	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	17	2	no	—
136	C₄×C₈	—	34	2	no	—
137	C₁₇	#12665	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	—	17	1	no	—
139	C₂²×C₄	—	34	1	no	—
140	M₁₆	—	17	4	no	—
141	M₁₆	—	34	2	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	1	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	1	no	—
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	17	2	no	—
151	C₂×C₈	—	34	2	no	—
152	C₂×C₈	C₂×#3633	1	1	yes	characteristic
153	C₄⋊C₄	—	17	2	no	—
154	C₄⋊C₄	C₂×#3633	1	2	yes	—
155	C₂²⋊C₄	—	34	2	no	—
156	C₄²	—	17	2	no	—
157	C₄²	—	34	2	no	—
158	C₄²	—	34	1	no	—
159	C₁₆	—	17	4	no	—
160	C₁₆	—	34	2	no	—
161	C₁₆	—	34	4	no	—
162	C₂³	—	17	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	17	1	no	—
165	C₂×C₄	—	17	1	no	—
166	C₂×C₄	—	17	1	no	—
167	C₂×C₄	—	17	1	no	—
168	C₂×C₄	—	17	1	no	—
169	C₂×C₄	—	34	2	no	—
170	C₂×C₄	—	34	2	no	—
171	C₂×C₄	—	68	2	no	—
172	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
173	C₈	—	17	1	no	—
174	C₈	—	17	1	no	—
175	C₈	—	17	1	no	—
176	C₈	—	17	1	no	—
177	C₈	—	17	2	no	—
178	C₈	—	34	4	no	—
179	C₈	C₄×#3633	1	2	yes	—
180	C₂²	—	17	1	no	—
181	C₂²	—	17	1	no	—
182	C₂²	—	34	2	no	—
183	C₂²	D₄×#3633	1	1	yes	characteristic
184	C₄	—	4	2	no	—
185	C₄	—	17	1	no	—
186	C₄	—	17	1	no	—
187	C₄	—	68	2	no	—
188	C₄	—	68	2	no	—
189	C₄	C₂×C₄×#3633	1	1	yes	characteristic
190	C₄	Q₈×#3633	1	1	yes	characteristic
191	C₂	—	34	1	no	—
192	C₂	—	34	1	no	—
193	C₂	C₄⋊C₄×#3633	1	1	yes	characteristic
194	C₂	#14546004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763914

C₂

yes

maximal, characteristic

#14545946

C₂

yes

maximal

C₂×#11763904

C₂

yes

maximal, characteristic

#14545864

C₂

yes

maximal

#14544587

C₂

yes

maximal, characteristic

C₂×#10587666

C₂²

yes

—

C₂×#10587665

C₂²

yes

characteristic

C₂×#10587665

C₂²

yes

characteristic

#11763914

C₄

yes

complemented

#11763904

C₄

yes

—

—

C₄

yes

characteristic

#11763774

C₂²

yes

—

C₂×C₈×#595

C₂²

yes

characteristic

#11763679

C₄

yes

—

#11762844

C₄

yes

characteristic

C₂×C₄×#595

C₂×C₄

yes

characteristic

C₂×#63243

C₂×C₄

yes

—

C₂×#63242

C₂³

yes

—

—

D₄

yes

—

#10587666

—

#10587666

C₂×C₄

yes

—

#10587665

—

#10587665

C₂×C₄

yes

—

#10587665

C₂×C₄

yes

—

#10587665

Q₈

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

#10587578

C₂×C₄

yes

—

#10587542

—

C₄×#3624

—

#10587535

C₈

yes

—

C₈×#595

C₂×C₄

yes

—

C₈×#595

C₂×C₄

yes

—

C₁₇×#106

C₈

yes

characteristic

#10587488

C₈

yes

characteristic

#10587478

C₈

yes

—

C₂²×#595

—

C₂×C₄×D₁₇

C₂²×C₄

yes

characteristic

C₂×#3624

—

C₂×#3624

C₂×D₄

yes

yes

—

C₄²

yes

—

#63243

—

#63243

C₂×C₈

yes

—

#63242

C₂²×C₄

yes

characteristic

#63242

C₂²×C₄

yes

characteristic

#63242

C₂×Q₈

yes

yes

—

—

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

#63227

—

C₄⋊Dic₁₇

C₂×C₈

yes

—

C₄×Dic₁₇

—

C₂×#3622

C₂×C₈

yes

characteristic

C₈×D₁₇

C₄²

yes

—

C₈×D₁₇

C₂×C₈

yes

—

—

C₂²×D₁₇

yes

—

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄×D₄

yes

characteristic

C₄×D₁₇

C₂×C₄²

yes

characteristic

C₄×D₁₇

C₄×Q₈

yes

characteristic

C₄×D₁₇

C₂²×C₈

yes

C₄×D₁₇

yes

—

C₂×C₄⋊C₄

yes

yes

C₄×D₄

yes

characteristic

#3624

C₄×Q₈

yes

characteristic

C₈×C₁₇

C₄×C₈

yes

—

#3622

C₄×C₈

yes

—

—

—

—

—

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#405

yes

characteristic

C₂×D₁₇

C₈×Q₈

yes

characteristic

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

C₄×C₁₇

C₈×Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄×C₄⋊C₄

yes

characteristic

Dic₁₇

C₈×D₄

yes

characteristic

C₂×M₃₂

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₂×#108

—

103

#427

—

104

#405

—

105

C₂×C₄×C₈

—

106

#364

—

107

#346

—

108

#346

—

109

C₄×C₁₆

—

110

#339

—

111

C₂×C₁₇

C₈×C₄⋊C₄

yes

characteristic

112

C₂×C₁₇

#2171

yes

—

113

D₁₇

—

114

D₁₇

—

115

C₂×M₁₆

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

#117

—

119

C₂×C₄²

—

120

M₃₂

—

121

M₃₂

—

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

#108

—

130

#107

—

131

#106

—

132

yes

133

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₁₇

#12665

yes

characteristic, complemented, Sylow

138

C₂²×C₄

—

139

C₂²×C₄

—

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂×#3633

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

C₂×#3633

yes

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₁₆

—

160

C₁₆

—

161

C₁₆

—

162

C₂³

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂²×#3633

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

C₄×#3633

yes

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

D₄×#3633

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

C₂×C₄×#3633

yes

characteristic

190

C₄

Q₈×#3633

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

C₄⋊C₄×#3633

yes

characteristic

194

C₂

#14546004

yes

—

	subgroup	G/H
center	C₂²	D₄×#3633
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₁₇×#106	C₈
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂²×D₁₇	#131
cosocle	C₂×#63242	C₂³
hypercenter	#106	#3633
hypocenter	C₁₇	#12665
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#3633

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#3633

Fitting

C₁₇×#106

C₈

socle

C₂²×C₁₇

C₈×D₄

C₂²×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
9. #11763914	C₄	semidirect
10. #11763904	C₄	semidirect
137. C₁₇	#12665	semidirect

H ≅ G/N

type

9. #11763914

C₄

semidirect

10. #11763904

C₄

semidirect

137. C₁₇

#12665

semidirect

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

subgroup	index
C₂×#11763914	2
#14545946	2
C₂×#11763904	2
#14545864	2
#14544587	2
#12665	17

subgroup

index

Subgroups — #17982485

Conjugacy classes of subgroups

Complements

Maximal subgroups