Group #17948521 — subgroups

Subgroups — #17948521

Order 4352.

302 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 86 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	#14545144	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	#14530582	C₂	1	1	yes	maximal, characteristic
4	C₄×#63242	C₄	1	1	yes	characteristic
5	C₂×C₈×#595	—	2	1	no	—
6	C₂×C₈×#595	—	2	1	no	—
7	C₄²×#595	C₄	1	1	yes	characteristic
8	C₄⋊#63216	C₄	1	2	yes	complemented
9	#11763704	—	2	1	no	—
10	#11763695	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
13	#11762588	—	2	1	no	—
14	C₁₆×Dic₁₇	—	2	1	no	—
15	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×#595	—	2	1	no	—
17	C₂×C₄×#595	—	2	1	no	—
18	C₂×C₄×#595	C₈	1	2	yes	—
19	C₂×#63242	—	2	1	no	—
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
22	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×#3624	—	2	1	no	—
24	C₄×#3624	C₈	1	2	yes	—
25	C₂×#63216	—	2	2	no	—
26	C₈×#595	—	2	2	no	—
27	C₈×#595	—	2	2	no	—
28	C₈×#595	—	2	4	no	—
29	C₂×C₁₆×C₁₇	—	2	1	no	—
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₇	D₄	1	1	yes	characteristic
32	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
33	C₂×#63214	—	2	1	no	—
34	C₄×#3622	C₂×C₄	1	1	yes	characteristic
35	C₂²×#595	—	2	2	no	—
36	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
39	C₂×#3624	—	2	2	no	—
40	C₄×#595	—	2	4	no	—
41	C₄×#595	—	2	2	no	—
42	C₄×#595	—	2	2	no	—
43	C₄×#595	—	2	2	no	—
44	C₄×#595	C₁₆	1	4	yes	complemented
45	#63242	—	2	2	no	—
46	#63242	—	2	2	no	—
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₄²	1	1	yes	characteristic
50	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
52	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
53	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
55	C₈×D₁₇	—	2	1	no	—
56	C₈×D₁₇	—	2	1	no	—
57	C₈×D₁₇	—	2	2	no	—
58	#63216	—	4	2	no	—
59	C₁₆×C₁₇	—	4	1	no	—
60	#63214	—	4	1	no	—
61	#7148	—	17	1	no	maximal, Sylow
62	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×#595	—	2	4	no	—
64	C₂×#595	—	2	4	no	—
65	C₂×#595	—	2	4	no	—
66	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
69	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
71	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
72	C₄×D₁₇	—	2	1	no	—
73	C₄×D₁₇	—	2	1	no	—
74	C₄×D₁₇	—	2	2	no	—
75	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
76	C₄×D₁₇	M₃₂	1	2	yes	—
77	#3624	—	2	4	no	—
78	C₈×C₁₇	—	2	1	no	—
79	C₈×C₁₇	#104	1	1	yes	characteristic
80	C₈×C₁₇	#104	1	1	yes	characteristic
81	#3622	—	2	1	no	—
82	#3622	#104	1	2	yes	—
83	#2162	—	17	1	no	—
84	#2162	—	17	1	no	—
85	C₄²×C₈	—	17	1	no	—
86	C₂²×C₁₇	#336	1	1	yes	characteristic
87	C₂×D₁₇	—	2	1	no	—
88	C₂×D₁₇	—	2	1	no	—
89	C₂×D₁₇	C₂²⋊C₁₆	1	2	yes	—
90	#595	—	2	8	no	—
91	C₄×C₁₇	—	2	1	no	—
92	C₄×C₁₇	#339	1	1	yes	characteristic
93	C₄×C₁₇	#339	1	1	yes	characteristic
94	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
95	C₄×C₁₇	#346	1	1	yes	characteristic
96	Dic₁₇	—	2	1	no	—
97	Dic₁₇	C₄⋊C₁₆	1	2	yes	—
98	Dic₁₇	#364	1	2	yes	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	—	34	1	no	—
102	C₂×C₄×C₈	—	34	1	no	—
103	C₄³	—	17	1	no	—
104	C₄⋊C₁₆	—	17	2	no	—
105	C₄⋊C₁₆	—	17	1	no	—
106	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
107	C₂²⋊C₁₆	—	34	1	no	—
108	C₂²⋊C₁₆	—	34	1	no	—
109	C₄×C₁₆	—	34	1	no	—
110	C₄×C₁₆	—	34	1	no	—
111	C₂×C₁₇	#1276	1	1	yes	characteristic
112	C₂×C₁₇	#1374	1	1	yes	characteristic
113	C₂×C₁₇	#1376	1	1	yes	characteristic
114	D₁₇	—	2	2	no	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	34	1	no	—
118	C₂×C₄²	—	17	1	no	—
119	C₂×C₄²	—	17	2	no	—
120	C₂×C₄²	—	34	1	no	—
121	C₂×C₄²	—	34	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	34	2	no	—
124	C₂×C₁₆	—	34	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	17	1	no	—
127	C₄×C₈	—	17	2	no	—
128	C₄×C₈	—	17	1	no	—
129	C₄×C₈	—	34	1	no	—
130	C₄×C₈	—	34	4	no	—
131	C₄×C₈	—	34	2	no	—
132	C₄×C₈	—	34	2	no	—
133	C₄×C₈	C₂×#595	1	1	yes	characteristic
134	C₁₇	#7148	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	—	17	1	no	—
136	C₂²×C₄	—	17	1	no	—
137	C₂²×C₄	—	17	1	no	—
138	C₂²×C₄	—	34	2	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	34	2	no	—
142	C₂×C₈	—	34	2	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	—	34	1	no	—
146	C₂×C₈	—	34	1	no	—
147	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
149	C₄²	—	17	1	no	—
150	C₄²	—	17	1	no	—
151	C₄²	—	17	1	no	—
152	C₄²	—	17	4	no	—
153	C₄²	—	34	2	no	—
154	C₄²	—	34	2	no	—
155	C₄²	—	34	2	no	—
156	C₄²	—	34	4	no	—
157	C₄²	C₄×#595	1	1	yes	characteristic
158	C₁₆	—	4	1	no	—
159	C₁₆	—	68	2	no	—
160	C₁₆	—	68	1	no	—
161	C₂³	—	17	1	no	—
162	C₂×C₄	—	17	1	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	17	2	no	—
165	C₂×C₄	—	17	2	no	—
166	C₂×C₄	—	17	1	no	—
167	C₂×C₄	—	34	2	no	—
168	C₂×C₄	—	34	4	no	—
169	C₂×C₄	—	34	4	no	—
170	C₂×C₄	—	34	4	no	—
171	C₂×C₄	—	34	1	no	—
172	C₂×C₄	—	34	1	no	—
173	C₂×C₄	C₈×#595	1	1	yes	characteristic
174	C₂×C₄	#10587532	1	1	yes	characteristic
175	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	17	2	no	—
178	C₈	—	34	4	no	—
179	C₈	—	34	1	no	—
180	C₈	#10587548	1	1	yes	characteristic
181	C₈	#10587550	1	1	yes	characteristic
182	C₂²	—	17	2	no	—
183	C₂²	—	34	1	no	—
184	C₂²	—	34	1	no	—
185	C₂²	#11763674	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	17	2	no	—
188	C₄	—	17	2	no	—
189	C₄	—	34	8	no	—
190	C₄	—	34	1	no	—
191	C₄	#11763679	1	1	yes	characteristic
192	C₄	#11763681	1	1	yes	characteristic
193	C₄	C₁₆×#595	1	1	yes	characteristic
194	C₄	#11763696	1	1	yes	characteristic
195	C₂	—	34	2	no	—
196	C₂	#14543611	1	1	yes	characteristic
197	C₂	#14543821	1	1	yes	characteristic
198	C₂	#14543827	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545144

C₂

yes

maximal, characteristic

C₄×C₈×#595

C₂

yes

maximal, characteristic

#14530582

C₂

yes

maximal, characteristic

C₄×#63242

C₄

yes

characteristic

C₂×C₈×#595

—

C₂×C₈×#595

—

C₄²×#595

C₄

yes

characteristic

C₄⋊#63216

C₄

yes

complemented

#11763704

—

#11763695

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊#63214

C₄

yes

characteristic, complemented

#11762588

—

C₁₆×Dic₁₇

—

C₁₇×C₄⋊C₁₆

C₄

yes

characteristic, complemented

C₂×C₄×#595

—

C₂×C₄×#595

—

C₂×C₄×#595

C₈

yes

—

C₂×#63242

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

C₄×#3624

—

C₄×#3624

C₈

yes

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

—

—

—

—

—

C₁₆

yes

complemented

#63242

—

#63242

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

#63216

—

C₁₆×C₁₇

—

#63214

—

#7148

—

maximal, Sylow

C₂²×D₁₇

C₂²⋊C₈

yes

—

—

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×C₁₆

yes

—

C₄×D₁₇

M₃₂

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

—

#2162

—

#2162

—

C₄²×C₈

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊C₁₆

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₄×C₁₆

yes

characteristic

C₄×C₁₇

#346

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄⋊C₁₆

yes

—

Dic₁₇

#364

yes

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

C₄³

—

104

C₄⋊C₁₆

—

105

C₄⋊C₁₆

—

106

C₄⋊C₁₆

#595

yes

characteristic, complemented

107

C₂²⋊C₁₆

—

108

C₂²⋊C₁₆

—

109

C₄×C₁₆

—

110

C₄×C₁₆

—

111

C₂×C₁₇

#1276

yes

characteristic

112

C₂×C₁₇

#1374

yes

characteristic

113

C₂×C₁₇

#1376

yes

characteristic

114

D₁₇

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×#595

yes

characteristic

134

C₁₇

#7148

yes

characteristic, complemented, Sylow

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₄⋊#595

yes

characteristic

148

C₂×C₈

C₂²⋊#595

yes

characteristic

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

C₄×#595

yes

characteristic

158

C₁₆

—

159

C₁₆

—

160

C₁₆

—

161

C₂³

—

162

C₂×C₄

—

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

—

173

C₂×C₄

C₈×#595

yes

characteristic

174

C₂×C₄

#10587532

yes

characteristic

175

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

#10587548

yes

characteristic

181

C₈

#10587550

yes

characteristic

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

#11763674

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#11763679

yes

characteristic

192

C₄

#11763681

yes

characteristic

193

C₄

C₁₆×#595

yes

characteristic

194

C₄

#11763696

yes

characteristic

195

C₂

—

196

C₂

#14543611

yes

characteristic

197

C₂

#14543821

yes

characteristic

198

C₂

#14543827

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#595
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×C₄⋊C₁₆	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#7148
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊#595

commutator

C₄×C₁₇

C₄×C₁₆

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×C₄⋊C₁₆

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₂²⋊C₈

cosocle

C₄×C₈×D₁₇

C₂²

C₄⋊C₁₆

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
8. C₄⋊#63216	C₄	semidirect
12. C₄⋊#63214	C₄	semidirect
15. C₁₇×C₄⋊C₁₆	C₄	semidirect
44. C₄×#595	C₁₆	semidirect
106. C₄⋊C₁₆	#595	semidirect
134. C₁₇	#7148	semidirect

H ≅ G/N

type

8. C₄⋊#63216

C₄

semidirect

12. C₄⋊#63214

C₄

semidirect

15. C₁₇×C₄⋊C₁₆

C₄

semidirect

44. C₄×#595

C₁₆

semidirect

106. C₄⋊C₁₆

#595

semidirect

134. C₁₇

#7148

semidirect

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

subgroup	index
#14545144	2
C₄×C₈×#595	2
#14530582	2
#7148	17

subgroup

index

#14545144

C₄×C₈×#595

#14530582

#7148

Subgroups — #17948521

Conjugacy classes of subgroups

Complements

Maximal subgroups