Group #16939388 — subgroups

Subgroups — #16939388

Order 4352.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#14529746	C₂	1	2	yes	maximal
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14528911	C₂	1	2	yes	maximal
4	#14526611	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	2	no	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	#11762679	C₂²	1	2	yes	—
9	#11762571	—	2	4	no	—
10	#11762571	—	2	4	no	—
11	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×#103	—	2	2	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#384	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₇	—	2	1	no	—
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₁₇×C₄⋊Q₈	—	2	2	no	—
18	C₁₇×C₄⋊Q₈	—	2	1	no	—
19	C₁₇×C₄⋊Q₈	Q₈	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂²×#3622	D₄	1	4	yes	—
28	C₁₇×#103	—	4	4	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	#10587489	—	4	4	no	—
31	C₄×#3622	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	2	no	—
34	C₂×Q₈×C₁₇	—	4	1	no	—
35	C₂×Q₈×C₁₇	—	4	2	no	—
36	C₂×Q₈×C₁₇	—	4	2	no	—
37	C₂²×C₄×C₁₇	—	4	1	no	—
38	C₂²×C₄×C₁₇	—	4	1	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	4	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	D₈	1	4	yes	—
45	C₄⋊C₄×C₁₇	—	2	4	no	—
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊C₄×C₁₇	—	4	4	no	—
49	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	1	yes	characteristic
51	C₄²×C₁₇	#41	1	2	yes	—
52	C₄⋊Dic₁₇	—	4	4	no	—
53	C₄⋊Dic₁₇	—	4	4	no	—
54	C₄⋊Dic₁₇	—	4	4	no	—
55	C₂×#3622	—	2	4	no	—
56	C₂×#3622	—	2	4	no	—
57	C₂×#3622	—	2	4	no	—
58	#10676	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	#127	1	1	yes	characteristic
60	Q₈×C₁₇	—	4	4	no	—
61	Q₈×C₁₇	—	8	2	no	—
62	C₂×C₄×C₁₇	—	4	2	no	—
63	C₂×C₄×C₁₇	—	8	1	no	—
64	C₂×C₄×C₁₇	—	8	1	no	—
65	C₂×C₄×C₁₇	—	8	1	no	—
66	C₂×C₄×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#123	1	2	yes	—
70	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₁₇	#135	1	4	yes	—
73	C₂×C₄×C₁₇	#137	1	4	yes	—
74	C₂×C₄×C₁₇	#137	1	2	yes	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₂×Dic₁₇	—	8	2	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	C₈×C₁₇	—	2	4	no	—
79	#3622	—	4	4	no	—
80	#1990	—	17	2	no	—
81	#1956	D₁₇	1	1	yes	characteristic
82	#1845	—	17	1	no	—
83	#1834	—	17	2	no	—
84	#1273	—	17	1	no	—
85	C₂²×C₁₇	#399	1	1	yes	characteristic
86	C₂²×C₁₇	#493	1	1	yes	characteristic
87	C₂²×C₁₇	#495	1	2	yes	—
88	C₂²×C₁₇	#497	1	2	yes	—
89	C₂²×C₁₇	#497	1	1	yes	characteristic
90	C₄×C₁₇	—	8	2	no	—
91	C₄×C₁₇	—	16	1	no	—
92	C₄×C₁₇	#474	1	2	yes	—
93	C₄×C₁₇	#479	1	4	yes	—
94	C₄×C₁₇	#480	1	2	yes	—
95	C₄×C₁₇	#485	1	4	yes	—
96	Dic₁₇	—	16	2	no	—
97	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
98	C₂×#107	—	34	2	no	—
99	C₂×#103	—	2	2	no	—
100	C₂×C₄×C₈	—	17	2	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#384	—	17	2	no	—
103	#384	C₂×D₁₇	1	1	yes	characteristic
104	#340	—	34	4	no	—
105	#340	—	34	4	no	—
106	C₂×C₁₇	#1709	1	1	yes	characteristic
107	C₂×C₁₇	#1718	1	2	yes	—
108	C₂×C₁₇	#1722	1	1	yes	characteristic
109	C₂×C₁₇	#2093	1	1	yes	characteristic
110	C₂×C₁₇	#2097	1	2	yes	—
111	C₂²×Q₈	—	2	1	no	—
112	C₂²×C₈	—	17	4	no	—
113	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
114	C₄⋊Q₈	—	2	1	no	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	Dic₃₄	1	2	yes	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	34	2	no	—
121	C₂×C₄⋊C₄	—	34	2	no	—
122	C₂×C₄⋊C₄	—	34	2	no	—
123	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
124	#107	—	68	4	no	—
125	#103	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	34	2	no	—
128	C₄×C₈	—	34	2	no	—
129	C₁₇	#10676	1	1	yes	characteristic, complemented, Sylow
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	68	2	no	—
137	C₂²×C₄	D₄×D₁₇	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	34	4	no	—
142	C₂×C₈	—	34	4	no	—
143	C₂×C₈	—	34	4	no	—
144	C₂×C₈	#63228	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	68	4	no	—
150	C₄⋊C₄	—	68	4	no	—
151	C₄⋊C₄	—	68	4	no	—
152	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
153	C₄²	#63252	1	1	yes	characteristic
154	C₄²	#63254	1	2	yes	—
155	C₂³	#10587626	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	—	136	2	no	—
165	C₂×C₄	—	136	2	no	—
166	C₂×C₄	#10587564	1	2	yes	—
167	C₂×C₄	#10587569	1	2	yes	—
168	C₂×C₄	#10587601	1	4	yes	—
169	C₂×C₄	#10587605	1	4	yes	—
170	C₂×C₄	#10587610	1	1	yes	characteristic
171	C₂×C₄	C₂×#63228	1	2	yes	—
172	C₂×C₄	#10587621	1	1	yes	characteristic
173	C₂×C₄	#10587637	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	68	4	no	—
176	C₂²	#11762745	1	1	yes	characteristic
177	C₂²	#11763185	1	1	yes	characteristic
178	C₂²	#11763194	1	2	yes	—
179	C₂²	#11763200	1	2	yes	—
180	C₂²	#11763204	1	1	yes	characteristic
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	272	2	no	—
184	C₄	#11763089	1	2	yes	—
185	C₄	#11763118	1	4	yes	—
186	C₄	#11763120	1	2	yes	—
187	C₄	#11763149	1	4	yes	—
188	C₂	#14528391	1	1	yes	characteristic
189	C₂	#14528433	1	2	yes	—
190	C₂	#14528453	1	1	yes	characteristic
191	C₂	#14530312	1	1	yes	characteristic
192	C₂	#14530330	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529746

C₂

yes

maximal

#14528944

C₂

yes

maximal, characteristic

#14528911

C₂

yes

maximal

#14526611

C₂

yes

maximal, characteristic

C₁₇×#1956

C₂

yes

maximal, characteristic

C₂×#10587489

—

C₂×C₄×#3622

C₂²

yes

—

#11762679

C₂²

yes

—

#11762571

—

#11762571

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₁₇×#103

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#384

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

Q₈

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

—

C₁₇×#103

—

C₄×C₈×C₁₇

—

#10587489

—

C₄×#3622

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#123

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

—

#1990

—

#1956

D₁₇

yes

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#497

yes

—

C₂²×C₁₇

#497

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#474

yes

—

C₄×C₁₇

#479

yes

—

C₄×C₁₇

#480

yes

—

C₄×C₁₇

#485

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₂×#107

—

C₂×#103

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

102

#384

—

103

#384

C₂×D₁₇

yes

characteristic

104

#340

—

105

#340

—

106

C₂×C₁₇

#1709

yes

characteristic

107

C₂×C₁₇

#1718

yes

—

108

C₂×C₁₇

#1722

yes

characteristic

109

C₂×C₁₇

#2093

yes

characteristic

110

C₂×C₁₇

#2097

yes

—

111

C₂²×Q₈

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²⋊D₁₇

yes

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

Dic₃₄

yes

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂²×D₁₇

yes

characteristic

124

#107

—

125

#103

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₁₇

#10676

yes

characteristic, complemented, Sylow

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

D₄×D₁₇

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

#63228

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Dic₃₄

yes

characteristic

153

C₄²

#63252

yes

characteristic

154

C₄²

#63254

yes

—

155

C₂³

#10587626

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

136

—

164

C₂×C₄

—

136

—

165

C₂×C₄

—

136

—

166

C₂×C₄

#10587564

yes

—

167

C₂×C₄

#10587569

yes

—

168

C₂×C₄

#10587601

yes

—

169

C₂×C₄

#10587605

yes

—

170

C₂×C₄

#10587610

yes

characteristic

171

C₂×C₄

C₂×#63228

yes

—

172

C₂×C₄

#10587621

yes

characteristic

173

C₂×C₄

#10587637

yes

—

174

C₈

—

175

C₈

—

176

C₂²

#11762745

yes

characteristic

177

C₂²

#11763185

yes

characteristic

178

C₂²

#11763194

yes

—

179

C₂²

#11763200

yes

—

180

C₂²

#11763204

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

272

—

184

C₄

#11763089

yes

—

185

C₄

#11763118

yes

—

186

C₄

#11763120

yes

—

187

C₄

#11763149

yes

—

188

C₂

#14528391

yes

characteristic

189

C₂

#14528433

yes

—

190

C₂

#14528453

yes

characteristic

191

C₂

#14530312

yes

characteristic

192

C₂

#14530330

yes

—

	subgroup	G/H
center	C₂³	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1956	C₂
socle, prime radical	C₂³×C₁₇	#127
hypercenter	#1956	D₁₇
hypocenter	C₁₇	#10676
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1956

C₂

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
129. C₁₇	#10676	semidirect

H ≅ G/N

type

129. C₁₇

#10676

semidirect

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

subgroup	index
#14529746	2
#14528944	2
#14528911	2
#14526611	2
C₁₇×#1956	2
#10676	17

subgroup

index

C₁₇×#1956

Subgroups — #16939388

Conjugacy classes of subgroups

Complements

Maximal subgroups