Group #16939439 — subgroups

Subgroups — #16939439

Order 4352.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#14528941	C₂	1	1	yes	maximal, characteristic
3	#14528940	C₂	1	2	yes	maximal
4	#14526611	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587488	—	2	2	no	—
7	C₂×#10587479	—	2	4	no	—
8	C₂×C₄×#3622	C₂²	1	2	yes	—
9	#11762679	C₂²	1	2	yes	—
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	#10587488	—	4	4	no	—
31	#10587479	—	4	8	no	—
32	C₄×#3622	—	2	2	no	—
33	C₄×#3622	—	2	2	no	—
34	C₂×Q₈×C₁₇	—	2	2	no	—
35	C₂×Q₈×C₁₇	—	4	1	no	—
36	C₂×Q₈×C₁₇	—	4	2	no	—
37	C₂×Q₈×C₁₇	—	4	2	no	—
38	C₂²×C₄×C₁₇	—	4	1	no	—
39	C₂²×C₄×C₁₇	—	4	1	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊C₄×C₁₇	—	4	4	no	—
49	C₄⋊C₄×C₁₇	—	4	2	no	—
50	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₇	#41	1	1	yes	characteristic
52	C₄²×C₁₇	#41	1	2	yes	—
53	C₄⋊Dic₁₇	—	4	4	no	—
54	C₄⋊Dic₁₇	—	4	4	no	—
55	C₄⋊Dic₁₇	—	4	4	no	—
56	C₂×#3622	—	2	4	no	—
57	C₂×#3622	—	2	4	no	—
58	C₂×#3622	SD₁₆	1	8	yes	—
59	#10685	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	#127	1	1	yes	characteristic
61	Q₈×C₁₇	—	4	4	no	—
62	Q₈×C₁₇	—	8	2	no	—
63	C₂×C₄×C₁₇	—	4	2	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	—	8	1	no	—
68	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#123	1	2	yes	—
71	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×SD₁₆	1	4	yes	—
73	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₁₇	#137	1	4	yes	—
76	C₂×Dic₁₇	—	8	2	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	C₂×Dic₁₇	—	8	2	no	—
79	C₈×C₁₇	—	2	4	no	—
80	#3622	—	2	8	no	—
81	#1990	—	17	2	no	—
82	#1956	D₁₇	1	1	yes	characteristic
83	#1844	—	17	1	no	—
84	#1844	—	17	2	no	—
85	#1273	—	17	1	no	—
86	C₂²×C₁₇	#399	1	1	yes	characteristic
87	C₂²×C₁₇	#492	1	2	yes	—
88	C₂²×C₁₇	#492	1	1	yes	characteristic
89	C₂²×C₁₇	#496	1	1	yes	characteristic
90	C₂²×C₁₇	#497	1	2	yes	—
91	C₄×C₁₇	—	8	2	no	—
92	C₄×C₁₇	—	16	1	no	—
93	C₄×C₁₇	#475	1	4	yes	—
94	C₄×C₁₇	#476	1	2	yes	—
95	C₄×C₁₇	#481	1	2	yes	—
96	C₄×C₁₇	#483	1	4	yes	—
97	Dic₁₇	—	16	2	no	—
98	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
99	C₂×#106	—	34	4	no	—
100	C₂×#106	—	34	2	no	—
101	C₂×#103	—	2	2	no	—
102	C₂×C₄×C₈	—	17	2	no	—
103	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
104	#384	—	17	2	no	—
105	#384	C₂×D₁₇	1	1	yes	characteristic
106	#340	—	34	4	no	—
107	C₂×C₁₇	#1705	1	1	yes	characteristic
108	C₂×C₁₇	#1714	1	2	yes	—
109	C₂×C₁₇	#1716	1	1	yes	characteristic
110	C₂×C₁₇	#2094	1	1	yes	characteristic
111	C₂×C₁₇	#2096	1	2	yes	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	17	4	no	—
114	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	Dic₃₄	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	34	2	no	—
122	C₂×C₄⋊C₄	—	34	2	no	—
123	C₂×C₄⋊C₄	—	34	2	no	—
124	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
125	#106	—	68	8	no	—
126	#106	—	68	4	no	—
127	#103	—	4	4	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	34	2	no	—
130	C₄×C₈	—	34	2	no	—
131	C₁₇	#10685	1	1	yes	characteristic, complemented, Sylow
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	68	2	no	—
139	C₂²×C₄	D₄×D₁₇	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	17	8	no	—
144	C₂×C₈	—	34	4	no	—
145	C₂×C₈	—	34	4	no	—
146	C₂×C₈	#63229	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	68	4	no	—
152	C₄⋊C₄	—	68	4	no	—
153	C₄⋊C₄	—	68	4	no	—
154	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
155	C₄²	#63252	1	1	yes	characteristic
156	C₄²	#63254	1	2	yes	—
157	C₂³	#10587626	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	136	2	no	—
166	C₂×C₄	—	136	2	no	—
167	C₂×C₄	—	136	2	no	—
168	C₂×C₄	#10587564	1	2	yes	—
169	C₂×C₄	#10587569	1	2	yes	—
170	C₂×C₄	SD₁₆×D₁₇	1	4	yes	—
171	C₂×C₄	#10587604	1	4	yes	—
172	C₂×C₄	#10587610	1	1	yes	characteristic
173	C₂×C₄	C₂×#63229	1	2	yes	—
174	C₂×C₄	#10587621	1	1	yes	characteristic
175	C₂×C₄	#10587635	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	34	8	no	—
178	C₂²	#11762745	1	1	yes	characteristic
179	C₂²	#11763180	1	2	yes	—
180	C₂²	#11763181	1	1	yes	characteristic
181	C₂²	#11763199	1	1	yes	characteristic
182	C₂²	#11763202	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	272	2	no	—
186	C₄	#11763092	1	4	yes	—
187	C₄	#11763099	1	2	yes	—
188	C₄	#11763125	1	2	yes	—
189	C₄	#11763138	1	4	yes	—
190	C₂	#14528370	1	1	yes	characteristic
191	C₂	#14528408	1	2	yes	—
192	C₂	#14528415	1	1	yes	characteristic
193	C₂	#14530317	1	1	yes	characteristic
194	C₂	#14530325	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529746

C₂

yes

maximal

#14528941

C₂

yes

maximal, characteristic

#14528940

C₂

yes

maximal

#14526611

C₂

yes

maximal, characteristic

C₁₇×#1956

C₂

yes

maximal, characteristic

C₂×#10587488

—

C₂×#10587479

—

C₂×C₄×#3622

C₂²

yes

—

#11762679

C₂²

yes

—

#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

—

—

—

—

—

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

SD₁₆

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

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

SD₁₆

yes

—

—

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×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

—

#1990

—

#1956

D₁₇

yes

—

—

—

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#492

yes

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#475

yes

—

C₄×C₁₇

#476

yes

—

C₄×C₁₇

#481

yes

—

C₄×C₁₇

#483

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₂×#106

—

100

C₂×#106

—

101

C₂×#103

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

104

#384

—

105

#384

C₂×D₁₇

yes

characteristic

106

#340

—

107

C₂×C₁₇

#1705

yes

characteristic

108

C₂×C₁₇

#1714

yes

—

109

C₂×C₁₇

#1716

yes

characteristic

110

C₂×C₁₇

#2094

yes

characteristic

111

C₂×C₁₇

#2096

yes

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₁₇

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₃₄

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₁₇

yes

characteristic

125

#106

—

126

#106

—

127

#103

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₁₇

#10685

yes

characteristic, complemented, Sylow

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

D₄×D₁₇

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#63229

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

C₂×Dic₃₄

yes

characteristic

155

C₄²

#63252

yes

characteristic

156

C₄²

#63254

yes

—

157

C₂³

#10587626

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

136

—

166

C₂×C₄

—

136

—

167

C₂×C₄

—

136

—

168

C₂×C₄

#10587564

yes

—

169

C₂×C₄

#10587569

yes

—

170

C₂×C₄

SD₁₆×D₁₇

yes

—

171

C₂×C₄

#10587604

yes

—

172

C₂×C₄

#10587610

yes

characteristic

173

C₂×C₄

C₂×#63229

yes

—

174

C₂×C₄

#10587621

yes

characteristic

175

C₂×C₄

#10587635

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#11762745

yes

characteristic

179

C₂²

#11763180

yes

—

180

C₂²

#11763181

yes

characteristic

181

C₂²

#11763199

yes

characteristic

182

C₂²

#11763202

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

272

—

186

C₄

#11763092

yes

—

187

C₄

#11763099

yes

—

188

C₄

#11763125

yes

—

189

C₄

#11763138

yes

—

190

C₂

#14528370

yes

characteristic

191

C₂

#14528408

yes

—

192

C₂

#14528415

yes

characteristic

193

C₂

#14530317

yes

characteristic

194

C₂

#14530325

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	#10685
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
131. C₁₇	#10685	semidirect

H ≅ G/N

type

131. C₁₇

#10685

semidirect

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

subgroup	index
#14529746	2
#14528941	2
#14528940	2
#14526611	2
C₁₇×#1956	2
#10685	17

subgroup

index

C₁₇×#1956

Subgroups — #16939439

Conjugacy classes of subgroups

Complements

Maximal subgroups