Group #16933036 — subgroups

Subgroups — #16933036

Order 4352.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14529557	C₂	1	2	yes	maximal
2	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
3	#14527073	C₂	1	1	yes	maximal, characteristic
4	#14526679	C₂	1	2	yes	maximal
5	C₁₇×#1567	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	1	yes	characteristic
7	C₂×#10587493	—	2	2	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	C₄×#63223	C₂²	1	1	yes	characteristic
10	#11762573	—	2	2	no	—
11	#11762560	C₄	1	2	yes	—
12	#11762560	C₄	1	2	yes	—
13	#11762531	—	2	1	no	—
14	#11762530	—	2	1	no	—
15	C₁₇×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#335	—	2	1	no	—
18	C₁₇×#335	—	2	1	no	—
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×#63223	D₄	1	1	yes	characteristic
24	C₂×#63223	D₄	1	1	yes	characteristic
25	C₂²×#3622	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₁₇×C₄⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₂²⋊#3622	Q₈	1	2	yes	—
33	#10587493	—	4	4	no	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄⋊#3622	—	2	1	no	—
36	C₄⋊#3622	C₂×C₄	1	4	yes	—
37	#10587474	—	2	1	no	—
38	C₄×#3622	—	2	1	no	—
39	C₄×#3622	—	2	1	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₁₇	—	4	2	no	—
44	M₁₆×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	4	1	no	—
47	C₂×C₈×C₁₇	—	4	2	no	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₇	#41	1	1	yes	characteristic
55	C₄²×C₁₇	#41	1	1	yes	characteristic
56	#63223	—	2	2	no	—
57	#63223	—	2	2	no	—
58	C₂×#3622	—	2	4	no	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	—	2	2	no	—
62	C₂×#3622	—	4	2	no	—
63	C₂×#3622	—	4	2	no	—
64	#9626	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	4	1	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₁₇	#137	1	2	yes	—
77	C₈×C₁₇	—	2	1	no	—
78	C₈×C₁₇	—	2	1	no	—
79	C₈×C₁₇	—	2	1	no	—
80	C₈×C₁₇	—	8	2	no	—
81	C₈×C₁₇	#108	1	2	yes	complemented
82	#3622	—	4	2	no	—
83	#3622	—	4	4	no	—
84	#3622	—	4	1	no	—
85	#3622	—	4	1	no	—
86	#1950	—	17	2	no	—
87	#1567	D₁₇	1	1	yes	characteristic
88	C₂×#334	—	17	1	no	—
89	#1451	—	17	1	no	—
90	#1301	—	17	2	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	—	2	1	no	—
93	C₂²×C₁₇	#389	1	1	yes	characteristic
94	C₂²×C₁₇	#501	1	2	yes	—
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
99	C₄×C₁₇	#430	1	1	yes	characteristic
100	C₄×C₁₇	#440	1	2	yes	—
101	C₄×C₁₇	#469	1	1	yes	characteristic
102	C₄×C₁₇	#470	1	1	yes	characteristic
103	#431	—	17	1	no	—
104	C₂×#108	—	34	2	no	—
105	#423	C₂×D₁₇	1	2	yes	—
106	C₂×C₄⋊C₈	—	17	2	no	—
107	C₄×M₁₆	—	17	1	no	—
108	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
109	#341	—	34	2	no	—
110	#335	—	2	1	no	—
111	#335	—	2	1	no	—
112	#334	—	17	2	no	—
113	#334	—	17	2	no	—
114	#322	—	34	1	no	—
115	#322	—	34	1	no	—
116	C₂×C₁₇	—	2	2	no	—
117	C₂×C₁₇	#1944	1	1	yes	characteristic
118	C₂×C₁₇	#1950	1	1	yes	characteristic
119	C₂×C₁₇	#1953	1	1	yes	characteristic
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	17	1	no	—
122	C₂×M₁₆	—	17	1	no	—
123	C₂²×C₈	—	34	2	no	—
124	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
125	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
127	#108	—	68	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	17	4	no	—
131	C₄⋊C₈	—	34	2	no	—
132	C₄⋊C₈	—	34	1	no	—
133	C₂²⋊C₈	—	17	2	no	—
134	#97	—	34	1	no	—
135	C₄×C₈	—	34	1	no	—
136	C₄×C₈	—	34	1	no	—
137	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
138	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
141	C₁₇	#9626	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
143	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
144	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	34	2	no	—
148	M₁₆	—	34	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	34	4	no	—
153	C₂×C₈	—	34	1	no	—
154	C₂×C₈	—	34	1	no	—
155	C₂×C₈	—	34	2	no	—
156	C₂×C₈	—	68	2	no	—
157	C₂×C₈	—	68	2	no	—
158	C₂×C₈	#63225	1	1	yes	characteristic
159	C₂×C₈	#63225	1	1	yes	characteristic
160	C₂×C₈	#63225	1	1	yes	characteristic
161	C₂×C₈	#63225	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
163	C₄²	#63254	1	1	yes	characteristic
164	C₄²	#63256	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	C₂³	#10587574	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	#10587573	1	1	yes	characteristic
171	C₂×C₄	#10587576	1	1	yes	characteristic
172	C₂×C₄	#10587579	1	1	yes	characteristic
173	C₂×C₄	#10587597	1	2	yes	—
174	C₂×C₄	#10587604	1	2	yes	—
175	C₂×C₄	C₂×#63225	1	1	yes	characteristic
176	C₂×C₄	#10587624	1	1	yes	characteristic
177	C₂×C₄	#10587632	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	8	2	no	—
182	C₈	—	68	2	no	—
183	C₈	—	68	1	no	—
184	C₈	—	68	1	no	—
185	C₈	—	68	4	no	—
186	C₈	#10587493	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#11762705	1	1	yes	characteristic
190	C₂²	#11763217	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#10587493	1	1	yes	characteristic
195	C₄	#11762857	1	1	yes	characteristic
196	C₄	#11762908	1	2	yes	—
197	C₄	#11763059	1	1	yes	characteristic
198	C₄	#11763064	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#14529525	1	1	yes	characteristic
201	C₂	#14529557	1	1	yes	characteristic
202	C₂	#14529572	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14529557

C₂

yes

maximal

C₂×#11762560

C₂

yes

maximal, characteristic

#14527073

C₂

yes

maximal, characteristic

#14526679

C₂

yes

maximal

C₁₇×#1567

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

characteristic

C₂×#10587493

—

C₂×C₄⋊#3622

C₂²

yes

—

C₄×#63223

C₂²

yes

characteristic

#11762573

—

#11762560

C₄

yes

—

#11762560

C₄

yes

—

#11762531

—

#11762530

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#335

—

C₁₇×#335

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

D₄

yes

characteristic

C₂×#63223

D₄

yes

characteristic

C₂²×#3622

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₂²⋊#3622

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

#10587474

—

C₄×#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

—

—

—

—

—

—

—

—

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

—

—

—

—

—

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#389

yes

characteristic

C₂²×C₁₇

#501

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

100

C₄×C₁₇

#440

yes

—

101

C₄×C₁₇

#469

yes

characteristic

102

C₄×C₁₇

#470

yes

characteristic

103

#431

—

104

C₂×#108

—

105

#423

C₂×D₁₇

yes

—

106

C₂×C₄⋊C₈

—

107

C₄×M₁₆

—

108

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

109

#341

—

110

#335

—

111

#335

—

112

#334

—

113

#334

—

114

#322

—

115

#322

—

116

C₂×C₁₇

—

117

C₂×C₁₇

#1944

yes

characteristic

118

C₂×C₁₇

#1950

yes

characteristic

119

C₂×C₁₇

#1953

yes

characteristic

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

Dic₃₄

yes

characteristic

125

C₂²×C₈

Dic₃₄

yes

characteristic

126

C₂×C₄²

C₂²×D₁₇

yes

characteristic

127

#108

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

C₄×D₁₇

yes

characteristic

138

C₄×C₈

C₄×D₁₇

yes

characteristic

139

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

140

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

141

C₁₇

#9626

yes

characteristic, complemented, Sylow

142

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

143

C₂²×C₄

D₄×D₁₇

yes

characteristic

144

C₂²×C₄

Q₈×D₁₇

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

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

#63225

yes

characteristic

159

C₂×C₈

#63225

yes

characteristic

160

C₂×C₈

#63225

yes

characteristic

161

C₂×C₈

#63225

yes

characteristic

162

C₄²

C₂×C₄×D₁₇

yes

characteristic

163

C₄²

#63254

yes

characteristic

164

C₄²

#63256

yes

characteristic

165

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

166

C₂³

#10587574

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#10587573

yes

characteristic

171

C₂×C₄

#10587576

yes

characteristic

172

C₂×C₄

#10587579

yes

characteristic

173

C₂×C₄

#10587597

yes

—

174

C₂×C₄

#10587604

yes

—

175

C₂×C₄

C₂×#63225

yes

characteristic

176

C₂×C₄

#10587624

yes

characteristic

177

C₂×C₄

#10587632

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

#10587493

yes

complemented

187

C₂²

—

188

C₂²

—

189

C₂²

#11762705

yes

characteristic

190

C₂²

#11763217

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₂×#10587493

yes

characteristic

195

C₄

#11762857

yes

characteristic

196

C₄

#11762908

yes

—

197

C₄

#11763059

yes

characteristic

198

C₄

#11763064

yes

characteristic

199

C₂

—

200

C₂

#14529525

yes

characteristic

201

C₂

#14529557

yes

characteristic

202

C₂

#14529572

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1567	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1567	D₁₇
hypocenter	C₁₇	#9626
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#10587624

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1567

C₂

socle

C₂²×C₁₇

#389

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
81. C₈×C₁₇	#108	semidirect
141. C₁₇	#9626	semidirect
186. C₈	#10587493	semidirect

H ≅ G/N

type

81. C₈×C₁₇

#108

semidirect

141. C₁₇

#9626

semidirect

186. C₈

#10587493

semidirect

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

subgroup	index
#14529557	2
C₂×#11762560	2
#14527073	2
#14526679	2
C₁₇×#1567	2
#9626	17

subgroup

index

C₁₇×#1567

Subgroups — #16933036

Conjugacy classes of subgroups

Complements

Maximal subgroups