Group #16933031 — subgroups

Subgroups — #16933031

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×#11762562	C₂	1	1	yes	maximal, characteristic
3	#14527073	C₂	1	1	yes	maximal, characteristic
4	#14526679	C₂	1	2	yes	maximal
5	C₁₇×#1566	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	#11762562	C₄	1	2	yes	—
12	#11762562	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×#334	—	2	1	no	—
18	C₁₇×#334	—	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	2	yes	—
24	C₂²×#3622	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²⋊#3622	Q₈	1	2	yes	—
32	#10587493	—	4	4	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	—	2	1	no	—
35	C₄⋊#3622	C₂×C₄	1	4	yes	—
36	#10587474	—	2	1	no	—
37	C₄×#3622	—	2	1	no	—
38	C₄×#3622	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₁₇	—	4	2	no	—
43	M₁₆×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	—	4	1	no	—
46	C₂×C₈×C₁₇	—	4	2	no	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
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	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₇	#41	1	1	yes	characteristic
54	C₄²×C₁₇	#41	1	1	yes	characteristic
55	#63223	—	2	2	no	—
56	#63223	—	2	2	no	—
57	C₂×#3622	—	2	4	no	—
58	C₂×#3622	—	2	2	no	—
59	C₂×#3622	—	2	2	no	—
60	C₂×#3622	—	4	2	no	—
61	C₂×#3622	—	4	2	no	—
62	#9625	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	4	1	no	—
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
73	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₁₇	#137	1	2	yes	—
75	C₈×C₁₇	—	2	1	no	—
76	C₈×C₁₇	—	2	1	no	—
77	C₈×C₁₇	—	2	1	no	—
78	C₈×C₁₇	—	8	2	no	—
79	C₈×C₁₇	#108	1	2	yes	complemented
80	#3622	—	4	2	no	—
81	#3622	—	4	4	no	—
82	#3622	—	4	2	no	—
83	#1950	—	17	2	no	—
84	#1566	D₁₇	1	1	yes	characteristic
85	C₂×#335	—	17	1	no	—
86	#1451	—	17	1	no	—
87	#1301	—	17	2	no	—
88	C₂²×C₁₇	—	2	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	#389	1	1	yes	characteristic
91	C₂²×C₁₇	#501	1	2	yes	—
92	C₄×C₁₇	—	2	1	no	—
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
96	C₄×C₁₇	#430	1	1	yes	characteristic
97	C₄×C₁₇	#441	1	1	yes	characteristic
98	C₄×C₁₇	#442	1	1	yes	characteristic
99	C₄×C₁₇	#468	1	2	yes	—
100	#431	—	17	1	no	—
101	C₂×#108	—	34	2	no	—
102	#423	C₂×D₁₇	1	2	yes	—
103	C₂×C₄⋊C₈	—	17	2	no	—
104	C₄×M₁₆	—	17	1	no	—
105	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
106	#341	—	34	2	no	—
107	#335	—	17	2	no	—
108	#335	—	17	2	no	—
109	#334	—	2	1	no	—
110	#334	—	2	1	no	—
111	#322	—	34	1	no	—
112	#322	—	34	1	no	—
113	C₂×C₁₇	—	2	2	no	—
114	C₂×C₁₇	#1943	1	1	yes	characteristic
115	C₂×C₁₇	#1950	1	1	yes	characteristic
116	C₂×C₁₇	#1953	1	1	yes	characteristic
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	17	2	no	—
119	C₂²×C₈	—	34	2	no	—
120	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
121	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	#108	—	68	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	17	4	no	—
127	C₄⋊C₈	—	34	2	no	—
128	C₄⋊C₈	—	34	1	no	—
129	C₂²⋊C₈	—	17	2	no	—
130	#97	—	34	1	no	—
131	C₄×C₈	—	34	1	no	—
132	C₄×C₈	—	34	1	no	—
133	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
134	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
137	C₁₇	#9625	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	34	2	no	—
144	M₁₆	—	34	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	34	4	no	—
149	C₂×C₈	—	34	2	no	—
150	C₂×C₈	—	34	2	no	—
151	C₂×C₈	—	68	2	no	—
152	C₂×C₈	—	68	2	no	—
153	C₂×C₈	#63225	1	1	yes	characteristic
154	C₂×C₈	#63225	1	1	yes	characteristic
155	C₂×C₈	#63225	1	1	yes	characteristic
156	C₂×C₈	#63225	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
158	C₄²	#63254	1	1	yes	characteristic
159	C₄²	#63256	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
161	C₂³	#10587574	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	#10587573	1	1	yes	characteristic
166	C₂×C₄	#10587576	1	1	yes	characteristic
167	C₂×C₄	#10587579	1	1	yes	characteristic
168	C₂×C₄	#10587600	1	2	yes	—
169	C₂×C₄	#10587601	1	2	yes	—
170	C₂×C₄	C₂×#63225	1	1	yes	characteristic
171	C₂×C₄	#10587624	1	1	yes	characteristic
172	C₂×C₄	#10587632	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	68	2	no	—
178	C₈	—	68	2	no	—
179	C₈	—	68	4	no	—
180	C₈	#10587493	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#11762705	1	1	yes	characteristic
184	C₂²	#11763216	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#10587493	1	1	yes	characteristic
189	C₄	#11762857	1	1	yes	characteristic
190	C₄	#11762915	1	1	yes	characteristic
191	C₄	#11762920	1	1	yes	characteristic
192	C₄	#11763052	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#14529520	1	1	yes	characteristic
195	C₂	#14529557	1	1	yes	characteristic
196	C₂	#14529572	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14529557

C₂

yes

maximal

C₂×#11762562

C₂

yes

maximal, characteristic

#14527073

C₂

yes

maximal, characteristic

#14526679

C₂

yes

maximal

C₁₇×#1566

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

characteristic

C₂×#10587493

—

C₂×C₄⋊#3622

C₂²

yes

—

C₄×#63223

C₂²

yes

characteristic

#11762573

—

#11762562

C₄

yes

—

#11762562

C₄

yes

—

#11762531

—

#11762530

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#334

—

C₁₇×#334

—

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

—

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

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

100

#431

—

101

C₂×#108

—

102

#423

C₂×D₁₇

yes

—

103

C₂×C₄⋊C₈

—

104

C₄×M₁₆

—

105

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

106

#341

—

107

#335

—

108

#335

—

109

#334

—

110

#334

—

111

#322

—

112

#322

—

113

C₂×C₁₇

—

114

C₂×C₁₇

#1943

yes

characteristic

115

C₂×C₁₇

#1950

yes

characteristic

116

C₂×C₁₇

#1953

yes

characteristic

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂²×C₈

—

120

C₂²×C₈

Dic₃₄

yes

characteristic

121

C₂²×C₈

Dic₃₄

yes

characteristic

122

C₂×C₄²

C₂²×D₁₇

yes

characteristic

123

#108

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₂²⋊C₈

—

130

#97

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₄×D₁₇

yes

characteristic

134

C₄×C₈

C₄×D₁₇

yes

characteristic

135

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

136

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

137

C₁₇

#9625

yes

characteristic, complemented, Sylow

138

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

139

C₂²×C₄

D₄×D₁₇

yes

characteristic

140

C₂²×C₄

Q₈×D₁₇

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#63225

yes

characteristic

154

C₂×C₈

#63225

yes

characteristic

155

C₂×C₈

#63225

yes

characteristic

156

C₂×C₈

#63225

yes

characteristic

157

C₄²

C₂×C₄×D₁₇

yes

characteristic

158

C₄²

#63254

yes

characteristic

159

C₄²

#63256

yes

characteristic

160

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

161

C₂³

#10587574

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#10587573

yes

characteristic

166

C₂×C₄

#10587576

yes

characteristic

167

C₂×C₄

#10587579

yes

characteristic

168

C₂×C₄

#10587600

yes

—

169

C₂×C₄

#10587601

yes

—

170

C₂×C₄

C₂×#63225

yes

characteristic

171

C₂×C₄

#10587624

yes

characteristic

172

C₂×C₄

#10587632

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

#10587493

yes

complemented

181

C₂²

—

182

C₂²

—

183

C₂²

#11762705

yes

characteristic

184

C₂²

#11763216

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×#10587493

yes

characteristic

189

C₄

#11762857

yes

characteristic

190

C₄

#11762915

yes

characteristic

191

C₄

#11762920

yes

characteristic

192

C₄

#11763052

yes

—

193

C₂

—

194

C₂

#14529520

yes

characteristic

195

C₂

#14529557

yes

characteristic

196

C₂

#14529572

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1566	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1566	D₁₇
hypocenter	C₁₇	#9625
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#10587624

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1566

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 ◁ #1566

N	H ≅ G/N	type
79. C₈×C₁₇	#108	semidirect
137. C₁₇	#9625	semidirect
180. C₈	#10587493	semidirect

H ≅ G/N

type

79. C₈×C₁₇

#108

semidirect

137. C₁₇

#9625

semidirect

180. 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×#11762562	2
#14527073	2
#14526679	2
C₁₇×#1566	2
#9625	17

subgroup

index

C₁₇×#1566

Subgroups — #16933031

Conjugacy classes of subgroups

Complements

Maximal subgroups