Group #16939099 — subgroups

Subgroups — #16939099

Order 4352.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14529765	C₂	1	1	yes	maximal, characteristic
2	#14528938	C₂	1	1	yes	maximal, characteristic
3	#14527439	C₂	1	2	yes	maximal
4	#14526680	C₂	1	2	yes	maximal
5	C₁₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#11763366	C₂²	1	1	yes	characteristic
7	#11762833	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#3622	—	2	1	no	—
9	C₂×C₄⋊#3622	—	2	1	no	—
10	C₂×C₄⋊#3622	C₂²	1	2	yes	—
11	#11762569	—	2	1	no	—
12	#11762569	—	2	1	no	—
13	#11762557	—	2	2	no	—
14	#11762556	—	2	2	no	—
15	C₁₇×#431	C₂²	1	2	yes	—
16	C₁₇×#376	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
20	#10587612	—	2	1	no	—
21	#10587610	—	2	2	no	—
22	C₂×#63223	—	2	1	no	—
23	C₂²×#3622	—	2	2	no	—
24	#10587556	—	2	1	no	—
25	#10587555	D₄	1	2	yes	—
26	C₄×Dic₃₄	—	2	2	no	—
27	C₁₇×C₄⋊C₈	—	2	2	no	—
28	C₁₇×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄⋊#3622	—	2	2	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	—	4	2	no	—
35	C₄⋊#3622	—	4	2	no	—
36	C₂²×C₄×C₁₇	—	2	1	no	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	#41	1	2	yes	—
40	C₂×Dic₃₄	—	2	2	no	—
41	C₂×C₈×C₁₇	—	4	2	no	—
42	C₄²×C₁₇	—	2	1	no	—
43	C₄²×C₁₇	—	2	1	no	—
44	C₄²×C₁₇	SD₁₆	1	2	yes	—
45	C₄²×C₁₇	Q₁₆	1	2	yes	—
46	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₇	#41	1	2	yes	—
48	C₄²×C₁₇	#41	1	1	yes	characteristic
49	C₂²⋊Dic₁₇	—	4	1	no	—
50	C₄⋊Dic₁₇	—	2	2	no	—
51	C₄⋊Dic₁₇	—	2	2	no	—
52	#63225	—	4	2	no	—
53	C₄×Dic₁₇	—	4	1	no	—
54	#63223	—	4	1	no	—
55	#63223	—	4	1	no	—
56	C₂×#3622	—	4	2	no	—
57	C₂×#3622	—	4	1	no	—
58	C₂×#3622	—	4	2	no	—
59	C₂×#3622	—	4	2	no	—
60	#10628	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	#123	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	4	2	no	—
65	C₂×C₄×C₁₇	—	4	2	no	—
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#123	1	2	yes	—
69	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#126	1	2	yes	—
71	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
75	C₂×Dic₁₇	—	4	2	no	—
76	Dic₃₄	—	8	2	no	—
77	C₈×C₁₇	—	8	2	no	—
78	#3622	—	8	2	no	—
79	#3622	—	8	1	no	—
80	#1995	—	17	1	no	—
81	#1843	—	17	1	no	—
82	#1542	—	17	2	no	—
83	#1301	—	17	2	no	—
84	#1274	D₁₇	1	1	yes	characteristic
85	C₂²×C₁₇	—	2	2	no	—
86	C₂²×C₁₇	#397	1	1	yes	characteristic
87	C₂²×C₁₇	#483	1	1	yes	characteristic
88	C₂²×C₁₇	#484	1	1	yes	characteristic
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	4	2	no	—
92	C₄×C₁₇	—	4	2	no	—
93	C₄×C₁₇	#460	1	1	yes	characteristic
94	C₄×C₁₇	#462	1	1	yes	characteristic
95	C₄×C₁₇	#487	1	1	yes	characteristic
96	C₄×C₁₇	#489	1	1	yes	characteristic
97	Dic₁₇	—	8	2	no	—
98	#533	—	17	1	no	—
99	#431	C₂×D₁₇	1	2	yes	—
100	#423	—	17	1	no	—
101	C₂×C₄⋊C₈	—	17	2	no	—
102	C₂×C₄⋊C₈	—	34	1	no	—
103	C₂×C₄⋊C₈	—	34	1	no	—
104	#376	C₂×D₁₇	1	1	yes	characteristic
105	#339	—	34	1	no	—
106	#339	—	34	1	no	—
107	#333	—	34	2	no	—
108	#332	—	34	2	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	#2072	1	1	yes	characteristic
112	C₂×C₁₇	#2080	1	2	yes	—
113	C₂×M₁₆	—	34	1	no	—
114	C₂²×C₈	—	34	2	no	—
115	C₄⋊Q₈	—	17	2	no	—
116	#125	—	34	1	no	—
117	C₂²⋊Q₈	—	34	2	no	—
118	C₄×Q₈	—	34	2	no	—
119	#117	—	34	1	no	—
120	C₂×C₄²	D₆₈	1	1	yes	characteristic
121	C₂×C₄²	D₆₈	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	34	2	no	—
125	C₄⋊C₈	—	34	2	no	—
126	C₄⋊C₈	—	34	2	no	—
127	C₄⋊C₈	—	68	2	no	—
128	C₄⋊C₈	—	68	2	no	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	#95	—	2	2	no	—
132	C₁₇	#10628	1	1	yes	characteristic, complemented, Sylow
133	C₂×Q₈	—	34	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
137	C₂²×C₄	#63254	1	2	yes	—
138	M₁₆	—	68	1	no	—
139	M₁₆	—	68	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	68	1	no	—
142	C₂×C₈	—	68	2	no	—
143	C₂×C₈	—	68	2	no	—
144	C₂×C₈	—	68	2	no	—
145	C₄⋊C₄	—	34	2	no	—
146	C₄⋊C₄	—	34	2	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₂²⋊C₄	—	68	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	68	1	no	—
152	C₄²	#63219	1	2	yes	—
153	C₄²	Dic₆₈	1	2	yes	—
154	C₄²	#63252	1	2	yes	—
155	C₄²	D₄×D₁₇	1	1	yes	characteristic
156	C₄²	#63256	1	1	yes	characteristic
157	C₂³	#10587572	1	1	yes	characteristic
158	Q₈	—	136	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	68	2	no	—
165	C₂×C₄	#10587561	1	1	yes	characteristic
166	C₂×C₄	#10587569	1	2	yes	—
167	C₂×C₄	#10587581	1	1	yes	characteristic
168	C₂×C₄	#10587584	1	2	yes	—
169	C₂×C₄	C₂×#63219	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
171	C₂×C₄	#10587594	1	1	yes	characteristic
172	C₂×C₄	#10587595	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	136	2	no	—
175	C₈	—	136	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#11762736	1	1	yes	characteristic
178	C₂²	#11763140	1	1	yes	characteristic
179	C₂²	#11763145	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	136	2	no	—
185	C₄	#11763010	1	1	yes	characteristic
186	C₄	#11763020	1	1	yes	characteristic
187	C₄	#11763160	1	1	yes	characteristic
188	C₄	#11763169	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#14530194	1	1	yes	characteristic
192	C₂	#14530236	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529765

C₂

yes

maximal, characteristic

#14528938

C₂

yes

maximal, characteristic

#14527439

C₂

yes

maximal

#14526680

C₂

yes

maximal

C₁₇×#1274

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

characteristic

#11762833

C₂²

yes

characteristic

C₂×C₄⋊#3622

—

C₂×C₄⋊#3622

—

C₂×C₄⋊#3622

C₂²

yes

—

—

—

—

—

C₁₇×#431

C₂²

yes

—

C₁₇×#376

C₂²

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

C₂²×#3622

—

#10587556

—

#10587555

D₄

yes

—

C₄×Dic₃₄

—

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

C₁₇×#95

—

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

SD₁₆

yes

—

C₄²×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄²×C₁₇

#41

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

—

—

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#126

yes

—

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#533

—

#431

C₂×D₁₇

yes

—

100

#423

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

—

104

#376

C₂×D₁₇

yes

characteristic

105

#339

—

106

#339

—

107

#333

—

108

#332

—

109

C₂×C₁₇

—

110

C₂×C₁₇

—

111

C₂×C₁₇

#2072

yes

characteristic

112

C₂×C₁₇

#2080

yes

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

#125

—

117

C₂²⋊Q₈

—

118

C₄×Q₈

—

119

#117

—

120

C₂×C₄²

D₆₈

yes

characteristic

121

C₂×C₄²

D₆₈

yes

characteristic

122

C₂×C₄²

C₂²×D₁₇

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₂²⋊C₈

—

130

C₄×C₈

—

131

#95

—

132

C₁₇

#10628

yes

characteristic, complemented, Sylow

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₆₈

yes

characteristic

137

C₂²×C₄

#63254

yes

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#63219

yes

—

153

C₄²

Dic₆₈

yes

—

154

C₄²

#63252

yes

—

155

C₄²

D₄×D₁₇

yes

characteristic

156

C₄²

#63256

yes

characteristic

157

C₂³

#10587572

yes

characteristic

158

Q₈

—

136

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#10587561

yes

characteristic

166

C₂×C₄

#10587569

yes

—

167

C₂×C₄

#10587581

yes

characteristic

168

C₂×C₄

#10587584

yes

—

169

C₂×C₄

C₂×#63219

yes

characteristic

170

C₂×C₄

C₂×Dic₆₈

yes

characteristic

171

C₂×C₄

#10587594

yes

characteristic

172

C₂×C₄

#10587595

yes

characteristic

173

C₈

—

174

C₈

—

136

—

175

C₈

—

136

—

176

C₂²

—

177

C₂²

#11762736

yes

characteristic

178

C₂²

#11763140

yes

characteristic

179

C₂²

#11763145

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

136

—

185

C₄

#11763010

yes

characteristic

186

C₄

#11763020

yes

characteristic

187

C₄

#11763160

yes

characteristic

188

C₄

#11763169

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#14530194

yes

characteristic

192

C₂

#14530236

yes

—

	subgroup	G/H
center	C₂×C₄	#10587561
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1274	C₂
socle	C₂²×C₁₇	#397
prime radical	C₂³×C₁₇	#123
hypercenter	#1274	D₁₇
hypocenter	C₁₇	#10628
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#10587561

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1274

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
132. C₁₇	#10628	semidirect

H ≅ G/N

type

132. C₁₇

#10628

semidirect

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

subgroup	index
#14529765	2
#14528938	2
#14527439	2
#14526680	2
C₁₇×#1274	2
#10628	17

subgroup

index

C₁₇×#1274

Subgroups — #16939099

Conjugacy classes of subgroups

Complements

Maximal subgroups