Group #16939098 — subgroups

Subgroups — #16939098

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	#14527442	C₂	1	2	yes	maximal
3	#14526681	C₂	1	2	yes	maximal
4	#14526613	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1843	C₂	1	1	yes	maximal, characteristic
6	#11763366	C₂²	1	1	yes	characteristic
7	#11762862	C₂²	1	2	yes	—
8	#11762833	C₂²	1	1	yes	characteristic
9	#11762569	—	2	1	no	—
10	#11762569	—	2	1	no	—
11	#11762557	—	2	2	no	—
12	#11762554	—	2	2	no	—
13	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
14	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
15	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₁₇×#376	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	—	2	2	no	—
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	#10587612	—	2	1	no	—
22	#10587610	—	2	2	no	—
23	C₂×#63223	—	2	1	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×C₄⋊C₈	—	2	2	no	—
29	C₁₇×C₄⋊C₈	—	4	2	no	—
30	C₁₇×C₄⋊C₈	—	4	2	no	—
31	C₁₇×#95	—	2	2	no	—
32	C₂²⋊#3622	—	2	2	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄×#3622	—	2	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	C₂×C₈×C₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	—	4	2	no	—
44	C₄²×C₁₇	—	2	1	no	—
45	C₄²×C₁₇	—	2	1	no	—
46	C₄²×C₁₇	SD₁₆	1	2	yes	—
47	C₄²×C₁₇	Q₁₆	1	2	yes	—
48	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	2	yes	—
51	C₂²⋊Dic₁₇	—	4	1	no	—
52	C₄⋊Dic₁₇	—	2	2	no	—
53	C₄⋊Dic₁₇	—	2	2	no	—
54	#63225	—	4	2	no	—
55	C₄×Dic₁₇	—	4	1	no	—
56	#63223	—	4	1	no	—
57	#63223	—	4	1	no	—
58	C₂×#3622	—	4	2	no	—
59	C₂×#3622	—	4	1	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	D₁₇	1	1	yes	characteristic
82	#1542	—	17	2	no	—
83	#1301	—	17	2	no	—
84	#1274	—	17	1	no	—
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	—	17	2	no	—
100	#423	—	17	1	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
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	—	2	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	C₂²⋊D₁₇	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	34	2	no	—
128	C₄⋊C₈	—	34	2	no	—
129	C₂²⋊C₈	—	34	2	no	—
130	C₄×C₈	—	34	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	#63252	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
138	M₁₆	—	68	1	no	—
139	M₁₆	—	68	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	68	2	no	—
144	C₂×C₈	—	68	1	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	#63229	1	2	yes	—
153	C₄²	#63231	1	2	yes	—
154	C₄²	D₄×D₁₇	1	1	yes	characteristic
155	C₄²	#63254	1	1	yes	characteristic
156	C₄²	#63256	1	2	yes	—
157	C₂³	#10587615	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	#10587580	1	2	yes	—
166	C₂×C₄	#10587584	1	2	yes	—
167	C₂×C₄	#10587618	1	1	yes	characteristic
168	C₂×C₄	C₂×#63229	1	1	yes	characteristic
169	C₂×C₄	#10587624	1	1	yes	characteristic
170	C₂×C₄	#10587628	1	1	yes	characteristic
171	C₂×C₄	C₂×#63231	1	1	yes	characteristic
172	C₂×C₄	#10587633	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	#11762735	1	1	yes	characteristic
178	C₂²	#11763139	1	1	yes	characteristic
179	C₂²	#11763144	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	#11763006	1	1	yes	characteristic
186	C₄	#11763016	1	1	yes	characteristic
187	C₄	#11763161	1	1	yes	characteristic
188	C₄	#11763171	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#14530193	1	1	yes	characteristic
192	C₂	#14530235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529765

C₂

yes

maximal, characteristic

#14527442

C₂

yes

maximal

#14526681

C₂

yes

maximal

#14526613

C₂

yes

maximal, characteristic

C₁₇×#1843

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

characteristic

#11762862

C₂²

yes

—

#11762833

C₂²

yes

—

—

—

—

C₂×C₁₇×C₄⋊C₈

—

C₂×C₁₇×C₄⋊C₈

—

C₂×C₁₇×C₄⋊C₈

C₂²

yes

—

C₁₇×#376

C₂²

yes

characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₃₄

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×#95

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

SD₁₆

yes

—

C₄²×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂×D₄

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

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

—

—

—

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

—

100

#423

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

C₂×D₁₇

yes

—

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

C₂²⋊D₁₇

yes

characteristic

121

C₂×C₄²

C₂²⋊D₁₇

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

#63252

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

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

#63229

yes

—

153

C₄²

#63231

yes

—

154

C₄²

D₄×D₁₇

yes

characteristic

155

C₄²

#63254

yes

characteristic

156

C₄²

#63256

yes

—

157

C₂³

#10587615

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

#10587580

yes

—

166

C₂×C₄

#10587584

yes

—

167

C₂×C₄

#10587618

yes

characteristic

168

C₂×C₄

C₂×#63229

yes

characteristic

169

C₂×C₄

#10587624

yes

characteristic

170

C₂×C₄

#10587628

yes

characteristic

171

C₂×C₄

C₂×#63231

yes

characteristic

172

C₂×C₄

#10587633

yes

characteristic

173

C₈

—

174

C₈

—

136

—

175

C₈

—

136

—

176

C₂²

—

177

C₂²

#11762735

yes

characteristic

178

C₂²

#11763139

yes

characteristic

179

C₂²

#11763144

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

136

—

185

C₄

#11763006

yes

characteristic

186

C₄

#11763016

yes

characteristic

187

C₄

#11763161

yes

characteristic

188

C₄

#11763171

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#14530193

yes

characteristic

192

C₂

#14530235

yes

—

	subgroup	G/H
center	C₂×C₄	#10587633
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1843	C₂
socle	C₂²×C₁₇	#397
prime radical	C₂³×C₁₇	#123
hypercenter	#1843	D₁₇
hypocenter	C₁₇	#10628
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#10587633

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1843

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

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
#14527442	2
#14526681	2
#14526613	2
C₁₇×#1843	2
#10628	17

subgroup

index

C₁₇×#1843

Subgroups — #16939098

Conjugacy classes of subgroups

Complements

Maximal subgroups