Group #17948979 — subgroups

Subgroups — #17948979

Order 4352.

202 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14545144	C₂	1	1	yes	maximal, characteristic
2	#14544342	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#335	C₂	1	1	yes	maximal, characteristic
4	#11763813	—	2	1	no	—
5	C₄⋊#63216	—	2	1	no	—
6	#11763704	C₄	1	2	yes	—
7	#11763695	—	2	1	no	—
8	#11763672	—	2	1	no	—
9	#11763672	C₄	1	2	yes	—
10	C₄⋊C₈×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762561	—	2	1	no	—
14	C₁₇×#335	C₄	1	1	yes	characteristic
15	C₂×#63243	—	4	1	no	—
16	C₂×C₈×D₁₇	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	C₄⋊#3624	—	2	2	no	—
21	C₄⋊#3624	—	4	1	no	—
22	C₂×#63216	—	2	2	no	—
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	#10587485	—	4	1	no	—
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₄⋊#3622	—	2	1	no	—
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
32	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
33	C₂×#3624	—	4	2	no	—
34	#63243	—	8	1	no	—
35	#63243	—	8	1	no	—
36	C₂×C₈×C₁₇	—	4	1	no	—
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
43	C₂×#3622	—	4	1	no	—
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₁₇	—	2	1	no	—
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	4	1	no	—
49	C₈×D₁₇	—	8	1	no	—
50	C₈×D₁₇	—	8	1	no	—
51	#63216	—	4	2	no	—
52	#7288	—	17	1	no	maximal, Sylow
53	C₂²×D₁₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
57	C₂×Dic₁₇	#102	1	1	yes	characteristic
58	C₂×Dic₁₇	#103	1	1	yes	characteristic
59	C₂×Dic₁₇	#106	1	1	yes	characteristic
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	4	1	no	—
63	C₄×D₁₇	D₁₆	1	1	yes	characteristic
64	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
65	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
66	C₄×D₁₇	Q₃₂	1	1	yes	characteristic
67	#3624	—	8	2	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	—	8	1	no	—
70	C₈×C₁₇	#108	1	1	yes	characteristic
71	C₈×C₁₇	#108	1	1	yes	characteristic
72	#3622	—	2	1	no	—
73	#3622	—	2	1	no	—
74	#3622	—	8	1	no	—
75	#2162	—	17	1	no	—
76	#1567	—	17	1	no	—
77	C₂×#335	—	17	1	no	—
78	C₂²×C₁₇	#340	1	1	yes	characteristic
79	C₂×D₁₇	—	2	1	no	—
80	C₂×D₁₇	—	2	1	no	—
81	C₂×D₁₇	#366	1	1	yes	characteristic
82	C₂×D₁₇	#367	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	#341	1	1	yes	characteristic
85	C₄×C₁₇	#344	1	1	yes	characteristic
86	C₄×C₁₇	#357	1	1	yes	characteristic
87	C₄×C₁₇	#358	1	1	yes	characteristic
88	Dic₁₇	—	2	1	no	—
89	Dic₁₇	—	2	1	no	—
90	Dic₁₇	#357	1	1	yes	characteristic
91	Dic₁₇	#358	1	1	yes	characteristic
92	#423	—	34	1	no	—
93	C₂×C₄⋊C₈	—	34	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	C₄⋊C₁₆	—	34	1	no	—
96	C₂²⋊C₁₆	—	17	2	no	—
97	C₄×C₁₆	—	34	1	no	—
98	#335	—	17	2	no	—
99	#335	—	17	1	no	—
100	#335	—	34	1	no	—
101	#335	—	34	1	no	—
102	#335	#595	1	1	yes	characteristic
103	C₂×C₁₇	#1301	1	1	yes	characteristic
104	C₂×C₁₇	#1380	1	1	yes	characteristic
105	C₂×C₁₇	#1388	1	1	yes	characteristic
106	D₁₇	—	2	1	no	—
107	D₁₇	—	2	1	no	—
108	C₂×M₁₆	—	68	1	no	—
109	C₂²×C₈	—	17	1	no	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	68	1	no	—
112	C₂×C₄²	—	17	1	no	—
113	C₂×C₁₆	—	34	2	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	34	2	no	—
116	C₄⋊C₈	—	34	1	no	—
117	C₄⋊C₈	—	68	1	no	—
118	C₄⋊C₈	—	68	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	17	1	no	—
122	C₄×C₈	C₂×#595	1	1	yes	characteristic
123	C₁₇	#7288	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	—	17	1	no	—
127	M₁₆	—	136	1	no	—
128	M₁₆	—	136	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	34	1	no	—
133	C₂×C₈	—	34	1	no	—
134	C₂×C₈	—	68	1	no	—
135	C₂×C₈	—	68	2	no	—
136	C₂×C₈	—	68	1	no	—
137	C₂×C₈	—	136	1	no	—
138	C₂×C₈	—	136	1	no	—
139	C₂×C₈	C₄×#595	1	1	yes	characteristic
140	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
141	C₄²	—	17	1	no	—
142	C₄²	—	17	1	no	—
143	C₄²	—	17	1	no	—
144	C₄²	C₂²⋊#595	1	1	yes	characteristic
145	C₁₆	—	68	2	no	—
146	C₂³	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	1	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	1	no	—
154	C₂×C₄	—	34	1	no	—
155	C₂×C₄	—	34	1	no	—
156	C₂×C₄	—	68	1	no	—
157	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
158	C₂×C₄	#10587547	1	1	yes	characteristic
159	C₂×C₄	#10587549	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	8	1	no	—
162	C₈	—	34	1	no	—
163	C₈	—	34	1	no	—
164	C₈	—	136	2	no	—
165	C₈	—	136	1	no	—
166	C₈	#10587535	1	1	yes	characteristic
167	C₈	#10587536	1	1	yes	characteristic
168	C₂²	—	17	1	no	—
169	C₂²	—	17	1	no	—
170	C₂²	—	34	1	no	—
171	C₂²	—	34	1	no	—
172	C₂²	#11763682	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	17	1	no	—
175	C₄	—	17	1	no	—
176	C₄	—	34	1	no	—
177	C₄	—	34	1	no	—
178	C₄	#11763685	1	1	yes	characteristic
179	C₄	#11763692	1	1	yes	characteristic
180	C₄	#11763721	1	1	yes	characteristic
181	C₄	#11763722	1	1	yes	characteristic
182	C₂	—	34	1	no	—
183	C₂	—	34	1	no	—
184	C₂	#14543676	1	1	yes	characteristic
185	C₂	#14543841	1	1	yes	characteristic
186	C₂	#14543863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545144

C₂

yes

maximal, characteristic

#14544342

C₂

yes

maximal, characteristic

D₁₇×#335

C₂

yes

maximal, characteristic

#11763813

—

C₄⋊#63216

—

#11763704

C₄

yes

—

#11763695

—

#11763672

—

#11763672

C₄

yes

—

C₄⋊C₈×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762561

—

C₁₇×#335

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

D₁₆

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

C₄×D₁₇

Q₃₂

yes

characteristic

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#335

—

#335

—

100

#335

—

101

#335

—

102

#335

#595

yes

characteristic

103

C₂×C₁₇

#1301

yes

characteristic

104

C₂×C₁₇

#1380

yes

characteristic

105

C₂×C₁₇

#1388

yes

characteristic

106

D₁₇

—

107

D₁₇

—

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂×C₄²

—

113

C₂×C₁₆

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#595

yes

characteristic

123

C₁₇

#7288

yes

characteristic, complemented, Sylow

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

M₁₆

—

136

—

128

M₁₆

—

136

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

136

—

138

C₂×C₈

—

136

—

139

C₂×C₈

C₄×#595

yes

characteristic

140

C₂×C₈

C₄⋊#595

yes

characteristic

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²⋊#595

yes

characteristic

145

C₁₆

—

146

C₂³

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

158

C₂×C₄

#10587547

yes

characteristic

159

C₂×C₄

#10587549

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

136

—

165

C₈

—

136

—

166

C₈

#10587535

yes

characteristic

167

C₈

#10587536

yes

characteristic

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#11763682

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#11763685

yes

characteristic

179

C₄

#11763692

yes

characteristic

180

C₄

#11763721

yes

characteristic

181

C₄

#11763722

yes

characteristic

182

C₂

—

183

C₂

—

184

C₂

#14543676

yes

characteristic

185

C₂

#14543841

yes

characteristic

186

C₂

#14543863

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#107
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#335	#595
hypocenter	C₁₇	#7288
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#335

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
123. C₁₇	#7288	semidirect

H ≅ G/N

type

123. C₁₇

#7288

semidirect

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

subgroup	index
#14545144	2
#14544342	2
D₁₇×#335	2
#7288	17

subgroup

index

#14545144

#14544342

D₁₇×#335

#7288

Subgroups — #17948979

Conjugacy classes of subgroups

Complements

Maximal subgroups