Group #16922296 — subgroups

Subgroups — #16922296

Order 4352.

444 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 159 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	#14528530	C₂	1	2	yes	maximal
2	#14526607	C₂	1	4	yes	maximal
3	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#3622	—	2	2	no	—
5	#11762564	—	2	2	no	—
6	#11762564	C₂²	1	4	yes	—
7	C₂×C₁₇×#125	C₄	1	1	yes	characteristic
8	C₁₇×#389	C₂²	1	2	yes	—
9	C₁₇×#387	—	2	2	no	—
10	C₁₇×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#125	—	2	2	no	—
15	C₁₇×#125	C₈	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₇	—	2	2	no	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	D₄	1	2	yes	—
24	C₂×C₄²×C₁₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	—	2	4	no	—
27	C₁₇×#95	—	2	2	no	—
28	C₁₇×#95	—	2	2	no	—
29	C₁₇×#95	—	2	2	no	—
30	C₄×#3622	—	2	8	no	—
31	C₂²×C₄×C₁₇	—	2	2	no	—
32	C₂²×C₄×C₁₇	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₁₇	—	2	8	no	—
39	C₄⋊C₄×C₁₇	—	2	2	no	—
40	C₄⋊C₄×C₁₇	—	4	2	no	—
41	C₄⋊C₄×C₁₇	—	4	4	no	—
42	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
43	C₄²×C₁₇	—	2	2	no	—
44	C₄²×C₁₇	—	2	4	no	—
45	C₄²×C₁₇	—	2	4	no	—
46	C₄²×C₁₇	—	4	1	no	—
47	C₄²×C₁₇	—	4	4	no	—
48	C₄²×C₁₇	C₂×C₈	1	2	yes	—
49	C₄²×C₁₇	M₁₆	1	2	yes	—
50	C₂×#3622	—	2	8	no	—
51	C₂×#3622	—	4	4	no	—
52	C₂×#3622	—	4	4	no	—
53	#7568	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	1	no	—
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	4	no	—
62	C₂×C₄×C₁₇	—	2	1	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	—	4	2	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	C₂²⋊C₈	1	4	yes	—
70	C₂×C₄×C₁₇	#101	1	2	yes	—
71	C₂×C₄×C₁₇	#104	1	4	yes	—
72	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#135	1	4	yes	—
75	#3622	—	4	8	no	—
76	C₄×#125	D₁₇	1	1	yes	characteristic
77	#1744	—	17	2	no	—
78	#1271	—	17	4	no	—
79	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₁₇	#410	1	1	yes	characteristic
81	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
82	C₂²×C₁₇	#416	1	2	yes	—
83	C₂²×C₁₇	C₂×#104	1	2	yes	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	—	2	4	no	—
87	C₄×C₁₇	—	2	4	no	—
88	C₄×C₁₇	—	4	1	no	—
89	C₄×C₁₇	—	4	4	no	—
90	C₄×C₁₇	—	8	2	no	—
91	C₄×C₁₇	#330	1	4	yes	—
92	C₂×#125	Dic₁₇	1	1	yes	characteristic
93	C₂×C₄×C₈	—	34	2	no	—
94	#389	C₂×D₁₇	1	2	yes	—
95	#387	—	2	2	no	—
96	#382	Dic₁₇	1	1	yes	characteristic
97	C₄×C₄⋊C₄	—	2	2	no	—
98	C₄×C₄⋊C₄	Dic₁₇	1	2	yes	—
99	C₄³	C₂×D₁₇	1	1	yes	characteristic
100	#336	—	17	4	no	—
101	#336	—	34	2	no	—
102	C₂×C₁₇	#1461	1	1	yes	characteristic
103	C₂×C₁₇	#1476	1	2	yes	—
104	C₂×C₁₇	#1500	1	2	yes	—
105	C₂×C₁₇	C₂×#330	1	1	yes	characteristic
106	C₂×C₁₇	#1544	1	1	yes	characteristic
107	C₂²×C₈	—	34	4	no	—
108	#125	—	2	2	no	—
109	#125	#3622	1	4	yes	complemented
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
119	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
120	C₄×C₈	—	34	8	no	—
121	#95	—	2	2	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	C₁₇	#7568	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
129	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
132	C₂×C₈	—	34	8	no	—
133	C₂×C₈	—	68	4	no	—
134	C₂×C₈	—	68	4	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	C₂×#3622	1	4	yes	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×#3622	1	2	yes	—
146	C₄²	#63223	1	2	yes	—
147	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂²⋊#3622	1	4	yes	—
163	C₂×C₄	#10587507	1	2	yes	—
164	C₂×C₄	#10587508	1	4	yes	—
165	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
166	C₂×C₄	C₂×#63223	1	1	yes	characteristic
167	C₂×C₄	#10587636	1	4	yes	—
168	C₈	—	68	8	no	—
169	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
170	C₂²	#11762783	1	1	yes	characteristic
171	C₂²	C₂×#10587507	1	1	yes	characteristic
172	C₂²	#11762805	1	2	yes	—
173	C₂²	C₂×#10587508	1	2	yes	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	2	no	—
181	C₄	#11762551	1	4	yes	—
182	C₂	#14527108	1	1	yes	characteristic
183	C₂	#14527166	1	2	yes	—
184	C₂	#14527270	1	2	yes	—
185	C₂	C₂×#11762551	1	1	yes	characteristic
186	C₂	#14527449	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528530

C₂

yes

maximal

#14526607

C₂

yes

maximal

C₄×C₁₇×#125

C₂

yes

maximal, characteristic

C₂×C₄×#3622

—

#11762564

—

#11762564

C₂²

yes

—

C₂×C₁₇×#125

C₄

yes

characteristic

C₁₇×#389

C₂²

yes

—

C₁₇×#387

—

C₁₇×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₇

—

C₄×C₄⋊C₄×C₁₇

C₄

yes

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#125

—

C₁₇×#125

C₈

yes

complemented

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

M₁₆

yes

—

—

—

—

—

C₂³×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

#101

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

#3622

—

C₄×#125

D₁₇

yes

characteristic

#1744

—

#1271

—

C₂²×C₁₇

C₂×C₂²⋊C₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

C₂×#104

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#330

yes

—

C₂×#125

Dic₁₇

yes

characteristic

C₂×C₄×C₈

—

#389

C₂×D₁₇

yes

—

#387

—

#382

Dic₁₇

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₁₇

yes

—

C₄³

C₂×D₁₇

yes

characteristic

100

#336

—

101

#336

—

102

C₂×C₁₇

#1461

yes

characteristic

103

C₂×C₁₇

#1476

yes

—

104

C₂×C₁₇

#1500

yes

—

105

C₂×C₁₇

C₂×#330

yes

characteristic

106

C₂×C₁₇

#1544

yes

characteristic

107

C₂²×C₈

—

108

#125

—

109

#125

#3622

yes

complemented

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

113

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

116

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

117

C₂×C₄²

C₂²⋊D₁₇

yes

—

118

C₂×C₄²

C₂²⋊D₁₇

yes

—

119

C₂×C₄²

C₂²×D₁₇

yes

characteristic

120

C₄×C₈

—

121

#95

—

122

#95

—

123

#95

—

124

C₁₇

#7568

yes

characteristic, complemented, Sylow

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²⋊Dic₁₇

yes

characteristic

128

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

129

C₂²×C₄

C₂²⋊Dic₁₇

yes

characteristic

130

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

131

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

C₂×#3622

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂×#3622

yes

—

146

C₄²

#63223

yes

—

147

C₂³

C₂×C₂²⋊Dic₁₇

yes

characteristic

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

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂²⋊#3622

yes

—

163

C₂×C₄

#10587507

yes

—

164

C₂×C₄

#10587508

yes

—

165

C₂×C₄

C₂²×#3622

yes

characteristic

166

C₂×C₄

C₂×#63223

yes

characteristic

167

C₂×C₄

#10587636

yes

—

168

C₈

—

169

C₂²

C₂×C₂²⋊#3622

yes

characteristic

170

C₂²

#11762783

yes

characteristic

171

C₂²

C₂×#10587507

yes

characteristic

172

C₂²

#11762805

yes

—

173

C₂²

C₂×#10587508

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#11762551

yes

—

182

C₂

#14527108

yes

characteristic

183

C₂

#14527166

yes

—

184

C₂

#14527270

yes

—

185

C₂

C₂×#11762551

yes

characteristic

186

C₂

#14527449

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#7568
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₄×C₁₇×#125

C₂

socle, prime radical

C₂³×C₁₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
15. C₁₇×#125	C₈	semidirect
109. #125	#3622	semidirect
124. C₁₇	#7568	semidirect

H ≅ G/N

type

15. C₁₇×#125

C₈

semidirect

109. #125

#3622

semidirect

124. C₁₇

#7568

semidirect

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

subgroup	index
#14528530	2
#14526607	2
C₄×C₁₇×#125	2
#7568	17

subgroup

index

#14528530

#14526607

C₄×C₁₇×#125

#7568

Subgroups — #16922296

Conjugacy classes of subgroups

Complements

Maximal subgroups