Group #16950504 — subgroups

Subgroups — #16950504

Order 4352.

394 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 394 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	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762638	C₂	1	2	yes	maximal, complemented
3	#14528819	C₂	1	2	yes	maximal
4	#14526878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₁₇	—	2	4	no	—
9	C₂×C₁₆×D₁₇	C₄	1	4	yes	—
10	C₂×#10587490	C₂²	1	2	yes	—
11	#11762772	—	2	2	no	—
12	#11762638	—	2	4	no	—
13	#11762573	—	2	2	no	—
14	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#108	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	—	2	2	no	—
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×#63223	—	2	2	no	—
20	C₂²×#3622	D₄	1	1	yes	characteristic
21	C₂×C₈×D₁₇	—	2	2	no	—
22	C₂×C₈×D₁₇	—	2	2	no	—
23	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×D₁₇	Q₈	1	2	yes	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
27	C₁₇×#108	—	2	2	no	—
28	C₁₇×#108	—	2	2	no	—
29	#10587491	—	4	4	no	—
30	#10587490	—	2	2	no	—
31	#10587490	—	2	2	no	—
32	C₂×#63214	C₂×C₄	1	2	yes	—
33	C₂×#63214	D₄	1	4	yes	—
34	C₁₆×D₁₇	—	2	8	no	—
35	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₁₇	—	2	2	no	—
39	C₂×C₄×D₁₇	—	2	2	no	—
40	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₁₇	D₈	1	2	yes	—
42	M₁₆×C₁₇	—	4	2	no	—
43	M₁₆×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	2	yes	—
49	#63223	—	4	2	no	—
50	#63223	—	4	2	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	—	4	2	no	—
53	C₂×#3622	C₄⋊C₄	1	2	yes	—
54	C₈×D₁₇	—	2	2	no	—
55	C₈×D₁₇	—	2	2	no	—
56	C₈×D₁₇	—	2	4	no	—
57	C₁₆×C₁₇	—	2	2	no	—
58	#63214	C₂²⋊C₄	1	4	yes	—
59	#12837	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₂²×D₁₇	—	2	2	no	—
62	C₂²×D₁₇	—	2	4	no	—
63	C₂²×D₁₇	#107	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#121	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	2	yes	—
68	C₂×Dic₁₇	—	2	2	no	—
69	C₂×Dic₁₇	#107	1	2	yes	—
70	C₄×D₁₇	—	2	2	no	—
71	C₄×D₁₇	—	2	2	no	—
72	C₄×D₁₇	—	2	4	no	—
73	C₈×C₁₇	—	8	2	no	—
74	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₁₇	#117	1	1	yes	characteristic
76	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
77	#3622	—	2	2	no	—
78	#3622	—	8	2	no	—
79	C₂³×C₁₆	—	17	1	no	—
80	C₂×#368	—	17	2	no	—
81	#1810	—	17	2	no	—
82	#1381	—	17	1	no	—
83	#1381	D₁₇	1	1	yes	characteristic, complemented
84	C₂²×C₁₇	—	2	2	no	—
85	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₁₇	#467	1	2	yes	—
87	C₂×D₁₇	—	2	8	no	—
88	C₂×D₁₇	—	2	2	no	—
89	C₂×D₁₇	—	2	2	no	—
90	C₂×D₁₇	#368	1	4	yes	complemented
91	C₄×C₁₇	#380	1	1	yes	characteristic
92	C₄×C₁₇	#427	1	1	yes	characteristic
93	C₄×C₁₇	#466	1	2	yes	—
94	Dic₁₇	—	2	2	no	—
95	C₂³×C₈	—	17	1	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂²×C₁₆	—	17	4	no	—
98	C₂²×C₁₆	—	34	4	no	—
99	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
100	C₂×#108	—	17	2	no	—
101	C₂×#108	C₂×D₁₇	1	2	yes	—
102	#407	—	34	2	no	—
103	#368	—	34	4	no	—
104	#341	—	34	2	no	—
105	C₂×C₁₇	—	2	2	no	—
106	C₂×C₁₇	#1808	1	1	yes	characteristic
107	C₂×C₁₇	C₂×#368	1	2	yes	—
108	D₁₇	—	2	4	no	—
109	C₂³×C₄	—	17	1	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	34	2	no	—
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	—	17	2	no	—
114	C₂²×C₈	—	17	2	no	—
115	C₂²×C₈	—	34	2	no	—
116	C₂²×C₈	—	34	2	no	—
117	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	—	17	4	no	—
121	C₂×C₁₆	—	34	8	no	—
122	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#108	—	34	2	no	—
126	#108	—	34	2	no	—
127	C₂²⋊C₈	—	68	4	no	—
128	C₁₇	#12837	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	—	17	1	no	—
130	C₂²×C₄	—	17	1	no	—
131	C₂²×C₄	—	17	2	no	—
132	C₂²×C₄	—	17	2	no	—
133	C₂²×C₄	—	34	2	no	—
134	C₂²×C₄	—	34	2	no	—
135	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	68	2	no	—
139	M₁₆	—	68	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	34	2	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	—	34	4	no	—
146	C₂×C₈	—	68	2	no	—
147	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
148	C₂×C₈	#63252	1	2	yes	—
149	C₂×C₈	D₄×D₁₇	1	2	yes	—
150	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
151	C₁₆	—	2	2	no	—
152	C₁₆	—	17	4	no	—
153	C₂³	—	17	2	no	—
154	C₂³	—	34	4	no	—
155	C₂³	—	34	2	no	—
156	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
157	C₂×C₄	—	17	2	no	—
158	C₂×C₄	—	34	2	no	—
159	C₂×C₄	—	34	4	no	—
160	C₂×C₄	—	34	2	no	—
161	C₂×C₄	—	34	2	no	—
162	C₂×C₄	#10587570	1	2	yes	—
163	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
164	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
165	C₂×C₄	#10587602	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	34	2	no	—
168	C₈	—	136	2	no	—
169	C₈	#10587558	1	1	yes	characteristic
170	C₈	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
171	C₈	#10587582	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	17	4	no	—
174	C₂²	—	34	8	no	—
175	C₂²	—	34	2	no	—
176	C₂²	—	34	2	no	—
177	C₂²	#107×D₁₇	1	1	yes	characteristic
178	C₂²	#11763049	1	2	yes	—
179	C₄	—	34	2	no	—
180	C₄	#11762668	1	1	yes	characteristic
181	C₄	#11762844	1	1	yes	characteristic
182	C₄	#11763044	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	34	4	no	—
185	C₂	#14528812	1	1	yes	characteristic
186	C₂	D₁₇×#368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₁₇

C₂

yes

maximal, characteristic

C₂×#11762638

C₂

yes

maximal, complemented

#14528819

C₂

yes

maximal

#14526878

C₂

yes

maximal, characteristic, complemented

C₁₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₁₇

C₂²

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

C₂×C₁₆×D₁₇

—

C₂×C₁₆×D₁₇

C₄

yes

—

C₂×#10587490

C₂²

yes

—

#11762772

—

#11762638

—

#11762573

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₂²×C₄×D₁₇

D₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂²×#3622

D₄

yes

characteristic

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

—

—

—

C₂×C₄

yes

—

C₂×#63214

D₄

yes

—

C₁₆×D₁₇

—

C₂³×D₁₇

Q₁₆

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

Q₁₆

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

—

C₂×C₄×D₁₇

D₈

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

C₂²⋊C₄

yes

—

#12837

—

maximal, Sylow

C₂³×C₁₇

C₂×Q₁₆

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

#107

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

#107

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

#117

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

#3622

—

#3622

—

C₂³×C₁₆

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂²×C₁₇

#467

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

100

C₂×#108

—

101

C₂×#108

C₂×D₁₇

yes

—

102

#407

—

103

#368

—

104

#341

—

105

C₂×C₁₇

—

106

C₂×C₁₇

#1808

yes

characteristic

107

C₂×C₁₇

C₂×#368

yes

—

108

D₁₇

—

109

C₂³×C₄

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂²×D₁₇

yes

characteristic

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

C₄×D₁₇

yes

—

123

#108

—

124

#108

—

125

#108

—

126

#108

—

127

C₂²⋊C₈

—

128

C₁₇

#12837

yes

characteristic, complemented, Sylow

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

D₄×D₁₇

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

148

C₂×C₈

#63252

yes

—

149

C₂×C₈

D₄×D₁₇

yes

—

150

C₂×C₈

Q₈×D₁₇

yes

characteristic

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

Q₁₆×D₁₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#10587570

yes

—

163

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

164

C₂×C₄

D₈×D₁₇

yes

characteristic

165

C₂×C₄

#10587602

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

136

—

169

C₈

#10587558

yes

characteristic

170

C₈

C₂²⋊C₄×D₁₇

yes

characteristic

171

C₈

#10587582

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

#107×D₁₇

yes

characteristic

178

C₂²

#11763049

yes

—

179

C₄

—

180

C₄

#11762668

yes

characteristic

181

C₄

#11762844

yes

characteristic

182

C₄

#11763044

yes

—

183

C₂

—

184

C₂

—

185

C₂

#14528812

yes

characteristic

186

C₂

D₁₇×#368

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1381	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1381	D₁₇
hypocenter	C₁₇	#12837
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₁₇

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1381

C₂

socle

C₂²×C₁₇

C₂×#107

prime radical

C₂³×D₁₇

Q₁₆

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#11762638	C₂	semidirect
4. #14526878	C₂	semidirect
5. C₁₇×#1381	C₂	semidirect
83. #1381	D₁₇	semidirect
90. C₂×D₁₇	#368	semidirect
128. C₁₇	#12837	semidirect

H ≅ G/N

type

2. C₂×#11762638

C₂

semidirect

4. #14526878

C₂

semidirect

5. C₁₇×#1381

C₂

semidirect

83. #1381

D₁₇

semidirect

90. C₂×D₁₇

#368

semidirect

128. C₁₇

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×#11762638	2
#14528819	2
#14526878	2
C₁₇×#1381	2
#12837	17

subgroup

index

C₂²×C₁₆×D₁₇

C₁₇×#1381

Subgroups — #16950504

Conjugacy classes of subgroups

Complements

Maximal subgroups