Group #16954215 — subgroups

Subgroups — #16954215

Order 4352.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#14530394	C₂	1	1	yes	maximal, characteristic
2	#14528819	C₂	1	2	yes	maximal
3	#14526895	C₂	1	2	yes	maximal, complemented
4	#14526888	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₂×#10587483	C₂²	1	1	yes	characteristic
8	C₂×#10587490	C₂²	1	2	yes	—
9	#11762772	—	2	2	no	—
10	#11762588	—	2	2	no	—
11	#11762573	—	2	2	no	—
12	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#108	C₂²	1	2	yes	—
14	C₁₇×#368	—	2	2	no	—
15	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×#63223	—	2	2	no	—
19	C₂²×#3622	D₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	—	2	1	no	—
21	C₂×C₈×D₁₇	—	2	2	no	—
22	C₂×C₈×D₁₇	—	2	1	no	—
23	C₂×C₈×D₁₇	—	2	2	no	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
26	C₁₇×#108	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	#10587491	—	4	4	no	—
29	#10587490	—	2	2	no	—
30	#10587490	—	2	2	no	—
31	#10587483	—	2	2	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
36	C₂×C₄×D₁₇	—	2	1	no	—
37	C₂×C₄×D₁₇	—	2	2	no	—
38	C₂×C₄×D₁₇	—	2	1	no	—
39	C₂×C₄×D₁₇	—	2	2	no	—
40	M₁₆×C₁₇	—	4	2	no	—
41	M₁₆×C₁₇	—	4	2	no	—
42	C₂×C₈×C₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₇	#41	1	2	yes	—
47	#63223	—	4	2	no	—
48	#63223	—	4	2	no	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	4	2	no	—
52	C₈×D₁₇	—	4	1	no	—
53	C₈×D₁₇	—	4	1	no	—
54	C₈×D₁₇	—	4	2	no	—
55	C₁₆×C₁₇	—	2	2	no	—
56	#63214	—	4	1	no	—
57	#13498	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×D₁₇	—	2	1	no	—
60	C₂²×D₁₇	—	2	2	no	—
61	C₂²×D₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#123	1	2	yes	—
64	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
65	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×Dic₁₇	—	2	1	no	—
67	C₂×Dic₁₇	—	2	2	no	—
68	C₄×D₁₇	—	4	1	no	—
69	C₄×D₁₇	—	4	1	no	—
70	C₄×D₁₇	—	4	2	no	—
71	C₈×C₁₇	—	8	2	no	—
72	C₈×C₁₇	#123	1	1	yes	characteristic
73	C₈×C₁₇	#124	1	1	yes	characteristic
74	C₈×C₁₇	#126	1	2	yes	—
75	#3622	—	4	1	no	—
76	#3622	—	8	2	no	—
77	#2161	D₁₇	1	1	yes	characteristic, complemented
78	#2112	—	17	1	no	—
79	#1810	—	17	2	no	—
80	#1388	—	17	2	no	—
81	#1385	—	17	1	no	—
82	C₂²×C₁₇	—	4	1	no	—
83	C₂²×C₁₇	#467	1	1	yes	characteristic
84	C₂²×C₁₇	#484	1	2	yes	—
85	C₂×D₁₇	—	2	2	no	—
86	C₂×D₁₇	—	2	4	no	—
87	C₂×D₁₇	—	4	1	no	—
88	C₂×D₁₇	—	4	2	no	—
89	C₂×D₁₇	—	4	1	no	—
90	C₄×C₁₇	#397	1	1	yes	characteristic
91	C₄×C₁₇	#469	1	1	yes	characteristic
92	C₄×C₁₇	#482	1	2	yes	—
93	Dic₁₇	—	4	1	no	—
94	C₂³×C₈	—	17	1	no	—
95	C₂×M₃₂	—	17	1	no	—
96	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
97	C₂×#108	—	17	2	no	—
98	C₂×#108	C₂×D₁₇	1	2	yes	—
99	#407	—	34	2	no	—
100	#368	—	2	2	no	—
101	C₂²⋊C₁₆	—	34	2	no	—
102	#341	—	34	2	no	—
103	C₂×C₁₇	—	2	1	no	—
104	C₂×C₁₇	—	2	2	no	—
105	C₂×C₁₇	#2076	1	1	yes	characteristic
106	D₁₇	—	4	2	no	—
107	C₂³×C₄	—	17	1	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	34	2	no	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	34	2	no	—
112	C₂²×C₈	—	34	1	no	—
113	C₂²×C₈	—	34	2	no	—
114	C₂²×C₈	—	34	1	no	—
115	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	M₃₂	—	2	2	no	—
117	M₃₂	—	34	2	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
120	#108	—	2	2	no	—
121	#108	—	2	2	no	—
122	#108	—	34	2	no	—
123	#108	—	34	2	no	—
124	C₂²⋊C₈	—	68	4	no	—
125	C₁₇	#13498	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴	—	17	1	no	—
127	C₂²×C₄	—	17	1	no	—
128	C₂²×C₄	—	34	2	no	—
129	C₂²×C₄	—	34	2	no	—
130	C₂²×C₄	—	34	1	no	—
131	C₂²×C₄	—	34	1	no	—
132	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	68	2	no	—
136	M₁₆	—	68	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	34	2	no	—
139	C₂×C₈	—	34	1	no	—
140	C₂×C₈	—	68	2	no	—
141	C₂×C₈	—	68	1	no	—
142	C₂×C₈	—	68	1	no	—
143	C₂×C₈	—	68	2	no	—
144	C₂×C₈	#63252	1	2	yes	—
145	C₂×C₈	#63254	1	1	yes	characteristic
146	C₂×C₈	#63256	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	68	1	no	—
150	C₂³	—	34	1	no	—
151	C₂³	—	34	2	no	—
152	C₂³	—	68	2	no	—
153	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
154	C₂×C₄	—	34	1	no	—
155	C₂×C₄	—	34	2	no	—
156	C₂×C₄	—	68	1	no	—
157	C₂×C₄	—	68	1	no	—
158	C₂×C₄	—	68	2	no	—
159	C₂×C₄	#10587580	1	2	yes	—
160	C₂×C₄	#10587597	1	2	yes	—
161	C₂×C₄	#10587605	1	1	yes	characteristic
162	C₂×C₄	#10587624	1	1	yes	characteristic
163	C₈	—	8	2	no	—
164	C₈	—	68	1	no	—
165	C₈	—	136	2	no	—
166	C₈	#10587584	1	2	yes	—
167	C₈	#10587615	1	1	yes	characteristic
168	C₈	#10587633	1	1	yes	characteristic
169	C₂²	—	4	1	no	—
170	C₂²	—	34	2	no	—
171	C₂²	—	34	4	no	—
172	C₂²	—	68	1	no	—
173	C₂²	—	68	2	no	—
174	C₂²	—	68	1	no	—
175	C₂²	#11763047	1	1	yes	characteristic
176	C₂²	#11763146	1	2	yes	—
177	C₄	—	68	1	no	—
178	C₄	#11762735	1	1	yes	characteristic
179	C₄	#11763059	1	1	yes	characteristic
180	C₄	#11763136	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	68	2	no	—
184	C₂	#14530213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530394

C₂

yes

maximal, characteristic

#14528819

C₂

yes

maximal

#14526895

C₂

yes

maximal, complemented

#14526888

C₂

yes

maximal, characteristic, complemented

C₁₇×#2161

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₁₇

C₂²

yes

characteristic

C₂×#10587483

C₂²

yes

characteristic

C₂×#10587490

C₂²

yes

—

#11762772

—

#11762588

—

#11762573

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₁₇×#368

—

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×C₈×D₁₇

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

—

—

—

—

—

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

—

C₂×C₄×D₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

—

—

—

—

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

—

#13498

—

maximal, Sylow

C₂³×C₁₇

C₂×Q₁₆

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

#3622

—

#3622

—

#2161

D₁₇

yes

characteristic, complemented

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#467

yes

characteristic

C₂²×C₁₇

#484

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₁₇

yes

—

#407

—

100

#368

—

101

C₂²⋊C₁₆

—

102

#341

—

103

C₂×C₁₇

—

104

C₂×C₁₇

—

105

C₂×C₁₇

#2076

yes

characteristic

106

D₁₇

—

107

C₂³×C₄

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²×D₁₇

yes

characteristic

116

M₃₂

—

117

M₃₂

—

118

C₂×C₁₆

—

119

C₂×C₁₆

C₂²⋊D₁₇

yes

—

120

#108

—

121

#108

—

122

#108

—

123

#108

—

124

C₂²⋊C₈

—

125

C₁₇

#13498

yes

characteristic, complemented, Sylow

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

D₄×D₁₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

#63252

yes

—

145

C₂×C₈

#63254

yes

characteristic

146

C₂×C₈

#63256

yes

—

147

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

Q₁₆×D₁₇

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#10587580

yes

—

160

C₂×C₄

#10587597

yes

—

161

C₂×C₄

#10587605

yes

characteristic

162

C₂×C₄

#10587624

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

136

—

166

C₈

#10587584

yes

—

167

C₈

#10587615

yes

characteristic

168

C₈

#10587633

yes

characteristic

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#11763047

yes

characteristic

176

C₂²

#11763146

yes

—

177

C₄

—

178

C₄

#11762735

yes

characteristic

179

C₄

#11763059

yes

characteristic

180

C₄

#11763136

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#14530213

yes

characteristic

	subgroup	G/H
center	C₄	#11763059
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2161	C₂
socle	C₂×C₁₇	#2076
prime radical	C₂³×D₁₇	Q₁₆
hypercenter	#2161	D₁₇
hypocenter	C₁₇	#13498
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#11763059

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2161

C₂

socle

C₂×C₁₇

#2076

prime radical

C₂³×D₁₇

Q₁₆

D₁₇

C₁₇

G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
3. #14526895	C₂	semidirect
4. #14526888	C₂	semidirect
5. C₁₇×#2161	C₂	semidirect
77. #2161	D₁₇	semidirect
125. C₁₇	#13498	semidirect

H ≅ G/N

type

3. #14526895

C₂

semidirect

4. #14526888

C₂

semidirect

5. C₁₇×#2161

C₂

semidirect

77. #2161

D₁₇

semidirect

125. C₁₇

#13498

semidirect

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

subgroup	index
#14530394	2
#14528819	2
#14526895	2
#14526888	2
C₁₇×#2161	2
#13498	17

subgroup

index

C₁₇×#2161

Subgroups — #16954215

Conjugacy classes of subgroups

Complements

Maximal subgroups