Group #16953973 — subgroups

Subgroups — #16953973

Order 4352.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#11763207	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762632	C₂	1	2	yes	maximal
3	#14529532	C₂	1	1	yes	maximal, characteristic
4	#14526890	C₂	1	2	yes	maximal
5	C₁₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	—	2	1	no	—
7	#11763207	—	2	4	no	—
8	C₂×#10587490	C₂²	1	1	yes	characteristic
9	#11762847	C₂²	1	1	yes	characteristic
10	C₂×#10587488	—	2	1	no	—
11	C₂×#10587488	C₂²	1	2	yes	—
12	#11762835	—	2	1	no	—
13	#11762632	—	2	4	no	—
14	#11762569	—	2	1	no	—
15	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₄	—	4	1	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
21	#10587612	—	2	1	no	—
22	C₂×C₄⋊Dic₁₇	—	2	2	no	—
23	C₂×C₄⋊Dic₁₇	—	4	1	no	—
24	C₂×#63223	—	2	1	no	—
25	#10587555	—	2	4	no	—
26	#10587555	—	4	2	no	—
27	C₁₇×M₃₂	—	2	2	no	—
28	C₁₇×M₃₂	Q₈	1	4	yes	complemented
29	C₂×C₁₆×C₁₇	—	2	2	no	—
30	C₄×C₈×C₁₇	—	2	2	no	—
31	#10587490	—	2	2	no	—
32	#10587489	Q₈	1	2	yes	complemented
33	#10587488	—	2	4	no	—
34	#10587488	—	2	2	no	—
35	#10587488	—	2	1	no	—
36	#10587488	D₄	1	4	yes	—
37	#10587485	—	4	2	no	—
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	4	2	no	—
41	C₂×Dic₃₄	—	4	4	no	—
42	C₂×Dic₃₄	—	8	4	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₁₇	—	4	1	no	—
53	C₄⋊Dic₁₇	—	2	4	no	—
54	C₄⋊Dic₁₇	—	2	2	no	—
55	C₄⋊Dic₁₇	—	4	4	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄×Dic₁₇	—	4	1	no	—
58	#63223	—	4	1	no	—
59	#63223	—	4	1	no	—
60	C₂×#3622	—	4	1	no	—
61	C₁₆×C₁₇	—	2	4	no	—
62	#13457	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	4	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
72	C₂×Dic₁₇	—	4	4	no	—
73	C₂×Dic₁₇	—	4	2	no	—
74	C₂×Dic₁₇	—	8	4	no	—
75	Dic₃₄	—	8	8	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
79	#3622	—	8	1	no	—
80	C₂×#498	—	17	1	no	—
81	C₂×#365	—	17	2	no	—
82	#1945	—	17	1	no	—
83	#1386	—	17	2	no	—
84	#1376	D₁₇	1	1	yes	characteristic
85	C₂²×C₁₇	—	2	2	no	—
86	C₂²×C₁₇	#450	1	1	yes	characteristic
87	C₂²×C₁₇	#476	1	1	yes	characteristic
88	C₂²×C₁₇	#501	1	1	yes	characteristic
89	C₄×C₁₇	—	2	4	no	—
90	C₄×C₁₇	#395	1	1	yes	characteristic
91	C₄×C₁₇	#448	1	1	yes	characteristic
92	C₄×C₁₇	#475	1	1	yes	characteristic
93	C₄×C₁₇	#498	1	1	yes	characteristic
94	Dic₁₇	—	8	4	no	—
95	Dic₁₇	—	8	2	no	—
96	C₂×C₄⋊Q₈	—	34	1	no	—
97	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
98	#498	—	34	4	no	—
99	C₂×#108	—	17	1	no	—
100	#427	—	17	1	no	—
101	C₂×#106	—	17	2	no	—
102	C₂×#106	—	34	1	no	—
103	#424	—	34	1	no	—
104	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
105	#365	—	34	4	no	—
106	#339	—	34	1	no	—
107	C₂×C₁₇	—	2	1	no	—
108	C₂×C₁₇	—	2	1	no	—
109	C₂×C₁₇	#2058	1	1	yes	characteristic
110	C₂×C₁₇	#2230	1	2	yes	—
111	C₂²×Q₈	—	68	1	no	—
112	C₂×M₁₆	—	34	1	no	—
113	C₂²×C₈	D₆₈	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
115	C₄⋊Q₈	—	34	4	no	—
116	C₄⋊Q₈	—	68	2	no	—
117	#117	—	34	1	no	—
118	C₂×C₄⋊C₄	—	34	2	no	—
119	C₂×C₄⋊C₄	—	68	1	no	—
120	C₂×C₄²	D₆₈	1	1	yes	characteristic
121	M₃₂	—	2	2	no	—
122	M₃₂	Dic₃₄	1	4	yes	complemented
123	C₂×C₁₆	—	2	2	no	—
124	#108	—	34	2	no	—
125	#107	—	17	2	no	—
126	#106	—	17	4	no	—
127	#106	—	34	4	no	—
128	#106	—	34	2	no	—
129	#106	—	34	1	no	—
130	C₄⋊C₈	—	68	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₁₇	#13457	1	1	yes	characteristic, complemented, Sylow
133	C₂×Q₈	—	68	4	no	—
134	C₂×Q₈	—	136	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	68	2	no	—
137	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
138	M₁₆	—	68	1	no	—
139	M₁₆	—	68	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	68	1	no	—
143	C₂×C₈	#63219	1	2	yes	—
144	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
145	C₂×C₈	D₄×D₁₇	1	2	yes	—
146	C₂×C₈	#63254	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
148	C₄⋊C₄	—	34	4	no	—
149	C₄⋊C₄	—	34	2	no	—
150	C₄⋊C₄	—	68	4	no	—
151	C₄⋊C₄	—	68	2	no	—
152	C₂²⋊C₄	—	68	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	68	1	no	—
155	C₄²	#63219	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	#10587594	1	1	yes	characteristic
158	Q₈	—	136	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	68	4	no	—
162	C₂×C₄	—	68	2	no	—
163	C₂×C₄	—	136	4	no	—
164	C₂×C₄	#10587555	1	1	yes	characteristic
165	C₂×C₄	#10587568	1	1	yes	characteristic
166	C₂×C₄	#10587583	1	1	yes	characteristic
167	C₂×C₄	C₂×#63219	1	1	yes	characteristic
168	C₂×C₄	C₂×#63219	1	1	yes	characteristic
169	C₂×C₄	#10587595	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	136	1	no	—
172	C₈	#10587564	1	2	yes	—
173	C₈	#10587574	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#11762962	1	1	yes	characteristic
176	C₂²	#11763100	1	1	yes	characteristic
177	C₂²	#11763218	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	136	4	no	—
180	C₄	—	136	2	no	—
181	C₄	#11762730	1	1	yes	characteristic
182	C₄	#11762950	1	1	yes	characteristic
183	C₄	#11763095	1	1	yes	characteristic
184	C₄	#11763207	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#14530118	1	1	yes	characteristic
188	C₂	#14530956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763207

C₂

yes

maximal, characteristic

C₂×#11762632

C₂

yes

maximal

#14529532

C₂

yes

maximal, characteristic

#14526890

C₂

yes

maximal

C₁₇×#1376

C₂

yes

maximal, characteristic

C₂×#10587555

—

#11763207

—

C₂×#10587490

C₂²

yes

characteristic

#11762847

C₂²

yes

characteristic

C₂×#10587488

—

C₂×#10587488

C₂²

yes

—

#11762835

—

#11762632

—

#11762569

—

C₂×C₁₇×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Dic₃₄

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

#10587612

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×#63223

—

#10587555

—

#10587555

—

C₁₇×M₃₂

—

C₁₇×M₃₂

Q₈

yes

complemented

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

—

#10587490

—

#10587489

Q₈

yes

—

—

—

D₄

yes

—

#10587485

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

SD₁₆

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#120

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

#137

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂×D₁₇

yes

—

#498

—

C₂×#108

—

100

#427

—

101

C₂×#106

—

102

C₂×#106

—

103

#424

—

104

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

105

#365

—

106

#339

—

107

C₂×C₁₇

—

108

C₂×C₁₇

—

109

C₂×C₁₇

#2058

yes

characteristic

110

C₂×C₁₇

#2230

yes

—

111

C₂²×Q₈

—

112

C₂×M₁₆

—

113

C₂²×C₈

D₆₈

yes

characteristic

114

C₂²×C₈

C₂²×D₁₇

yes

characteristic

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

#117

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

D₆₈

yes

characteristic

121

M₃₂

—

122

M₃₂

Dic₃₄

yes

complemented

123

C₂×C₁₆

—

124

#108

—

125

#107

—

126

#106

—

127

#106

—

128

#106

—

129

#106

—

130

C₄⋊C₈

—

131

C₄×C₈

—

132

C₁₇

#13457

yes

characteristic, complemented, Sylow

133

C₂×Q₈

—

134

C₂×Q₈

—

136

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₆₈

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#63219

yes

—

144

C₂×C₈

C₂×Dic₃₄

yes

—

145

C₂×C₈

D₄×D₁₇

yes

—

146

C₂×C₈

#63254

yes

characteristic

147

C₂×C₈

Q₈×D₁₇

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

#63219

yes

—

156

C₁₆

—

157

C₂³

#10587594

yes

characteristic

158

Q₈

—

136

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

136

—

164

C₂×C₄

#10587555

yes

characteristic

165

C₂×C₄

#10587568

yes

characteristic

166

C₂×C₄

#10587583

yes

characteristic

167

C₂×C₄

C₂×#63219

yes

characteristic

168

C₂×C₄

C₂×#63219

yes

characteristic

169

C₂×C₄

#10587595

yes

characteristic

170

C₈

—

171

C₈

—

136

—

172

C₈

#10587564

yes

—

173

C₈

#10587574

yes

—

174

C₂²

—

175

C₂²

#11762962

yes

characteristic

176

C₂²

#11763100

yes

characteristic

177

C₂²

#11763218

yes

characteristic

178

C₄

—

179

C₄

—

136

—

180

C₄

—

136

—

181

C₄

#11762730

yes

characteristic

182

C₄

#11762950

yes

characteristic

183

C₄

#11763095

yes

characteristic

184

C₄

#11763207

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#14530118

yes

characteristic

188

C₂

#14530956

yes

—

	subgroup	G/H
center	C₂²	#11763100
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1376	C₂
socle	C₂²×C₁₇	#476
prime radical	C₂³×C₁₇	AGL(1,ℤ/8)
hypercenter	#1376	D₁₇
hypocenter	C₁₇	#13457
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#11763100

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1376

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
28. C₁₇×M₃₂	Q₈	semidirect
32. #10587489	Q₈	semidirect
122. M₃₂	Dic₃₄	semidirect
132. C₁₇	#13457	semidirect

H ≅ G/N

type

28. C₁₇×M₃₂

Q₈

semidirect

32. #10587489

Q₈

semidirect

122. M₃₂

Dic₃₄

semidirect

132. C₁₇

#13457

semidirect

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

subgroup	index
C₂×#11763207	2
C₂×#11762632	2
#14529532	2
#14526890	2
C₁₇×#1376	2
#13457	17

subgroup

index

C₁₇×#1376

Subgroups — #16953973

Conjugacy classes of subgroups

Complements

Maximal subgroups