Group #16953982 — subgroups

Subgroups — #16953982

Order 4352.

318 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14539308	C₂	1	1	yes	maximal, characteristic
2	#14530558	C₂	1	2	yes	maximal
3	#14526885	C₂	1	2	yes	maximal
4	#14526864	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1946	C₂	1	1	yes	maximal, characteristic
6	#11763366	—	2	1	no	—
7	C₂×#10587483	C₂²	1	2	yes	—
8	#11763218	—	2	4	no	—
9	#11762847	—	2	1	no	—
10	#11762847	C₂²	1	2	yes	—
11	#11762633	—	2	4	no	—
12	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
13	C₁₇×#427	C₂²	1	1	yes	characteristic
14	C₁₇×#424	—	2	1	no	—
15	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
16	C₁₇×#339	—	2	1	no	—
17	C₂×M₁₆×C₁₇	—	2	1	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₁₇×#117	—	2	1	no	—
21	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
22	#10587612	—	2	2	no	—
23	#10587610	—	4	2	no	—
24	#10587556	—	2	2	no	—
25	#10587555	—	2	2	no	—
26	C₄×Dic₃₄	—	4	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	C₁₇×#107	—	2	1	no	—
29	C₁₇×#106	Q₈	1	2	yes	complemented
30	C₁₇×C₄⋊C₈	—	4	2	no	—
31	C₁₇×#97	—	2	2	no	—
32	#10587489	—	2	2	no	—
33	#10587489	—	2	2	no	—
34	#10587488	—	2	2	no	—
35	#10587488	D₄	1	4	yes	—
36	#10587483	—	2	2	no	—
37	#10587483	—	2	2	no	—
38	C₂×#63214	Q₈	1	4	yes	complemented
39	C₂²×C₄×C₁₇	—	2	1	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃₄	—	4	2	no	—
42	M₁₆×C₁₇	—	4	1	no	—
43	M₁₆×C₁₇	—	4	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	4	1	no	—
47	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₁₇	—	2	2	no	—
53	C₂²⋊C₄×C₁₇	—	4	1	no	—
54	C₄²×C₁₇	—	2	1	no	—
55	C₄²×C₁₇	—	4	1	no	—
56	C₄²×C₁₇	SD₁₆	1	2	yes	—
57	C₂²⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	2	4	no	—
59	C₄⋊Dic₁₇	—	4	2	no	—
60	#63225	—	4	4	no	—
61	C₄×Dic₁₇	—	4	2	no	—
62	#63214	—	2	4	no	—
63	#13459	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	—	8	1	no	—
68	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
74	C₂×Dic₁₇	—	4	4	no	—
75	Dic₃₄	—	8	4	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	—	8	1	no	—
78	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
80	#3162	—	17	1	no	—
81	#2155	—	17	2	no	—
82	#1946	D₁₇	1	1	yes	characteristic
83	#1384	—	17	2	no	—
84	#1375	—	17	1	no	—
85	C₂²×C₁₇	—	4	1	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	—	4	2	no	—
90	C₄×C₁₇	—	8	2	no	—
91	C₄×C₁₇	#395	1	1	yes	characteristic
92	C₄×C₁₇	#448	1	1	yes	characteristic
93	C₄×C₁₇	#475	1	1	yes	characteristic
94	C₄×C₁₇	#498	1	1	yes	characteristic
95	Dic₁₇	—	8	4	no	—
96	#533	—	34	1	no	—
97	C₂×M₃₂	—	17	2	no	—
98	#501	—	34	4	no	—
99	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
100	#427	—	17	2	no	—
101	#427	—	34	1	no	—
102	#427	C₂×D₁₇	1	1	yes	characteristic
103	#424	—	2	1	no	—
104	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
105	#365	—	34	4	no	—
106	#339	—	2	1	no	—
107	C₂×C₁₇	—	2	1	no	—
108	C₂×C₁₇	—	2	1	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	#2058	1	1	yes	characteristic
111	C₂×M₁₆	—	2	1	no	—
112	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	C₄⋊Q₈	—	34	2	no	—
115	#125	—	34	2	no	—
116	C₂²⋊Q₈	—	68	2	no	—
117	C₄×Q₈	—	68	2	no	—
118	#117	—	2	1	no	—
119	#117	—	34	2	no	—
120	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
121	M₃₂	—	34	2	no	—
122	M₃₂	—	34	2	no	—
123	C₂×C₁₆	—	17	4	no	—
124	#108	—	2	2	no	—
125	#107	—	2	1	no	—
126	#107	—	34	2	no	—
127	#107	—	34	2	no	—
128	#106	—	17	4	no	—
129	#106	—	34	2	no	—
130	#106	Dic₃₄	1	2	yes	complemented
131	C₄⋊C₈	—	4	2	no	—
132	#97	—	2	2	no	—
133	C₁₇	#13459	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	68	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	M₁₆	—	4	1	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	#63229	1	2	yes	—
143	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
144	C₂×C₈	#63252	1	1	yes	characteristic
145	C₂×C₈	D₄×D₁₇	1	2	yes	—
146	C₂×C₈	Q₈×D₁₇	1	2	yes	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	34	4	no	—
149	C₄⋊C₄	—	68	4	no	—
150	C₄⋊C₄	—	68	2	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₂²⋊C₄	—	68	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	—	68	2	no	—
156	C₄²	#63230	1	2	yes	—
157	C₁₆	—	34	4	no	—
158	C₂³	#10587635	1	1	yes	characteristic
159	Q₈	—	136	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	68	4	no	—
164	C₂×C₄	#10587610	1	1	yes	characteristic
165	C₂×C₄	C₂×#63229	1	1	yes	characteristic
166	C₂×C₄	#10587623	1	1	yes	characteristic
167	C₂×C₄	C₂×#63230	1	1	yes	characteristic
168	C₂×C₄	#10587630	1	1	yes	characteristic
169	C₂×C₄	#10587637	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₈	#10587574	1	2	yes	—
173	C₈	#10587582	1	2	yes	—
174	C₂²	—	4	1	no	—
175	C₂²	#11762959	1	1	yes	characteristic
176	C₂²	#11763099	1	1	yes	characteristic
177	C₂²	#11763219	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	136	4	no	—
181	C₄	#11762729	1	1	yes	characteristic
182	C₄	#11762947	1	1	yes	characteristic
183	C₄	#11763094	1	1	yes	characteristic
184	C₄	#11763205	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#14530117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14539308

C₂

yes

maximal, characteristic

#14530558

C₂

yes

maximal

#14526885

C₂

yes

maximal

#14526864

C₂

yes

maximal, characteristic

C₁₇×#1946

C₂

yes

maximal, characteristic

#11763366

—

C₂×#10587483

C₂²

yes

—

#11763218

—

#11762847

—

#11762847

C₂²

yes

—

#11762633

—

C₂×C₁₇×#108

C₂²

yes

characteristic

C₁₇×#427

C₂²

yes

characteristic

C₁₇×#424

—

C₂×C₁₇×#97

C₂²

yes

characteristic

C₁₇×#339

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

—

C₂×C₄²×C₁₇

D₄

yes

—

—

—

—

C₄×Dic₃₄

—

C₁₇×#108

—

C₁₇×#107

—

C₁₇×#106

Q₈

yes

complemented

C₁₇×C₄⋊C₈

—

C₁₇×#97

—

—

—

—

D₄

yes

—

#10587483

—

#10587483

—

C₂×#63214

Q₈

yes

complemented

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

SD₁₆

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

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

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

—

#3162

—

#2155

—

#1946

D₁₇

yes

characteristic

#1384

—

#1375

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#533

—

C₂×M₃₂

—

#501

—

C₂×#108

C₂×D₁₇

yes

characteristic

100

#427

—

101

#427

—

102

#427

C₂×D₁₇

yes

characteristic

103

#424

—

104

C₂×#97

C₂×D₁₇

yes

characteristic

105

#365

—

106

#339

—

107

C₂×C₁₇

—

108

C₂×C₁₇

—

109

C₂×C₁₇

—

110

C₂×C₁₇

#2058

yes

characteristic

111

C₂×M₁₆

—

112

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

113

C₂²×C₈

C₂²×D₁₇

yes

characteristic

114

C₄⋊Q₈

—

115

#125

—

116

C₂²⋊Q₈

—

117

C₄×Q₈

—

118

#117

—

119

#117

—

120

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

121

M₃₂

—

122

M₃₂

—

123

C₂×C₁₆

—

124

#108

—

125

#107

—

126

#107

—

127

#107

—

128

#106

—

129

#106

—

130

#106

Dic₃₄

yes

complemented

131

C₄⋊C₈

—

132

#97

—

133

C₁₇

#13459

yes

characteristic, complemented, Sylow

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#63229

yes

—

143

C₂×C₈

C₂×Dic₃₄

yes

characteristic

144

C₂×C₈

#63252

yes

characteristic

145

C₂×C₈

D₄×D₁₇

yes

—

146

C₂×C₈

Q₈×D₁₇

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

#63230

yes

—

157

C₁₆

—

158

C₂³

#10587635

yes

characteristic

159

Q₈

—

136

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#10587610

yes

characteristic

165

C₂×C₄

C₂×#63229

yes

characteristic

166

C₂×C₄

#10587623

yes

characteristic

167

C₂×C₄

C₂×#63230

yes

characteristic

168

C₂×C₄

#10587630

yes

characteristic

169

C₂×C₄

#10587637

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

#10587574

yes

—

173

C₈

#10587582

yes

—

174

C₂²

—

175

C₂²

#11762959

yes

characteristic

176

C₂²

#11763099

yes

characteristic

177

C₂²

#11763219

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

136

—

181

C₄

#11762729

yes

characteristic

182

C₄

#11762947

yes

characteristic

183

C₄

#11763094

yes

characteristic

184

C₄

#11763205

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

#14530117

yes

characteristic

	subgroup	G/H
center	C₄	#11763205
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1946	C₂
socle	C₂×C₁₇	#2058
prime radical	C₂³×C₁₇	AGL(1,ℤ/8)
hypercenter	#1946	D₁₇
hypocenter	C₁₇	#13459
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#11763205

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1946

C₂

C₂×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
29. C₁₇×#106	Q₈	semidirect
38. C₂×#63214	Q₈	semidirect
130. #106	Dic₃₄	semidirect
133. C₁₇	#13459	semidirect

H ≅ G/N

type

29. C₁₇×#106

Q₈

semidirect

38. C₂×#63214

Q₈

semidirect

130. #106

Dic₃₄

semidirect

133. C₁₇

#13459

semidirect

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

subgroup	index
#14539308	2
#14530558	2
#14526885	2
#14526864	2
C₁₇×#1946	2
#13459	17

subgroup

index

C₁₇×#1946

Subgroups — #16953982

Conjugacy classes of subgroups

Complements

Maximal subgroups