Group #16953974 — subgroups

Subgroups — #16953974

Order 4352.

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

normal subgroups only

This group has 324 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×#11762631	C₂	1	2	yes	maximal
2	#14529532	C₂	1	1	yes	maximal, characteristic
3	#14526891	C₂	1	2	yes	maximal
4	#14526866	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#499	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587490	C₂²	1	1	yes	characteristic
8	#11762847	C₂²	1	1	yes	characteristic
9	#11762835	—	2	1	no	—
10	#11762631	—	2	4	no	—
11	#11762569	—	2	1	no	—
12	C₂×C₁₇×#125	—	2	1	no	—
13	C₁₇×#499	—	2	4	no	—
14	C₂×C₁₇×#107	—	2	1	no	—
15	C₂×C₁₇×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₁₇×#125	—	2	4	no	—
20	C₁₇×#125	—	4	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	#10587612	—	2	1	no	—
25	C₂×#63223	—	2	1	no	—
26	C₁₇×#107	—	2	4	no	—
27	C₁₇×#106	—	2	2	no	—
28	C₁₇×#106	D₄	1	4	yes	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	#10587490	—	2	2	no	—
31	#10587489	Q₈	1	2	yes	—
32	#10587488	—	2	1	no	—
33	#10587485	—	4	2	no	—
34	#10587483	—	2	2	no	—
35	#10587483	Q₈	1	4	yes	—
36	C₂×#63214	—	2	2	no	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	—	4	2	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	—	2	1	no	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₁₇	—	2	4	no	—
48	C₄⋊C₄×C₁₇	—	4	8	no	—
49	C₄⋊C₄×C₁₇	—	4	2	no	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₁₇	—	4	1	no	—
53	C₄⋊Dic₁₇	—	2	2	no	—
54	C₄×Dic₁₇	—	4	1	no	—
55	#63223	—	4	1	no	—
56	#63223	—	4	1	no	—
57	C₂×#3622	—	4	1	no	—
58	#63214	—	2	4	no	—
59	#13458	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	4	2	no	—
63	C₂×C₄×C₁₇	—	4	4	no	—
64	C₂×C₄×C₁₇	—	8	4	no	—
65	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
71	C₂×Dic₁₇	—	4	2	no	—
72	C₈×C₁₇	—	2	2	no	—
73	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
75	#3622	—	8	1	no	—
76	C₂×#499	D₁₇	1	1	yes	characteristic
77	C₂×#365	—	17	2	no	—
78	#1945	—	17	1	no	—
79	#1386	—	17	2	no	—
80	#1376	—	17	1	no	—
81	C₂²×C₁₇	—	2	2	no	—
82	C₂²×C₁₇	#450	1	1	yes	characteristic
83	C₂²×C₁₇	#476	1	1	yes	characteristic
84	C₂²×C₁₇	#501	1	1	yes	characteristic
85	C₄×C₁₇	—	4	2	no	—
86	C₄×C₁₇	—	8	4	no	—
87	C₄×C₁₇	#395	1	1	yes	characteristic
88	C₄×C₁₇	#448	1	1	yes	characteristic
89	C₄×C₁₇	#475	1	1	yes	characteristic
90	C₄×C₁₇	#498	1	1	yes	characteristic
91	Dic₁₇	—	8	2	no	—
92	C₂×#125	—	2	1	no	—
93	C₂×M₃₂	—	17	2	no	—
94	#499	—	2	4	no	—
95	C₂×#108	—	17	1	no	—
96	#427	—	17	1	no	—
97	C₂×#107	—	2	1	no	—
98	C₂×#106	C₂×D₁₇	1	2	yes	—
99	#424	—	34	1	no	—
100	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
101	#365	—	34	4	no	—
102	#339	—	34	1	no	—
103	C₂×C₁₇	—	2	1	no	—
104	C₂×C₁₇	—	2	1	no	—
105	C₂×C₁₇	#2058	1	1	yes	characteristic
106	C₂×C₁₇	#2231	1	2	yes	—
107	C₂×M₁₆	—	34	1	no	—
108	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
110	#125	—	2	4	no	—
111	#125	—	4	2	no	—
112	#117	—	34	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
116	M₃₂	—	17	4	no	—
117	M₃₂	—	34	2	no	—
118	C₂×C₁₆	—	34	2	no	—
119	#108	—	34	2	no	—
120	#107	—	2	4	no	—
121	#107	—	17	2	no	—
122	#106	—	2	2	no	—
123	#106	—	34	1	no	—
124	#106	C₂²⋊D₁₇	1	4	yes	—
125	C₄⋊C₈	—	68	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₁₇	#13458	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
131	M₁₆	—	68	1	no	—
132	M₁₆	—	68	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	68	1	no	—
136	C₂×C₈	#63230	1	2	yes	—
137	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₁₇	1	2	yes	—
139	C₂×C₈	#63256	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	34	2	no	—
145	C₂²⋊C₄	—	68	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	68	1	no	—
148	C₄²	#63229	1	2	yes	—
149	C₁₆	—	34	4	no	—
150	C₂³	#10587618	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	68	2	no	—
156	C₂×C₄	C₂×#63229	1	1	yes	characteristic
157	C₂×C₄	C₂×#63230	1	1	yes	characteristic
158	C₂×C₄	#10587628	1	1	yes	characteristic
159	C₂×C₄	#10587630	1	1	yes	characteristic
160	C₂×C₄	#10587632	1	1	yes	characteristic
161	C₂×C₄	#10587639	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	136	1	no	—
164	C₈	#10587630	1	2	yes	—
165	C₈	#10587632	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#11762960	1	1	yes	characteristic
168	C₂²	#11763096	1	1	yes	characteristic
169	C₂²	#11763215	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	136	2	no	—
173	C₄	#11762728	1	1	yes	characteristic
174	C₄	#11762948	1	1	yes	characteristic
175	C₄	#11763091	1	1	yes	characteristic
176	C₄	#11763205	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#14530114	1	1	yes	characteristic
180	C₂	#14530957	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762631

C₂

yes

maximal

#14529532

C₂

yes

maximal, characteristic

#14526891

C₂

yes

maximal

#14526866

C₂

yes

maximal, characteristic

C₂×C₁₇×#499

C₂

yes

maximal, characteristic

C₂×#10587483

C₂²

yes

—

C₂×#10587490

C₂²

yes

characteristic

#11762847

C₂²

yes

—

—

—

C₂×C₁₇×#125

—

C₁₇×#499

—

C₂×C₁₇×#107

—

C₂×C₁₇×#106

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#125

—

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

#10587612

—

C₂×#63223

—

C₁₇×#107

—

C₁₇×#106

—

C₁₇×#106

D₄

yes

—

C₄×C₈×C₁₇

—

#10587490

—

#10587489

Q₈

yes

—

—

—

—

Q₈

yes

—

C₂×#63214

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

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

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

SD₁₆

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

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

—

C₈×C₁₇

—

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

—

#3622

—

C₂×#499

D₁₇

yes

—

—

—

—

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

—

C₂×#125

—

C₂×M₃₂

—

—

—

—

—

C₂×D₁₇

yes

—

#424

—

100

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

101

#365

—

102

#339

—

103

C₂×C₁₇

—

104

C₂×C₁₇

—

105

C₂×C₁₇

#2058

yes

characteristic

106

C₂×C₁₇

#2231

yes

—

107

C₂×M₁₆

—

108

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

109

C₂²×C₈

C₂²×D₁₇

yes

characteristic

110

#125

—

111

#125

—

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

116

M₃₂

—

117

M₃₂

—

118

C₂×C₁₆

—

119

#108

—

120

#107

—

121

#107

—

122

#106

—

123

#106

—

124

#106

C₂²⋊D₁₇

yes

—

125

C₄⋊C₈

—

126

C₄×C₈

—

127

C₁₇

#13458

yes

characteristic, complemented, Sylow

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

#63230

yes

—

137

C₂×C₈

Q₈×D₁₇

yes

characteristic

138

C₂×C₈

Q₈×D₁₇

yes

—

139

C₂×C₈

#63256

yes

characteristic

140

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

#63229

yes

—

149

C₁₆

—

150

C₂³

#10587618

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

C₂×#63229

yes

characteristic

157

C₂×C₄

C₂×#63230

yes

characteristic

158

C₂×C₄

#10587628

yes

characteristic

159

C₂×C₄

#10587630

yes

characteristic

160

C₂×C₄

#10587632

yes

characteristic

161

C₂×C₄

#10587639

yes

characteristic

162

C₈

—

163

C₈

—

136

—

164

C₈

#10587630

yes

—

165

C₈

#10587632

yes

—

166

C₂²

—

167

C₂²

#11762960

yes

characteristic

168

C₂²

#11763096

yes

characteristic

169

C₂²

#11763215

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

136

—

173

C₄

#11762728

yes

characteristic

174

C₄

#11762948

yes

characteristic

175

C₄

#11763091

yes

characteristic

176

C₄

#11763205

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#14530114

yes

characteristic

180

C₂

#14530957

yes

—

	subgroup	G/H
center	C₂²	#11763096
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#499	C₂
socle	C₂²×C₁₇	#476
prime radical	C₂³×C₁₇	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₁₇
hypocenter	C₁₇	#13458
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#11763096

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#499

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
127. C₁₇	#13458	semidirect

H ≅ G/N

type

127. C₁₇

#13458

semidirect

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

subgroup	index
C₂×#11762631	2
#14529532	2
#14526891	2
#14526866	2
C₂×C₁₇×#499	2
#13458	17

subgroup

index

C₂×C₁₇×#499

Subgroups — #16953974

Conjugacy classes of subgroups

Complements

Maximal subgroups