Group #16950517 — subgroups

Subgroups — #16950517

Order 4352.

354 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11762639	C₂	1	2	yes	maximal, complemented
2	#14528834	C₂	1	2	yes	maximal
3	#14526888	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	C₂²	1	4	yes	—
6	#11762793	—	2	2	no	—
7	#11762639	—	2	4	no	—
8	#11762567	—	2	2	no	—
9	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
10	C₁₇×#504	—	2	4	no	—
11	C₂×C₁₇×M₃₂	C₄	1	2	yes	—
12	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	C₄	1	2	yes	—
14	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
16	C₁₇×#131	—	2	2	no	—
17	C₁₇×#131	—	2	2	no	—
18	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63223	—	2	4	no	—
23	C₁₇×M₃₂	—	2	4	no	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	—	2	4	no	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
30	#10587507	—	4	2	no	—
31	#10587504	—	4	2	no	—
32	#10587490	—	2	4	no	—
33	#10587490	—	2	4	no	—
34	#41×C₁₇	—	2	2	no	—
35	#41×C₁₇	—	2	2	no	—
36	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
37	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
38	C₂×D₄×C₁₇	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	2	no	—
42	M₁₆×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	2	yes	—
50	#63223	—	4	4	no	—
51	#63223	—	4	4	no	—
52	C₂×#3622	—	4	4	no	—
53	C₁₆×C₁₇	—	2	2	no	—
54	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
55	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
56	#12840	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	#107	1	2	yes	—
58	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
59	Q₈×C₁₇	—	2	2	no	—
60	D₄×C₁₇	—	2	4	no	—
61	D₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	#107	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#121	1	2	yes	—
67	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#135	1	2	yes	—
69	C₈×C₁₇	—	2	2	no	—
70	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₁₇	#117	1	1	yes	characteristic
72	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
73	#3622	—	8	4	no	—
74	C₂×#504	D₁₇	1	1	yes	characteristic
75	C₂×#368	—	17	2	no	—
76	#1813	—	17	2	no	—
77	#1385	—	17	2	no	—
78	C₂²×C₁₇	—	2	2	no	—
79	C₂²×C₁₇	—	4	1	no	—
80	C₂²×C₁₇	—	4	2	no	—
81	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
82	C₂²×C₁₇	#467	1	2	yes	—
83	C₄×C₁₇	—	2	2	no	—
84	C₄×C₁₇	#380	1	1	yes	characteristic
85	C₄×C₁₇	#427	1	1	yes	characteristic
86	C₄×C₁₇	#466	1	2	yes	—
87	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
88	#504	—	2	4	no	—
89	C₂×M₃₂	Dic₁₇	1	2	yes	—
90	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
91	C₂²×C₁₆	Dic₁₇	1	2	yes	—
92	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
93	C₂×#108	—	17	4	no	—
94	#413	—	34	2	no	—
95	#368	—	34	4	no	—
96	#338	—	34	2	no	—
97	C₂×C₁₇	—	2	1	no	—
98	C₂×C₁₇	—	2	2	no	—
99	C₂×C₁₇	—	4	2	no	—
100	C₂×C₁₇	#1808	1	1	yes	characteristic
101	C₂×#41	D₆₈	1	1	yes	characteristic
102	#131	—	2	2	no	—
103	#131	—	2	2	no	—
104	C₂×M₁₆	—	34	4	no	—
105	C₂×M₁₆	Dic₃₄	1	2	yes	—
106	C₂×M₁₆	D₆₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
108	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
109	M₃₂	—	2	2	no	—
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
113	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
114	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
115	C₂×C₁₆	C₂²⋊D₁₇	1	4	yes	—
116	#108	—	34	4	no	—
117	#108	—	34	4	no	—
118	#101	—	68	2	no	—
119	#100	—	68	2	no	—
120	C₁₇	#12840	1	1	yes	characteristic, complemented, Sylow
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
124	C₂×D₄	D₁₃₆	1	1	yes	characteristic
125	C₂×D₄	Dic₆₈	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
127	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	68	4	no	—
131	M₁₆	—	68	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	68	4	no	—
135	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
136	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
137	C₂×C₈	#63252	1	2	yes	—
138	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
140	C₁₆	—	2	2	no	—
141	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
142	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
143	C₂³	#10587489	1	2	yes	—
144	C₂³	C₂×Dic₆₈	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#10587489	1	2	yes	—
151	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
152	C₂×C₄	#10587590	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
154	C₂×C₄	#10587616	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	136	4	no	—
157	C₈	#10587610	1	2	yes	—
158	C₈	#10587612	1	1	yes	characteristic
159	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×#10587489	1	1	yes	characteristic
164	C₂²	#11763050	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	#11762669	1	1	yes	characteristic
167	C₄	#11762847	1	1	yes	characteristic
168	C₄	#11763045	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	#14528813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762639

C₂

yes

maximal, complemented

#14528834

C₂

yes

maximal

#14526888

C₂

yes

maximal, complemented

C₂×C₁₇×#504

C₂

yes

maximal, characteristic

C₂×#10587490

C₂²

yes

—

#11762793

—

#11762639

—

#11762567

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₁₇×#504

—

C₂×C₁₇×M₃₂

C₄

yes

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₄

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×#41×C₁₇

D₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂×M₁₆×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₁₇×M₃₂

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

—

—

—

—

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

D₈

yes

characteristic

C₂×D₄×C₁₇

D₈

yes

characteristic

C₂×D₄×C₁₇

Q₁₆

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

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

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₂²⋊C₄

yes

—

C₁₆×C₁₇

C₂²⋊C₄

yes

—

#12840

—

maximal, Sylow

C₂³×C₁₇

#107

yes

—

C₂³×C₁₇

C₂×Q₁₆

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#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

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

#117

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

#3622

—

C₂×#504

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂²×C₁₇

#467

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂×#131

C₂×D₁₇

yes

characteristic

#504

—

C₂×M₃₂

Dic₁₇

yes

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

Dic₁₇

yes

—

C₂²×C₁₆

C₂×D₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

100

C₂×C₁₇

#1808

yes

characteristic

101

C₂×#41

D₆₈

yes

characteristic

102

#131

—

103

#131

—

104

C₂×M₁₆

—

105

C₂×M₁₆

Dic₃₄

yes

—

106

C₂×M₁₆

D₆₈

yes

characteristic

107

C₂²×C₈

C₂×Dic₁₇

yes

—

108

C₂²×C₈

C₂²×D₁₇

yes

characteristic

109

M₃₂

—

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

C₂×Dic₁₇

yes

—

113

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

114

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

115

C₂×C₁₆

C₂²⋊D₁₇

yes

—

116

#108

—

117

#108

—

118

#101

—

119

#100

—

120

C₁₇

#12840

yes

characteristic, complemented, Sylow

121

#41

—

122

#41

—

123

C₂×Q₈

D₁₃₆

yes

characteristic

124

C₂×D₄

D₁₃₆

yes

characteristic

125

C₂×D₄

Dic₆₈

yes

—

126

C₂²×C₄

C₄⋊Dic₁₇

yes

—

127

C₂²×C₄

C₂×D₆₈

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₄⋊Dic₁₇

yes

—

136

C₂×C₈

C₂×Dic₃₄

yes

characteristic

137

C₂×C₈

#63252

yes

—

138

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

140

C₁₆

—

141

C₁₆

C₂²⋊Dic₁₇

yes

—

142

C₁₆

C₂²⋊Dic₁₇

yes

—

143

C₂³

#10587489

yes

—

144

C₂³

C₂×Dic₆₈

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#10587489

yes

—

151

C₂×C₄

C₂×D₁₃₆

yes

characteristic

152

C₂×C₄

#10587590

yes

—

153

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

154

C₂×C₄

#10587616

yes

—

155

C₈

—

156

C₈

—

136

—

157

C₈

#10587610

yes

—

158

C₈

#10587612

yes

characteristic

159

C₈

C₂×C₂²⋊Dic₁₇

yes

characteristic

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×#10587489

yes

characteristic

164

C₂²

#11763050

yes

—

165

C₄

—

166

C₄

#11762669

yes

characteristic

167

C₄

#11762847

yes

characteristic

168

C₄

#11763045

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#14528813

yes

characteristic

	subgroup	G/H
center	C₄	#11762847
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂×C₁₇	#1808
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#12840
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#11762847

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#504

C₂

socle

C₂×C₁₇

#1808

prime radical

C₂×#41×C₁₇

D₄

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. C₂×#11762639	C₂	semidirect
3. #14526888	C₂	semidirect
120. C₁₇	#12840	semidirect

H ≅ G/N

type

1. C₂×#11762639

C₂

semidirect

3. #14526888

C₂

semidirect

120. C₁₇

#12840

semidirect

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

subgroup	index
C₂×#11762639	2
#14528834	2
#14526888	2
C₂×C₁₇×#504	2
#12840	17

subgroup

index

C₂×C₁₇×#504

Subgroups — #16950517

Conjugacy classes of subgroups

Complements

Maximal subgroups