Group #16952921 — subgroups

Subgroups — #16952921

Order 4352.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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×#11763211	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526876	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587556	—	2	1	no	—
6	#11763211	—	2	2	no	—
7	#11763211	C₄	1	4	yes	complemented
8	C₂×#10587489	C₂²	1	2	yes	—
9	C₂×#10587489	C₂²	1	2	yes	—
10	C₂×#10587488	—	2	1	no	—
11	C₂×#10587488	—	2	1	no	—
12	#11762675	—	2	1	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₁₇	—	2	2	no	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂×#63225	—	4	2	no	—
22	#10587556	—	2	4	no	—
23	#10587556	—	4	2	no	—
24	C₂×C₁₆×C₁₇	—	2	2	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	Q₈	1	4	yes	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	D₄	1	2	yes	—
29	#10587502	—	4	2	no	—
30	#10587489	—	2	2	no	—
31	#10587489	—	2	4	no	—
32	#10587489	C₂×C₄	1	4	yes	—
33	#10587488	—	2	4	no	—
34	#10587488	—	4	2	no	—
35	C₂²×C₄×C₁₇	D₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	2	yes	—
46	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₁₇	SD₁₆	1	2	yes	—
48	C₄⋊Dic₁₇	—	2	4	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	C₄⋊Dic₁₇	—	4	2	no	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	#63225	—	4	8	no	—
53	C₁₆×C₁₇	—	2	4	no	—
54	#13281	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	2	4	no	—
57	C₂×C₄×C₁₇	#102	1	4	yes	—
58	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₇	—	4	4	no	—
65	C₂×Dic₁₇	—	8	4	no	—
66	C₂×Dic₁₇	—	8	2	no	—
67	C₂×Dic₁₇	—	8	4	no	—
68	C₈×C₁₇	#103	1	4	yes	—
69	C₈×C₁₇	C₄×Q₈	1	2	yes	—
70	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×#499	—	17	1	no	—
72	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
73	#1846	—	17	1	no	—
74	#1380	—	17	4	no	—
75	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
76	C₂²×C₁₇	#486	1	1	yes	characteristic
77	C₂²×C₁₇	#500	1	1	yes	characteristic
78	C₂²×C₁₇	#508	1	4	yes	—
79	C₄×C₁₇	—	2	4	no	—
80	C₄×C₁₇	#391	1	1	yes	characteristic
81	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
82	C₄×C₁₇	#487	1	1	yes	characteristic
83	C₄×C₁₇	#498	1	1	yes	characteristic
84	Dic₁₇	—	8	4	no	—
85	Dic₁₇	—	16	2	no	—
86	C₂×#125	—	34	1	no	—
87	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
88	#499	—	17	4	no	—
89	#499	—	34	2	no	—
90	C₂×#107	—	17	2	no	—
91	C₂×#107	—	17	2	no	—
92	C₂×#106	—	34	1	no	—
93	C₂×#106	—	34	1	no	—
94	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	#382	—	34	1	no	—
96	C₄×C₁₆	—	2	2	no	—
97	C₂×C₁₇	#1990	1	1	yes	characteristic
98	C₂×C₁₇	#2138	1	2	yes	—
99	C₂×C₁₇	#2242	1	2	yes	—
100	C₂×C₁₇	#2253	1	2	yes	—
101	C₂²×C₈	D₆₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	#125	—	34	4	no	—
104	#125	—	68	2	no	—
105	C₂×C₄⋊C₄	—	34	2	no	—
106	C₂×C₄⋊C₄	—	34	2	no	—
107	C₂×C₄⋊C₄	—	68	2	no	—
108	C₂×C₄²	D₆₈	1	1	yes	characteristic
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	Dic₃₄	1	4	yes	—
112	#107	—	17	4	no	—
113	#107	—	34	2	no	—
114	#107	—	34	4	no	—
115	#106	—	34	4	no	—
116	#106	—	68	2	no	—
117	C₄×C₈	C₄×D₁₇	1	2	yes	—
118	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
119	#95	—	68	2	no	—
120	C₁₇	#13281	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	68	2	no	—
122	C₂²×C₄	—	68	2	no	—
123	C₂²×C₄	D₁₃₆	1	2	yes	—
124	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
125	C₂×C₈	#63219	1	2	yes	—
126	C₂×C₈	Dic₆₈	1	2	yes	—
127	C₂×C₈	#63227	1	2	yes	—
128	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
130	C₂×C₈	#63252	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
132	C₄⋊C₄	—	34	4	no	—
133	C₄⋊C₄	—	68	8	no	—
134	C₄⋊C₄	—	68	2	no	—
135	C₄⋊C₄	—	68	2	no	—
136	C₄⋊C₄	—	68	2	no	—
137	C₄²	#63219	1	2	yes	—
138	C₄²	#63227	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	68	4	no	—
143	C₂×C₄	—	136	4	no	—
144	C₂×C₄	—	136	2	no	—
145	C₂×C₄	—	136	4	no	—
146	C₂×C₄	#10587492	1	4	yes	—
147	C₂×C₄	#10587555	1	1	yes	characteristic
148	C₂×C₄	#10587561	1	1	yes	characteristic
149	C₂×C₄	C₂×#63219	1	1	yes	characteristic
150	C₂×C₄	C₂×#63219	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
152	C₂×C₄	C₂×#63227	1	1	yes	characteristic
153	C₈	#10587487	1	4	yes	—
154	C₈	C₄×Dic₃₄	1	2	yes	—
155	C₈	#10587610	1	2	yes	—
156	C₂²	C₂×#10587492	1	1	yes	characteristic
157	C₂²	#11763156	1	1	yes	characteristic
158	C₂²	#11763214	1	1	yes	characteristic
159	C₂²	#11763241	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	272	2	no	—
162	C₄	—	136	4	no	—
163	C₄	#11762715	1	1	yes	characteristic
164	C₄	C₂×#10587487	1	1	yes	characteristic
165	C₄	#11763160	1	1	yes	characteristic
166	C₄	#11763207	1	1	yes	characteristic
167	C₂	#14529745	1	1	yes	characteristic
168	C₂	#14530492	1	2	yes	—
169	C₂	#14531016	1	2	yes	—
170	C₂	#14531063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763211

C₂

yes

maximal, characteristic

#14528948

C₂

yes

maximal, characteristic

#14526876

C₂

yes

maximal

C₂×C₄×C₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂×#10587556

—

#11763211

—

#11763211

C₄

yes

complemented

C₂×#10587489

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

C₂×#10587488

—

C₂×#10587488

—

#11762675

—

C₂²×C₁₆×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×#63225

—

#10587556

—

#10587556

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

Q₈

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#10587488

—

#10587488

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

SD₁₆

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₁₆×C₁₇

—

#13281

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#102

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#124

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

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

C₄×Q₈

yes

—

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₂×#499

—

C₂×C₄×C₁₆

D₁₇

yes

characteristic

#1846

—

#1380

—

C₂²×C₁₇

yes

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×#125

—

C₂²×C₁₆

C₂×D₁₇

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#382

—

C₄×C₁₆

—

C₂×C₁₇

#1990

yes

characteristic

C₂×C₁₇

#2138

yes

—

C₂×C₁₇

#2242

yes

—

100

C₂×C₁₇

#2253

yes

—

101

C₂²×C₈

D₆₈

yes

characteristic

102

C₂²×C₈

C₂²×D₁₇

yes

characteristic

103

#125

—

104

#125

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

D₆₈

yes

characteristic

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

Dic₃₄

yes

—

112

#107

—

113

#107

—

114

#107

—

115

#106

—

116

#106

—

117

C₄×C₈

C₄×D₁₇

yes

—

118

C₄×C₈

C₂²⋊D₁₇

yes

—

119

#95

—

120

C₁₇

#13281

yes

characteristic, complemented, Sylow

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

D₁₃₆

yes

—

124

C₂²×C₄

C₂×D₆₈

yes

characteristic

125

C₂×C₈

#63219

yes

—

126

C₂×C₈

Dic₆₈

yes

—

127

C₂×C₈

#63227

yes

—

128

C₂×C₈

C₂×Dic₃₄

yes

—

129

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

130

C₂×C₈

#63252

yes

—

131

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

#63219

yes

—

138

C₄²

#63227

yes

—

139

C₁₆

—

140

C₂³

C₂×D₁₃₆

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

136

—

144

C₂×C₄

—

136

—

145

C₂×C₄

—

136

—

146

C₂×C₄

#10587492

yes

—

147

C₂×C₄

#10587555

yes

characteristic

148

C₂×C₄

#10587561

yes

characteristic

149

C₂×C₄

C₂×#63219

yes

characteristic

150

C₂×C₄

C₂×#63219

yes

characteristic

151

C₂×C₄

C₂×Dic₆₈

yes

characteristic

152

C₂×C₄

C₂×#63227

yes

characteristic

153

C₈

#10587487

yes

—

154

C₈

C₄×Dic₃₄

yes

—

155

C₈

#10587610

yes

—

156

C₂²

C₂×#10587492

yes

characteristic

157

C₂²

#11763156

yes

characteristic

158

C₂²

#11763214

yes

characteristic

159

C₂²

#11763241

yes

—

160

C₄

—

161

C₄

—

272

—

162

C₄

—

136

—

163

C₄

#11762715

yes

characteristic

164

C₄

C₂×#10587487

yes

characteristic

165

C₄

#11763160

yes

characteristic

166

C₄

#11763207

yes

characteristic

167

C₂

#14529745

yes

characteristic

168

C₂

#14530492

yes

—

169

C₂

#14531016

yes

—

170

C₂

#14531063

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#13281
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₁₃₆

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₄×C₁₆×C₁₇

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₈

cosocle

C₂²×C₈×C₁₇

C₂³

hypercenter

C₂×C₄×C₁₆

D₁₇

hypocenter

C₁₇

#13281

LCS

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
7. #11763211	C₄	semidirect
120. C₁₇	#13281	semidirect

H ≅ G/N

type

7. #11763211

C₄

semidirect

120. C₁₇

#13281

semidirect

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

subgroup	index
C₂×#11763211	2
#14528948	2
#14526876	2
C₂×C₄×C₁₆×C₁₇	2
#13281	17

subgroup

index

C₂×C₄×C₁₆×C₁₇

Subgroups — #16952921

Conjugacy classes of subgroups

Complements

Maximal subgroups