Group #17961690 — subgroups

Subgroups — #17961690

Order 4352.

374 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 120 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	#14545751	C₂	1	1	yes	maximal, characteristic
2	#14545119	C₂	1	2	yes	maximal, complemented
3	#14545014	C₂	1	1	yes	maximal, characteristic
4	#14543834	C₂	1	2	yes	maximal, complemented
5	#14528586	C₂	1	1	yes	maximal, characteristic
6	#11763867	—	2	4	no	—
7	C₂×#10587538	—	2	2	no	—
8	C₂×#10587538	C₂²	1	1	yes	characteristic
9	C₂×#10587538	C₂²	1	2	yes	—
10	C₂²×#63216	C₂²	1	1	yes	characteristic
11	C₄⋊#63216	—	2	4	no	—
12	C₂²⋊#63216	—	2	2	no	—
13	C₂×#10587634	C₂²	1	1	yes	characteristic
14	#11762793	C₄	1	1	yes	characteristic
15	#11762765	C₂²	1	2	yes	—
16	#11762579	C₄	1	2	yes	—
17	C₁₇×#410	C₄	1	1	yes	characteristic
18	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₁₇×#117	C₂×C₄	1	2	yes	—
20	#10587634	—	2	2	no	—
21	#10587634	—	2	2	no	—
22	C₂×#63223	C₂×C₄	1	2	yes	—
23	C₂×#63223	C₂×C₄	1	1	yes	characteristic
24	C₂×#63223	Q₈	1	2	yes	—
25	C₂²×#3622	D₄	1	2	yes	—
26	C₂²×#3622	C₂³	1	1	yes	characteristic
27	#10587538	—	2	4	no	—
28	#10587538	—	2	2	no	—
29	#10587538	—	2	2	no	—
30	#10587538	—	2	2	no	—
31	C₂×#63216	—	2	4	no	—
32	C₂×#63216	—	2	2	no	—
33	C₂×#63216	—	2	1	no	—
34	C₂×#63216	—	2	1	no	—
35	C₂×#63216	D₄	1	4	yes	—
36	C₁₇×#99	C₈	1	4	yes	—
37	#10587507	—	2	1	no	—
38	#10587504	—	2	1	no	—
39	#10587474	—	2	2	no	—
40	C₄×#3622	—	2	2	no	—
41	#41×C₁₇	—	2	2	no	—
42	#41×C₁₇	—	2	2	no	—
43	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₁₇	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₇	—	2	2	no	—
49	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
50	C₄²×C₁₇	—	2	2	no	—
51	#63223	—	2	2	no	—
52	#63223	—	2	4	no	—
53	#63223	—	2	2	no	—
54	#63223	—	2	2	no	—
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	C₄⋊C₄	1	2	yes	—
59	C₂×#3622	C₂×D₄	1	2	yes	—
60	C₂×#3622	C₂×D₄	1	1	yes	characteristic
61	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
62	C₂×#3622	#41	1	2	yes	—
63	#63216	—	2	4	no	—
64	#63216	—	4	1	no	—
65	#63216	—	4	1	no	—
66	#12915	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
68	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	Q₈×C₁₇	—	2	2	no	—
70	D₄×C₁₇	—	2	4	no	—
71	D₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	4	no	—
75	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#131	1	2	yes	—
80	#3622	—	2	2	no	—
81	#3622	—	4	2	no	—
82	#3622	#120	1	1	yes	characteristic
83	#3622	C₂²⋊Q₈	1	2	yes	—
84	#3622	#123	1	1	yes	characteristic
85	#3407	—	17	1	no	—
86	#2151	—	17	2	no	—
87	#2113	—	17	1	no	—
88	#1757	—	17	1	no	—
89	#1378	—	17	2	no	—
90	C₂²×C₁₇	—	2	2	no	—
91	C₂²×C₁₇	—	4	1	no	—
92	C₂²×C₁₇	—	4	2	no	—
93	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₁₇	C₈×D₄	1	2	yes	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	4	4	no	—
97	C₄×C₁₇	#385	1	1	yes	characteristic
98	C₄×C₁₇	#424	1	1	yes	characteristic
99	C₄×C₁₇	#435	1	2	yes	—
100	C₂×#131	—	17	1	no	—
101	#504	—	34	4	no	—
102	C₂×M₃₂	—	17	2	no	—
103	C₂×M₃₂	—	17	1	no	—
104	C₂×M₃₂	—	34	2	no	—
105	C₂²×C₁₆	—	17	1	no	—
106	#413	—	17	1	no	—
107	#410	#595	1	1	yes	characteristic
108	#405	—	17	2	no	—
109	C₄⋊C₁₆	—	34	4	no	—
110	C₂²⋊C₁₆	—	34	2	no	—
111	#344	—	17	2	no	—
112	C₂×C₁₇	—	2	2	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	—	4	2	no	—
115	C₂×C₁₇	#1850	1	1	yes	characteristic
116	C₂×#41	C₂×#595	1	1	yes	characteristic
117	#131	—	34	2	no	—
118	#131	—	34	2	no	—
119	C₂×M₁₆	—	17	2	no	—
120	C₂×M₁₆	—	17	1	no	—
121	C₂×M₁₆	—	17	2	no	—
122	C₂²×C₈	—	17	1	no	—
123	C₂²×C₈	—	17	2	no	—
124	#117	C₂×#595	1	2	yes	—
125	M₃₂	—	34	2	no	—
126	M₃₂	—	34	2	no	—
127	M₃₂	—	34	4	no	—
128	M₃₂	—	34	2	no	—
129	C₂×C₁₆	—	17	4	no	—
130	C₂×C₁₆	—	34	4	no	—
131	C₂×C₁₆	—	34	1	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	2	no	—
134	#101	—	34	1	no	—
135	#100	—	34	1	no	—
136	#99	#3624	1	4	yes	—
137	#97	—	34	2	no	—
138	C₄×C₈	—	34	2	no	—
139	C₁₇	#12915	1	1	yes	characteristic, complemented, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#63243	1	1	yes	characteristic
143	C₂×D₄	#63243	1	1	yes	characteristic
144	C₂×D₄	C₂×#3624	1	2	yes	—
145	C₂²×C₄	C₄⋊#595	1	2	yes	—
146	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
147	M₁₆	—	34	4	no	—
148	M₁₆	—	34	2	no	—
149	M₁₆	—	34	2	no	—
150	M₁₆	—	34	2	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	2	no	—
155	C₂×C₈	—	17	2	no	—
156	C₂×C₈	—	34	2	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	C₂×#3624	1	4	yes	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	34	4	no	—
163	C₁₆	—	68	1	no	—
164	C₁₆	—	68	1	no	—
165	C₂³	C₄⋊#3624	1	2	yes	—
166	C₂³	C₂²×#3624	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₄⋊#3624	1	2	yes	—
174	C₂×C₄	C₂×#63243	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
176	C₂×C₄	#10587678	1	2	yes	—
177	C₂×C₄	D₄×#595	1	2	yes	—
178	C₈	—	17	1	no	—
179	C₈	—	17	1	no	—
180	C₈	—	17	2	no	—
181	C₈	—	34	2	no	—
182	C₈	—	68	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
187	C₂²	D₄×#3624	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	#11763756	1	1	yes	characteristic
191	C₄	#11763815	1	1	yes	characteristic
192	C₄	#11763838	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#14544727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545751

C₂

yes

maximal, characteristic

#14545119

C₂

yes

maximal, complemented

#14545014

C₂

yes

maximal, characteristic

#14543834

C₂

yes

maximal, complemented

#14528586

C₂

yes

maximal, characteristic

#11763867

—

C₂×#10587538

—

C₂×#10587538

C₂²

yes

characteristic

C₂×#10587538

C₂²

yes

—

C₂²×#63216

C₂²

yes

—

—

C₂²

yes

characteristic

#11762793

C₄

yes

characteristic

#11762765

C₂²

yes

—

#11762579

C₄

yes

—

C₁₇×#410

C₄

yes

characteristic

C₂×#41×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#117

C₂×C₄

yes

—

#10587634

—

#10587634

—

C₂×#63223

C₂×C₄

yes

—

C₂×#63223

C₂×C₄

yes

characteristic

C₂×#63223

Q₈

yes

—

C₂²×#3622

D₄

yes

—

C₂²×#3622

C₂³

yes

—

—

—

—

—

—

—

—

D₄

yes

—

C₁₇×#99

C₈

yes

—

—

—

—

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

M₁₆

yes

characteristic

C₂×D₄×C₁₇

C₂×C₈

yes

—

C₂×D₄×C₁₇

M₁₆

yes

characteristic

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#3622

C₂×D₄

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₂×#3622

#41

yes

—

—

—

—

—

C₂³×C₁₇

C₄⋊C₈

yes

—

C₂³×C₁₇

C₂²×C₈

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

#131

yes

—

—

—

yes

C₂²⋊Q₈

yes

—

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

C₈×D₄

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

100

C₂×#131

—

101

#504

—

102

C₂×M₃₂

—

103

C₂×M₃₂

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

—

106

#413

—

107

yes

108

—

109

C₄⋊C₁₆

—

110

C₂²⋊C₁₆

—

111

#344

—

112

C₂×C₁₇

—

113

C₂×C₁₇

—

114

C₂×C₁₇

—

115

C₂×C₁₇

#1850

yes

characteristic

116

C₂×#41

C₂×#595

yes

characteristic

117

#131

—

118

#131

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

#117

C₂×#595

yes

—

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#101

—

135

#100

—

136

#99

#3624

yes

—

137

#97

—

138

C₄×C₈

—

139

C₁₇

#12915

yes

characteristic, complemented, Sylow

140

#41

—

141

#41

—

142

C₂×Q₈

#63243

yes

characteristic

143

C₂×D₄

#63243

yes

characteristic

144

C₂×D₄

C₂×#3624

yes

—

145

C₂²×C₄

C₄⋊#595

yes

—

146

C₂²×C₄

C₂²×#595

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

C₂×#3624

yes

—

161

C₄²

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

C₄⋊#3624

yes

—

166

C₂³

C₂²×#3624

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₄⋊#3624

yes

—

174

C₂×C₄

C₂×#63243

yes

characteristic

175

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

176

C₂×C₄

#10587678

yes

—

177

C₂×C₄

D₄×#595

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×C₄⋊#3624

yes

characteristic

187

C₂²

D₄×#3624

yes

—

188

C₄

—

189

C₄

—

190

C₄

#11763756

yes

characteristic

191

C₄

#11763815

yes

characteristic

192

C₄

#11763838

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#14544727

yes

characteristic

	subgroup	G/H
center	C₄	#11763756
commutator	C₂³×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#410	C₄
socle	C₂×C₁₇	#1850
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂²×#3622	C₂³
hypercenter	#410	#595
hypocenter	C₁₇	#12915
LCS	G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#11763756

commutator

C₂³×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₁₇×#410

C₄

socle

C₂×C₁₇

#1850

prime radical

C₂×#41×C₁₇

C₂×C₄

cosocle

C₂²×#3622

C₂³

C₁₇

G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #14545119	C₂	semidirect
4. #14543834	C₂	semidirect
139. C₁₇	#12915	semidirect

H ≅ G/N

type

2. #14545119

C₂

semidirect

4. #14543834

C₂

semidirect

139. C₁₇

#12915

semidirect

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

subgroup	index
#14545751	2
#14545119	2
#14545014	2
#14543834	2
#14528586	2
#12915	17

subgroup

index

Subgroups — #17961690

Conjugacy classes of subgroups

Complements

Maximal subgroups