Group #14528786 — subgroups

Subgroups — #14528786

Order 2176.

308 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587481	C₂	1	2	yes	maximal
4	#11762571	C₂	1	2	yes	maximal
5	C₁₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
8	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂×#63225	—	2	1	no	—
13	C₂×C₄×Dic₁₇	—	2	1	no	—
14	C₂²×#3622	C₂²	1	2	yes	—
15	C₄×Dic₃₄	—	2	2	no	—
16	C₄×Dic₃₄	C₄	1	4	yes	complemented
17	#10587481	—	2	4	no	—
18	C₄⋊#3622	—	2	2	no	—
19	C₂²×C₄×C₁₇	—	2	2	no	—
20	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₇	—	2	1	no	—
24	C₂²×Dic₁₇	—	2	1	no	—
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×Dic₃₄	—	2	1	no	—
27	C₂×Dic₃₄	—	2	1	no	—
28	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₄	D₄	1	2	yes	—
30	C₂×Dic₃₄	Q₈	1	2	yes	—
31	C₄⋊C₄×C₁₇	—	2	2	no	—
32	C₄⋊C₄×C₁₇	—	2	2	no	—
33	C₄⋊C₄×C₁₇	—	4	2	no	—
34	C₄⋊C₄×C₁₇	—	4	2	no	—
35	C₄²×C₁₇	C₂×C₄	1	2	yes	—
36	C₄²×C₁₇	D₄	1	2	yes	—
37	C₄⋊Dic₁₇	—	2	1	no	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	#63225	—	2	4	no	—
40	C₄×Dic₁₇	—	2	4	no	—
41	C₂×#3622	—	2	2	no	—
42	C₂×#3622	—	2	2	no	—
43	C₂×#3622	—	2	2	no	—
44	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	1	no	—
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	4	2	no	—
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
60	C₂×Dic₁₇	—	2	2	no	—
61	C₂×Dic₁₇	—	2	2	no	—
62	C₂×Dic₁₇	—	2	2	no	—
63	C₂×Dic₁₇	—	2	2	no	—
64	C₂×Dic₁₇	—	4	1	no	—
65	C₂×Dic₁₇	—	4	1	no	—
66	Dic₃₄	—	2	2	no	—
67	Dic₃₄	—	2	2	no	—
68	Dic₃₄	—	2	2	no	—
69	Dic₃₄	C₄⋊C₄	1	4	yes	complemented
70	#3622	—	4	2	no	—
71	#1803	—	17	1	no	maximal, Sylow
72	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₇	#121	1	1	yes	characteristic
74	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₁₇	#137	1	1	yes	characteristic
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	—	8	2	no	—
81	C₄×C₁₇	#103	1	4	yes	complemented
82	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₁₇	#117	1	1	yes	characteristic
84	C₄×C₁₇	#121	1	1	yes	characteristic
85	C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₁₇	—	2	4	no	—
87	Dic₁₇	—	4	2	no	—
88	C₂×C₄×Q₈	—	17	1	no	—
89	C₂×C₄⋊C₈	—	17	1	no	—
90	C₂×#103	—	17	2	no	—
91	#384	D₁₇	1	1	yes	characteristic
92	#340	—	17	2	no	—
93	C₂×C₁₇	#380	1	1	yes	characteristic
94	C₂×C₁₇	C₂×#103	1	1	yes	characteristic
95	C₂×C₁₇	#417	1	1	yes	characteristic
96	C₂×C₁₇	#460	1	1	yes	characteristic
97	C₂×C₁₇	#462	1	1	yes	characteristic
98	C₂×C₁₇	#475	1	1	yes	characteristic
99	C₂×C₁₇	#477	1	1	yes	characteristic
100	C₂²×Q₈	—	17	1	no	—
101	C₂²×C₈	—	17	2	no	—
102	C₄×Q₈	—	17	4	no	—
103	C₄×Q₈	—	34	2	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	17	1	no	—
107	C₂×C₄⋊C₄	—	34	1	no	—
108	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
109	C₂×C₄²	—	34	1	no	—
110	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
111	C₄⋊C₈	—	34	2	no	—
112	#103	—	34	4	no	—
113	C₁₇	#1803	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	17	2	no	—
115	C₂×Q₈	—	17	2	no	—
116	C₂×Q₈	—	17	2	no	—
117	C₂×Q₈	—	34	1	no	—
118	C₂×Q₈	—	34	1	no	—
119	C₂×Q₈	—	34	1	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	34	1	no	—
122	C₂²×C₄	—	34	1	no	—
123	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
126	C₂×C₈	—	34	2	no	—
127	C₂×C₈	—	34	2	no	—
128	C₂×C₈	—	34	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	17	2	no	—
134	C₄⋊C₄	—	34	1	no	—
135	C₄⋊C₄	—	34	4	no	—
136	C₄²	—	34	4	no	—
137	C₄²	C₄×D₁₇	1	2	yes	—
138	C₄²	D₆₈	1	2	yes	—
139	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	Q₈	—	17	4	no	—
141	Q₈	—	34	2	no	—
142	Q₈	—	34	2	no	—
143	Q₈	—	34	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	34	2	no	—
150	C₂×C₄	—	34	2	no	—
151	C₂×C₄	—	34	2	no	—
152	C₂×C₄	—	34	2	no	—
153	C₂×C₄	—	68	1	no	—
154	C₂×C₄	—	68	1	no	—
155	C₂×C₄	#63227	1	2	yes	—
156	C₂×C₄	#63227	1	2	yes	—
157	C₂×C₄	#63229	1	2	yes	—
158	C₂×C₄	#63231	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
160	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
161	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
162	C₂×C₄	#63254	1	1	yes	characteristic
163	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
164	C₂×C₄	#63256	1	1	yes	characteristic
165	C₈	—	68	2	no	—
166	C₂²	C₂×#63227	1	1	yes	characteristic
167	C₂²	#10587618	1	1	yes	characteristic
168	C₂²	C₂×#63229	1	1	yes	characteristic
169	C₂²	#10587624	1	1	yes	characteristic
170	C₂²	#10587628	1	1	yes	characteristic
171	C₂²	C₂×#63231	1	1	yes	characteristic
172	C₂²	#10587632	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	8	2	no	—
175	C₄	—	34	4	no	—
176	C₄	—	68	2	no	—
177	C₄	#10587481	1	4	yes	complemented
178	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
179	C₄	#10587578	1	1	yes	characteristic
180	C₄	#10587581	1	1	yes	characteristic
181	C₄	#10587583	1	1	yes	characteristic
182	C₂	#11762666	1	1	yes	characteristic
183	C₂	C₂×#10587481	1	1	yes	characteristic
184	C₂	#11762808	1	1	yes	characteristic
185	C₂	#11763006	1	1	yes	characteristic
186	C₂	#11763016	1	1	yes	characteristic
187	C₂	#11763091	1	1	yes	characteristic
188	C₂	#11763101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₃₄

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂

yes

maximal, characteristic

C₂×#10587481

C₂

yes

maximal

#11762571

C₂

yes

maximal

C₁₇×#384

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×#63225

—

C₂×C₄×Dic₁₇

—

C₂²×#3622

C₂²

yes

—

C₄×Dic₃₄

—

C₄×Dic₃₄

C₄

yes

complemented

#10587481

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×Dic₃₄

D₄

yes

—

C₂×Dic₃₄

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

C₄⋊C₄

yes

—

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#103

yes

complemented

C₄×C₁₇

C₂×C₄⋊C₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

#384

D₁₇

yes

characteristic

#340

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

100

C₂²×Q₈

—

101

C₂²×C₈

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×D₁₇

yes

characteristic

111

C₄⋊C₈

—

112

#103

—

113

C₁₇

#1803

yes

characteristic, complemented, Sylow

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

124

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

125

C₂²×C₄

C₂²×D₁₇

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

C₄×D₁₇

yes

—

138

C₄²

D₆₈

yes

—

139

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

#63227

yes

—

156

C₂×C₄

#63227

yes

—

157

C₂×C₄

#63229

yes

—

158

C₂×C₄

#63231

yes

—

159

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

160

C₂×C₄

C₂×D₆₈

yes

characteristic

161

C₂×C₄

D₄×D₁₇

yes

characteristic

162

C₂×C₄

#63254

yes

characteristic

163

C₂×C₄

Q₈×D₁₇

yes

characteristic

164

C₂×C₄

#63256

yes

characteristic

165

C₈

—

166

C₂²

C₂×#63227

yes

characteristic

167

C₂²

#10587618

yes

characteristic

168

C₂²

C₂×#63229

yes

characteristic

169

C₂²

#10587624

yes

characteristic

170

C₂²

#10587628

yes

characteristic

171

C₂²

C₂×#63231

yes

characteristic

172

C₂²

#10587632

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#10587481

yes

complemented

178

C₄

C₄⋊C₄×D₁₇

yes

characteristic

179

C₄

#10587578

yes

characteristic

180

C₄

#10587581

yes

characteristic

181

C₄

#10587583

yes

characteristic

182

C₂

#11762666

yes

characteristic

183

C₂

C₂×#10587481

yes

characteristic

184

C₂

#11762808

yes

characteristic

185

C₂

#11763006

yes

characteristic

186

C₂

#11763016

yes

characteristic

187

C₂

#11763091

yes

characteristic

188

C₂

#11763101

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#384	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#384	D₁₇
hypocenter	C₁₇	#1803
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#384

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
16. C₄×Dic₃₄	C₄	semidirect
69. Dic₃₄	C₄⋊C₄	semidirect
81. C₄×C₁₇	#103	semidirect
113. C₁₇	#1803	semidirect
177. C₄	#10587481	semidirect

H ≅ G/N

type

16. C₄×Dic₃₄

C₄

semidirect

69. Dic₃₄

C₄⋊C₄

semidirect

81. C₄×C₁₇

#103

semidirect

113. C₁₇

#1803

semidirect

177. C₄

#10587481

semidirect

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

subgroup	index
C₂×C₄×Dic₃₄	2
C₂×C₄⋊#3622	2
C₂×#10587481	2
#11762571	2
C₁₇×#384	2
#1803	17

subgroup

index

C₂×C₄×Dic₃₄

C₂×C₄⋊#3622

C₂×#10587481

#11762571

C₁₇×#384

#1803

Subgroups — #14528786

Conjugacy classes of subgroups

Complements

Maximal subgroups