Group #16933078 — subgroups

Subgroups — #16933078

Order 4352.

268 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14527573	C₂	1	2	yes	maximal
2	#14527540	C₂	1	1	yes	maximal, characteristic
3	C₈×#63223	C₂	1	1	yes	maximal, characteristic
4	#14526675	C₂	1	2	yes	maximal
5	C₁₇×#1564	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	1	yes	characteristic
7	#11762833	C₂²	1	2	yes	—
8	C₄×#63223	C₂²	1	1	yes	characteristic
9	#11762562	—	2	2	no	—
10	#11762560	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762560	—	2	2	no	—
13	#11762531	—	2	1	no	—
14	C₈×#3622	—	2	1	no	—
15	C₁₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#335	C₄	1	2	yes	—
18	C₁₇×#334	C₄	1	2	yes	—
19	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×#63223	—	2	1	no	—
23	C₂×#63223	—	2	2	no	—
24	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
25	C₁₇×C₂²⋊C₈	—	2	2	no	—
26	C₄×C₈×C₁₇	—	2	2	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²⋊#3622	—	2	1	no	—
32	C₄⋊#3622	—	2	2	no	—
33	C₄⋊#3622	—	2	2	no	—
34	C₄⋊#3622	D₄	1	2	yes	—
35	#10587474	Q₈	1	2	yes	—
36	C₄×#3622	—	2	1	no	—
37	C₄×#3622	—	2	1	no	—
38	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	—	4	1	no	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	#63223	—	2	4	no	—
53	#63223	—	8	2	no	—
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	—	4	2	no	—
57	#9633	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	#125	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	—	2	1	no	—
72	C₈×C₁₇	—	4	4	no	—
73	C₈×C₁₇	#108	1	2	yes	—
74	#3622	—	2	4	no	—
75	#3622	—	4	2	no	—
76	#3622	—	8	2	no	—
77	#1574	—	17	2	no	—
78	#1566	—	17	1	no	—
79	#1564	D₁₇	1	1	yes	characteristic
80	C₈×M₁₆	—	17	1	no	—
81	#1299	—	17	2	no	—
82	C₂²×C₁₇	—	2	1	no	—
83	C₂²×C₁₇	—	2	1	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	#389	1	1	yes	characteristic
86	C₄×C₁₇	—	2	1	no	—
87	C₄×C₁₇	—	2	1	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
90	C₄×C₁₇	#430	1	1	yes	characteristic
91	C₄×C₁₇	#443	1	2	yes	—
92	C₄×C₁₇	#471	1	2	yes	—
93	#431	—	17	1	no	—
94	#431	C₂×D₁₇	1	2	yes	—
95	#423	—	17	2	no	—
96	C₄×M₁₆	—	17	1	no	—
97	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
98	#335	—	34	2	no	—
99	#335	Dic₁₇	1	2	yes	—
100	#334	—	34	2	no	—
101	#334	—	34	1	no	—
102	#334	—	34	1	no	—
103	#334	Dic₁₇	1	2	yes	—
104	#322	—	34	1	no	—
105	C₈²	—	34	1	no	—
106	C₂×C₁₇	—	4	1	no	—
107	C₂×C₁₇	#1945	1	1	yes	characteristic
108	C₂×C₁₇	#1950	1	1	yes	characteristic
109	C₂×C₁₇	#1951	1	1	yes	characteristic
110	C₂×M₁₆	—	34	1	no	—
111	C₂×M₁₆	—	34	2	no	—
112	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
113	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
114	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
115	C₄⋊C₈	—	17	2	no	—
116	C₄⋊C₈	—	34	2	no	—
117	C₄⋊C₈	—	34	2	no	—
118	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	34	1	no	—
121	#97	—	17	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	34	1	no	—
124	C₄×C₈	—	34	1	no	—
125	C₄×C₈	D₆₈	1	1	yes	characteristic
126	C₄×C₈	D₆₈	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
129	C₁₇	#9633	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
131	C₂²×C₄	#63254	1	1	yes	characteristic
132	C₂²×C₄	#63256	1	1	yes	characteristic
133	M₁₆	—	34	4	no	—
134	M₁₆	—	136	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	34	2	no	—
139	C₂×C₈	—	34	2	no	—
140	C₂×C₈	—	68	2	no	—
141	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
145	C₄²	C₂×D₆₈	1	1	yes	characteristic
146	C₄²	D₄×D₁₇	1	1	yes	characteristic
147	C₄²	Q₈×D₁₇	1	1	yes	characteristic
148	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
149	C₂³	#10587576	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	#10587574	1	1	yes	characteristic
155	C₂×C₄	#10587581	1	1	yes	characteristic
156	C₂×C₄	#10587583	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
158	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
159	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	4	no	—
164	C₈	—	34	4	no	—
165	C₈	—	68	2	no	—
166	C₈	—	136	2	no	—
167	C₈	#10587490	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#11762708	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#10587490	1	1	yes	characteristic
176	C₄	#11762858	1	1	yes	characteristic
177	C₄	#11762926	1	2	yes	—
178	C₄	#11763073	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#14529533	1	1	yes	characteristic
181	C₂	#14529560	1	1	yes	characteristic
182	C₂	#14529565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14527573

C₂

yes

maximal

#14527540

C₂

yes

maximal, characteristic

C₈×#63223

C₂

yes

maximal, characteristic

#14526675

C₂

yes

maximal

C₁₇×#1564

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

characteristic

#11762833

C₂²

yes

—

C₄×#63223

C₂²

yes

—

—

—

—

—

—

C₁₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#335

C₄

yes

—

C₁₇×#334

C₄

yes

—

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂×#63223

—

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₂²⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

D₄

yes

—

#10587474

Q₈

yes

—

C₄×#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×Q₈

yes

C₂²×C₄×C₁₇

yes

C₂²×C₄×C₁₇

yes

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#108

yes

—

—

—

—

—

—

D₁₇

yes

characteristic

C₈×M₁₆

—

#1299

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#389

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₄×C₁₇

#471

yes

—

#431

—

#431

C₂×D₁₇

yes

—

#423

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#335

—

#335

Dic₁₇

yes

—

100

#334

—

101

#334

—

102

#334

—

103

#334

Dic₁₇

yes

—

104

#322

—

105

C₈²

—

106

C₂×C₁₇

—

107

C₂×C₁₇

#1945

yes

characteristic

108

C₂×C₁₇

#1950

yes

characteristic

109

C₂×C₁₇

#1951

yes

characteristic

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

Dic₃₄

yes

characteristic

113

C₂²×C₈

Dic₃₄

yes

characteristic

114

C₂×C₄²

C₂²×D₁₇

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

C₂×Dic₁₇

yes

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

D₆₈

yes

characteristic

126

C₄×C₈

D₆₈

yes

characteristic

127

C₄×C₈

C₂×Dic₁₇

yes

characteristic

128

C₄×C₈

C₂×Dic₁₇

yes

characteristic

129

C₁₇

#9633

yes

characteristic, complemented, Sylow

130

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

131

C₂²×C₄

#63254

yes

characteristic

132

C₂²×C₄

#63256

yes

characteristic

133

M₁₆

—

134

M₁₆

—

136

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

142

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

143

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

144

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

145

C₄²

C₂×D₆₈

yes

characteristic

146

C₄²

D₄×D₁₇

yes

characteristic

147

C₄²

Q₈×D₁₇

yes

characteristic

148

C₄²

C₂²×Dic₁₇

yes

characteristic

149

C₂³

#10587576

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#10587574

yes

characteristic

155

C₂×C₄

#10587581

yes

characteristic

156

C₂×C₄

#10587583

yes

characteristic

157

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

158

C₂×C₄

D₄×Dic₁₇

yes

characteristic

159

C₂×C₄

Q₈×Dic₁₇

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

136

—

167

C₈

#10587490

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#11762708

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#10587490

yes

characteristic

176

C₄

#11762858

yes

characteristic

177

C₄

#11762926

yes

—

178

C₄

#11763073

yes

—

179

C₂

—

180

C₂

#14529533

yes

characteristic

181

C₂

#14529560

yes

characteristic

182

C₂

#14529565

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587581
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1564	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	#125
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1564	D₁₇
hypocenter	C₁₇	#9633
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#10587581

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1564

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
129. C₁₇	#9633	semidirect

H ≅ G/N

type

129. C₁₇

#9633

semidirect

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

subgroup	index
#14527573	2
#14527540	2
C₈×#63223	2
#14526675	2
C₁₇×#1564	2
#9633	17

subgroup

index

C₁₇×#1564

Subgroups — #16933078

Conjugacy classes of subgroups

Complements

Maximal subgroups