Group #16933757 — subgroups

Subgroups — #16933757

Order 4352.

414 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 151 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×#11763207	C₂	1	1	yes	maximal, characteristic
2	#14527470	C₂	1	2	yes	maximal
3	#14527466	C₂	1	2	yes	maximal
4	#14527067	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	2	yes	—
7	#11763207	—	2	1	no	—
8	#11763207	—	2	1	no	—
9	#11763207	C₄	1	4	yes	—
10	#11762862	C₂²	1	2	yes	—
11	C₂×#10587488	—	2	2	no	—
12	#11762557	—	2	4	no	—
13	#11762554	—	2	4	no	—
14	#11762530	—	2	1	no	—
15	C₈×#3622	—	2	1	no	—
16	C₂×C₁₇×#108	—	2	2	no	—
17	C₁₇×#423	C₂²	1	1	yes	characteristic
18	C₁₇×#423	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₃₄	—	2	2	no	—
21	C₂×M₁₆×C₁₇	—	2	1	no	—
22	C₂×M₁₆×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	D₄	1	2	yes	—
24	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	#10587555	—	2	2	no	—
29	#10587555	—	2	2	no	—
30	#10587555	C₂×C₄	1	4	yes	—
31	C₁₇×#108	—	4	4	no	—
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₁₇×C₄⋊C₈	—	2	2	no	—
34	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₂²⋊#3622	—	2	2	no	—
38	#10587488	—	2	4	no	—
39	#10587488	—	2	4	no	—
40	C₄⋊#3622	D₄	1	4	yes	—
41	C₄×#3622	—	2	2	no	—
42	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	#41	1	2	yes	—
44	C₂²×Dic₁₇	—	4	2	no	—
45	C₂×Dic₃₄	—	2	4	no	—
46	C₂×Dic₃₄	—	4	2	no	—
47	C₂×Dic₃₄	—	8	4	no	—
48	M₁₆×C₁₇	—	4	2	no	—
49	M₁₆×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	—	4	1	no	—
51	C₂×C₈×C₁₇	—	4	1	no	—
52	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
55	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
56	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₁₇	—	2	4	no	—
60	C₄⋊Dic₁₇	—	2	4	no	—
61	C₄⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₂×#3622	—	2	4	no	—
64	#9767	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₁₇	#121	1	2	yes	—
70	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
73	C₂×Dic₁₇	—	4	4	no	—
74	C₂×Dic₁₇	—	8	4	no	—
75	Dic₃₄	—	8	4	no	—
76	Dic₃₄	—	8	4	no	—
77	C₈×C₁₇	—	8	1	no	—
78	C₈×C₁₇	—	8	1	no	—
79	C₈×C₁₇	#103	1	4	yes	—
80	C₈×C₁₇	#103	1	4	yes	—
81	#3622	—	4	4	no	—
82	C₂×#498	—	17	1	no	—
83	#1848	D₁₇	1	1	yes	characteristic
84	#1549	—	17	2	no	—
85	#1548	—	17	2	no	—
86	#1450	—	17	1	no	—
87	C₂²×C₁₇	—	2	2	no	—
88	C₂²×C₁₇	#390	1	1	yes	characteristic
89	C₂²×C₁₇	#487	1	2	yes	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	C₂×#103	1	2	yes	—
92	C₄×C₁₇	#471	1	4	yes	—
93	C₄×C₁₇	#492	1	1	yes	characteristic
94	C₄×C₁₇	#494	1	1	yes	characteristic
95	Dic₁₇	—	8	4	no	—
96	C₂×C₄⋊Q₈	—	17	2	no	—
97	#498	—	17	4	no	—
98	#498	—	34	1	no	—
99	#498	—	34	1	no	—
100	#431	—	17	2	no	—
101	C₂×#108	—	2	2	no	—
102	C₂×#106	—	34	2	no	—
103	#423	C₂×D₁₇	1	1	yes	characteristic
104	#423	C₂×D₁₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
106	#333	—	34	4	no	—
107	#332	—	34	4	no	—
108	#322	—	34	1	no	—
109	C₈²	—	34	1	no	—
110	C₂×C₁₇	—	2	2	no	—
111	C₂×C₁₇	#1956	1	1	yes	characteristic
112	C₂×C₁₇	#1977	1	2	yes	—
113	C₂²×Q₈	—	34	2	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
117	C₄⋊Q₈	—	17	4	no	—
118	C₄⋊Q₈	—	34	2	no	—
119	C₄⋊Q₈	—	34	2	no	—
120	C₄⋊Q₈	—	34	2	no	—
121	C₂×C₄⋊C₄	—	34	2	no	—
122	C₂×C₄⋊C₄	—	34	2	no	—
123	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	#106	—	34	4	no	—
126	#106	—	34	4	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	17	4	no	—
130	C₂²⋊C₈	—	34	2	no	—
131	C₄×C₈	—	34	2	no	—
132	C₄×C₈	C₄×D₁₇	1	2	yes	—
133	C₄×C₈	D₆₈	1	1	yes	characteristic
134	C₄×C₈	D₆₈	1	1	yes	characteristic
135	C₁₇	#9767	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	34	4	no	—
137	C₂×Q₈	—	68	2	no	—
138	C₂×Q₈	—	136	4	no	—
139	C₂²×C₄	—	68	2	no	—
140	C₂²×C₄	#63256	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	34	4	no	—
147	C₂×C₈	#63227	1	2	yes	—
148	C₂×C₈	#63227	1	2	yes	—
149	C₂×C₈	#63229	1	4	yes	—
150	C₂×C₈	#63231	1	4	yes	—
151	C₄⋊C₄	—	34	4	no	—
152	C₄⋊C₄	—	34	4	no	—
153	C₄⋊C₄	—	68	2	no	—
154	C₄⋊C₄	—	68	2	no	—
155	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
156	C₄²	C₂×D₆₈	1	1	yes	characteristic
157	C₄²	D₄×D₁₇	1	2	yes	—
158	C₂³	#10587633	1	1	yes	characteristic
159	Q₈	—	136	4	no	—
160	Q₈	—	136	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	68	4	no	—
163	C₂×C₄	—	136	4	no	—
164	C₂×C₄	#10587579	1	2	yes	—
165	C₂×C₄	#10587581	1	2	yes	—
166	C₂×C₄	C₂×#63227	1	1	yes	characteristic
167	C₂×C₄	C₂×#63229	1	2	yes	—
168	C₂×C₄	#10587626	1	1	yes	characteristic
169	C₂×C₄	C₂×#63231	1	2	yes	—
170	C₈	—	8	1	no	—
171	C₈	—	8	1	no	—
172	C₈	—	68	4	no	—
173	C₈	#10587481	1	4	yes	—
174	C₈	#10587481	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#11762710	1	1	yes	characteristic
177	C₂²	#11763161	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	136	4	no	—
180	C₄	C₂×#10587481	1	2	yes	—
181	C₄	#11763073	1	4	yes	—
182	C₄	#11763181	1	1	yes	characteristic
183	C₄	#11763188	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#14529585	1	1	yes	characteristic
186	C₂	#14529685	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763207

C₂

yes

maximal, characteristic

#14527470

C₂

yes

maximal

#14527466

C₂

yes

maximal

#14527067

C₂

yes

maximal, characteristic

C₁₇×#1848

C₂

yes

maximal, characteristic

C₂×#10587555

C₂²

yes

—

#11763207

—

#11763207

—

#11763207

C₄

yes

—

#11762862

C₂²

yes

—

—

—

—

—

—

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Dic₃₄

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

—

—

—

C₂×C₄

yes

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₂²⋊#3622

—

#10587488

—

#10587488

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

#103

yes

—

#3622

—

C₂×#498

—

#1848

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

C₄×C₁₇

#471

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×C₄⋊Q₈

—

#498

—

#498

—

#498

—

100

#431

—

101

C₂×#108

—

102

C₂×#106

—

103

#423

C₂×D₁₇

yes

characteristic

104

#423

C₂×D₁₇

yes

characteristic

105

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

106

#333

—

107

#332

—

108

#322

—

109

C₈²

—

110

C₂×C₁₇

—

111

C₂×C₁₇

#1956

yes

characteristic

112

C₂×C₁₇

#1977

yes

—

113

C₂²×Q₈

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂²×C₈

C₂²⋊D₁₇

yes

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂²×D₁₇

yes

characteristic

124

#108

—

125

#106

—

126

#106

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

C₄×C₈

C₄×D₁₇

yes

—

133

C₄×C₈

D₆₈

yes

characteristic

134

C₄×C₈

D₆₈

yes

characteristic

135

C₁₇

#9767

yes

characteristic, complemented, Sylow

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

136

—

139

C₂²×C₄

—

140

C₂²×C₄

#63256

yes

—

141

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

#63227

yes

—

148

C₂×C₈

#63227

yes

—

149

C₂×C₈

#63229

yes

—

150

C₂×C₈

#63231

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₂×C₄×D₁₇

yes

characteristic

156

C₄²

C₂×D₆₈

yes

characteristic

157

C₄²

D₄×D₁₇

yes

—

158

C₂³

#10587633

yes

characteristic

159

Q₈

—

136

—

160

Q₈

—

136

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

136

—

164

C₂×C₄

#10587579

yes

—

165

C₂×C₄

#10587581

yes

—

166

C₂×C₄

C₂×#63227

yes

characteristic

167

C₂×C₄

C₂×#63229

yes

—

168

C₂×C₄

#10587626

yes

characteristic

169

C₂×C₄

C₂×#63231

yes

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

#10587481

yes

—

174

C₈

#10587481

yes

—

175

C₂²

—

176

C₂²

#11762710

yes

characteristic

177

C₂²

#11763161

yes

—

178

C₄

—

179

C₄

—

136

—

180

C₄

C₂×#10587481

yes

—

181

C₄

#11763073

yes

—

182

C₄

#11763181

yes

characteristic

183

C₄

#11763188

yes

characteristic

184

C₂

—

185

C₂

#14529585

yes

characteristic

186

C₂

#14529685

yes

—

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#9767
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#11762710

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1848

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
135. C₁₇	#9767	semidirect

H ≅ G/N

type

135. C₁₇

#9767

semidirect

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

subgroup	index
C₂×#11763207	2
#14527470	2
#14527466	2
#14527067	2
C₁₇×#1848	2
#9767	17

subgroup

index

C₁₇×#1848

Subgroups — #16933757

Conjugacy classes of subgroups

Complements

Maximal subgroups