Group #16933760 — subgroups

Subgroups — #16933760

Order 4352.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#14528956	C₂	1	1	yes	maximal, characteristic
2	#14527469	C₂	1	2	yes	maximal
3	#14527465	C₂	1	2	yes	maximal
4	#14527067	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#498	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×#10587490	—	2	2	no	—
8	#11762833	C₂²	1	1	yes	characteristic
9	#11762833	C₂²	1	1	yes	characteristic
10	#11762558	—	2	4	no	—
11	#11762555	—	2	4	no	—
12	#11762530	—	2	1	no	—
13	C₈×#3622	—	2	1	no	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₇×#498	—	2	1	no	—
16	C₁₇×#498	—	2	1	no	—
17	C₁₇×#498	C₄	1	4	yes	—
18	C₂×C₁₇×#106	—	2	2	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	D₄	1	2	yes	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	2	2	no	—
25	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×#63223	—	2	1	no	—
30	C₂×#63223	—	2	1	no	—
31	C₁₇×#106	—	2	4	no	—
32	C₁₇×#106	—	2	4	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₂²⋊#3622	—	2	2	no	—
37	#10587490	—	4	4	no	—
38	C₄⋊#3622	—	2	2	no	—
39	C₄⋊#3622	—	2	2	no	—
40	C₄⋊#3622	D₄	1	4	yes	—
41	C₄×#3622	—	2	2	no	—
42	C₂×Q₈×C₁₇	—	2	4	no	—
43	C₂×Q₈×C₁₇	—	4	2	no	—
44	C₂×Q₈×C₁₇	—	8	4	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	#41	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
51	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₄⋊C₄×C₁₇	—	2	4	no	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₂×D₄	1	2	yes	—
58	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
59	#63223	—	4	2	no	—
60	#63223	—	4	2	no	—
61	C₂×#3622	—	2	4	no	—
62	C₂×#3622	—	4	1	no	—
63	C₂×#3622	—	4	1	no	—
64	#9767	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	#124	1	1	yes	characteristic
66	Q₈×C₁₇	—	8	4	no	—
67	Q₈×C₁₇	—	8	4	no	—
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	4	4	no	—
70	C₂×C₄×C₁₇	—	8	4	no	—
71	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₁₇	#121	1	2	yes	—
74	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
77	C₈×C₁₇	#103	1	4	yes	—
78	C₈×C₁₇	#103	1	4	yes	—
79	#3622	—	4	4	no	—
80	#3622	—	8	1	no	—
81	#3622	—	8	1	no	—
82	C₂×#498	D₁₇	1	1	yes	characteristic
83	#1848	—	17	1	no	—
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	—	8	4	no	—
92	C₄×C₁₇	C₂×#103	1	2	yes	—
93	C₄×C₁₇	#471	1	4	yes	—
94	C₄×C₁₇	#492	1	1	yes	characteristic
95	C₄×C₁₇	#494	1	1	yes	characteristic
96	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
97	#498	—	2	1	no	—
98	#498	—	2	1	no	—
99	#498	Dic₁₇	1	4	yes	—
100	#431	—	17	2	no	—
101	C₂×#108	—	34	2	no	—
102	C₂×#106	—	2	2	no	—
103	#423	—	17	1	no	—
104	#423	—	17	1	no	—
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	—	2	2	no	—
114	C₂×M₁₆	—	34	1	no	—
115	C₂×M₁₆	—	34	1	no	—
116	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
124	#108	—	68	4	no	—
125	#106	—	2	4	no	—
126	#106	—	2	4	no	—
127	C₄⋊C₈	—	17	4	no	—
128	C₄⋊C₈	—	34	2	no	—
129	C₄⋊C₈	—	34	2	no	—
130	C₂²⋊C₈	—	34	2	no	—
131	C₄×C₈	—	34	2	no	—
132	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
133	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₁₇	#9767	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	#63254	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
142	M₁₆	—	68	2	no	—
143	M₁₆	—	68	2	no	—
144	C₂×C₈	—	34	4	no	—
145	C₂×C₈	—	68	1	no	—
146	C₂×C₈	—	68	1	no	—
147	C₂×C₈	#63230	1	4	yes	—
148	C₂×C₈	#63231	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	D₄×D₁₇	1	2	yes	—
156	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
157	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
158	C₂³	#10587622	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
165	C₂×C₄	#10587624	1	2	yes	—
166	C₂×C₄	#10587626	1	1	yes	characteristic
167	C₂×C₄	C₂×#63230	1	2	yes	—
168	C₂×C₄	C₂×#63231	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
170	C₈	—	68	4	no	—
171	C₈	—	136	1	no	—
172	C₈	—	136	1	no	—
173	C₈	#10587506	1	4	yes	—
174	C₈	#10587506	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#11762712	1	1	yes	characteristic
177	C₂²	#11763159	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#10587506	1	2	yes	—
181	C₄	#11763069	1	4	yes	—
182	C₄	#11763183	1	1	yes	characteristic
183	C₄	#11763188	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#14529587	1	1	yes	characteristic
186	C₂	#14529689	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal, characteristic

#14527469

C₂

yes

maximal

#14527465

C₂

yes

maximal

#14527067

C₂

yes

maximal, characteristic

C₂×C₁₇×#498

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

C₂×#10587490

—

#11762833

C₂²

yes

characteristic

#11762833

C₂²

yes

—

—

—

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#498

—

C₁₇×#498

—

C₁₇×#498

C₄

yes

—

C₂×C₁₇×#106

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂×#63223

—

C₁₇×#106

—

C₁₇×#106

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₂²⋊#3622

—

—

—

—

D₄

yes

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

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

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

#103

yes

—

—

—

—

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

C₄×C₁₇

#471

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

#498

—

#498

—

#498

Dic₁₇

yes

—

100

#431

—

101

C₂×#108

—

102

C₂×#106

—

103

#423

—

104

#423

—

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

C₂×Dic₁₇

yes

—

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

C₂×Dic₁₇

yes

—

133

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

134

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

135

C₁₇

#9767

yes

characteristic, complemented, Sylow

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

#63254

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

#63230

yes

—

148

C₂×C₈

#63231

yes

—

149

C₂×C₈

C₂²⋊Dic₁₇

yes

—

150

C₂×C₈

C₂²⋊Dic₁₇

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

D₄×D₁₇

yes

—

156

C₄²

C₂²×Dic₁₇

yes

characteristic

157

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

158

C₂³

#10587622

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

D₄×Dic₁₇

yes

—

165

C₂×C₄

#10587624

yes

—

166

C₂×C₄

#10587626

yes

characteristic

167

C₂×C₄

C₂×#63230

yes

—

168

C₂×C₄

C₂×#63231

yes

—

169

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

170

C₈

—

171

C₈

—

136

—

172

C₈

—

136

—

173

C₈

#10587506

yes

—

174

C₈

#10587506

yes

—

175

C₂²

—

176

C₂²

#11762712

yes

characteristic

177

C₂²

#11763159

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#10587506

yes

—

181

C₄

#11763069

yes

—

182

C₄

#11763183

yes

characteristic

183

C₄

#11763188

yes

characteristic

184

C₂

—

185

C₂

#14529587

yes

characteristic

186

C₂

#14529689

yes

—

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#498	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#498	D₁₇
hypocenter	C₁₇	#9767
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#11762712

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#498

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

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
#14528956	2
#14527469	2
#14527465	2
#14527067	2
C₂×C₁₇×#498	2
#9767	17

subgroup

index

C₂×C₁₇×#498

Subgroups — #16933760

Conjugacy classes of subgroups

Complements

Maximal subgroups