Group #16939342 — subgroups

Subgroups — #16939342

Order 4352.

354 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14528938	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527560	C₂	1	2	yes	maximal, complemented
3	#14527183	C₂	1	2	yes	maximal
4	#14526613	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1963	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	—	2	2	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	#11762561	—	2	4	no	—
10	D₄⋊#3622	—	2	4	no	—
11	C₁₇×#532	C₂²	1	1	yes	characteristic
12	C₁₇×#423	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#104	—	2	2	no	—
14	C₁₇×#376	C₂²	1	1	yes	characteristic
15	C₂×#41×C₁₇	—	2	1	no	—
16	C₂×M₁₆×C₁₇	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
18	C₁₇×#127	Q₈	1	1	yes	characteristic
19	C₁₇×#124	—	2	1	no	—
20	C₁₇×#121	—	2	2	no	—
21	C₄×D₄×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	D₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂²×#3622	—	2	2	no	—
25	C₁₇×C₄⋊C₈	—	2	2	no	—
26	C₁₇×#104	—	4	4	no	—
27	C₁₇×#95	—	2	2	no	—
28	C₂²⋊#3622	—	2	2	no	—
29	C₄⋊#3622	—	2	2	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄⋊#3622	—	4	4	no	—
33	C₄×#3622	D₄	1	4	yes	—
34	#41×C₁₇	—	4	2	no	—
35	#41×C₁₇	—	4	2	no	—
36	C₂×Q₈×C₁₇	—	2	1	no	—
37	C₂×D₄×C₁₇	—	2	1	no	—
38	C₂×D₄×C₁₇	—	2	2	no	—
39	C₂×D₄×C₁₇	—	4	1	no	—
40	C₂²×C₄×C₁₇	—	2	2	no	—
41	C₂²×C₄×C₁₇	—	4	1	no	—
42	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	#41	1	2	yes	—
44	M₁₆×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	—	4	1	no	—
46	C₄⋊C₄×C₁₇	—	2	2	no	—
47	C₂²⋊C₄×C₁₇	—	4	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	D₈	1	4	yes	—
50	C₄²×C₁₇	C₂×D₄	1	2	yes	—
51	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₇	#41	1	1	yes	characteristic
53	C₂×#3622	—	2	4	no	—
54	C₂×#3622	—	4	2	no	—
55	C₂×#3622	—	4	4	no	—
56	#10667	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	—	4	1	no	—
58	C₂³×C₁₇	#123	1	1	yes	characteristic
59	Q₈×C₁₇	—	4	2	no	—
60	D₄×C₁₇	—	4	4	no	—
61	D₄×C₁₇	—	4	2	no	—
62	D₄×C₁₇	—	8	2	no	—
63	C₂×C₄×C₁₇	—	2	4	no	—
64	C₂×C₄×C₁₇	—	4	2	no	—
65	C₂×C₄×C₁₇	—	4	4	no	—
66	C₂×C₄×C₁₇	—	4	1	no	—
67	C₂×C₄×C₁₇	—	8	1	no	—
68	C₂×C₄×C₁₇	—	8	1	no	—
69	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#123	1	2	yes	—
72	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₁₇	#137	1	2	yes	—
75	C₈×C₁₇	—	8	1	no	—
76	#3622	—	4	4	no	—
77	#3622	—	8	2	no	—
78	#1963	D₁₇	1	1	yes	characteristic
79	#1843	—	17	1	no	—
80	#1571	—	17	2	no	—
81	#1480	—	17	2	no	—
82	#1274	—	17	1	no	—
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	—	8	1	no	—
85	C₂²×C₁₇	—	8	2	no	—
86	C₂²×C₁₇	#399	1	1	yes	characteristic
87	C₂²×C₁₇	#480	1	2	yes	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	4	4	no	—
91	C₄×C₁₇	—	8	1	no	—
92	C₄×C₁₇	#474	1	2	yes	—
93	C₄×C₁₇	#493	1	1	yes	characteristic
94	C₄×C₁₇	#497	1	1	yes	characteristic
95	#532	C₂×D₁₇	1	1	yes	characteristic
96	#431	—	17	2	no	—
97	#423	C₂×D₁₇	1	1	yes	characteristic
98	C₂×C₄⋊C₈	—	17	2	no	—
99	C₂×C₄⋊C₈	—	34	2	no	—
100	C₂×#104	—	2	2	no	—
101	#376	C₂×D₁₇	1	1	yes	characteristic
102	#335	—	34	4	no	—
103	D₄⋊C₈	—	34	4	no	—
104	C₂×C₁₇	—	2	2	no	—
105	C₂×C₁₇	—	16	1	no	—
106	C₂×C₁₇	#2093	1	1	yes	characteristic
107	C₂×C₁₇	#2098	1	2	yes	—
108	C₂×#41	—	2	1	no	—
109	C₂×M₁₆	—	2	1	no	—
110	C₂²×C₈	—	34	2	no	—
111	C₄⋊Q₈	Dic₃₄	1	1	yes	characteristic
112	#127	Dic₃₄	1	1	yes	characteristic
113	#124	—	2	1	no	—
114	#121	—	2	2	no	—
115	C₄×D₄	—	2	2	no	—
116	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	34	2	no	—
120	C₄⋊C₈	—	34	2	no	—
121	C₄⋊C₈	—	34	2	no	—
122	C₄⋊C₈	—	68	4	no	—
123	#104	—	4	4	no	—
124	C₂²⋊C₈	—	34	2	no	—
125	C₄×C₈	—	17	4	no	—
126	#95	—	2	2	no	—
127	C₁₇	#10667	1	1	yes	characteristic, complemented, Sylow
128	#41	—	4	2	no	—
129	#41	—	4	2	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	4	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	#63254	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	34	4	no	—
141	C₂×C₈	—	68	4	no	—
142	C₂×C₈	—	68	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#63228	1	4	yes	—
147	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
148	C₄²	#63252	1	1	yes	characteristic
149	C₄²	D₄×D₁₇	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#10587621	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	4	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#10587564	1	2	yes	—
163	C₂×C₄	#10587569	1	2	yes	—
164	C₂×C₄	#10587610	1	1	yes	characteristic
165	C₂×C₄	C₂×#63228	1	2	yes	—
166	C₂×C₄	#10587626	1	1	yes	characteristic
167	C₂×C₄	#10587637	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	68	4	no	—
170	C₈	—	136	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#11762745	1	1	yes	characteristic
175	C₂²	#11763120	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	8	1	no	—
180	C₄	#11763089	1	2	yes	—
181	C₄	#11763185	1	1	yes	characteristic
182	C₄	#11763204	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#14530312	1	1	yes	characteristic
186	C₂	#14530339	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528938

C₂

yes

maximal, characteristic, complemented

#14527560

C₂

yes

maximal, complemented

#14527183

C₂

yes

maximal

#14526613

C₂

yes

maximal, characteristic, complemented

C₁₇×#1963

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

C₂×C₄⋊#3622

—

C₂×C₄⋊#3622

C₂²

yes

—

#11762561

—

D₄⋊#3622

—

C₁₇×#532

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₂×C₁₇×#104

—

C₁₇×#376

C₂²

yes

characteristic

C₂×#41×C₁₇

—

C₂×M₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

Q₈

yes

characteristic

C₁₇×#127

Q₈

yes

characteristic

C₁₇×#124

—

C₁₇×#121

—

C₄×D₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₁₇×C₄⋊C₈

—

C₁₇×#104

—

C₁₇×#95

—

C₂²⋊#3622

—

—

—

—

—

D₄

yes

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₈

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

C₄²×C₁₇

yes

—

—

—

—

C₂³×C₁₇

—

C₂³×C₁₇

#123

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₈×C₁₇

—

#3622

—

#3622

—

#1963

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#399

yes

characteristic

C₂²×C₁₇

#480

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#474

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₂×D₁₇

yes

characteristic

#431

—

#423

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

100

C₂×#104

—

101

#376

C₂×D₁₇

yes

characteristic

102

#335

—

103

D₄⋊C₈

—

104

C₂×C₁₇

—

105

C₂×C₁₇

—

106

C₂×C₁₇

#2093

yes

characteristic

107

C₂×C₁₇

#2098

yes

—

108

C₂×#41

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₄⋊Q₈

Dic₃₄

yes

characteristic

112

#127

Dic₃₄

yes

characteristic

113

#124

—

114

#121

—

115

C₄×D₄

—

116

C₂×C₄²

C₂²⋊D₁₇

yes

—

117

C₂×C₄²

C₂²×D₁₇

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#104

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

C₁₇

#10667

yes

characteristic, complemented, Sylow

128

#41

—

129

#41

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

#63254

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

#63228

yes

—

147

C₄²

C₂×Dic₃₄

yes

characteristic

148

C₄²

#63252

yes

characteristic

149

C₄²

D₄×D₁₇

yes

—

150

C₂³

—

151

C₂³

#10587621

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#10587564

yes

—

163

C₂×C₄

#10587569

yes

—

164

C₂×C₄

#10587610

yes

characteristic

165

C₂×C₄

C₂×#63228

yes

—

166

C₂×C₄

#10587626

yes

characteristic

167

C₂×C₄

#10587637

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

136

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11762745

yes

characteristic

175

C₂²

#11763120

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11763089

yes

—

181

C₄

#11763185

yes

characteristic

182

C₄

#11763204

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#14530312

yes

characteristic

186

C₂

#14530339

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1963	C₂
socle	C₂²×C₁₇	#399
prime radical	C₁₇×#532	C₂²
hypercenter	#1963	D₁₇
hypocenter	C₁₇	#10667
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1963

C₂

socle

C₂²×C₁₇

#399

prime radical

C₁₇×#532

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #14528938	C₂	semidirect
2. #14527560	C₂	semidirect
4. #14526613	C₂	semidirect
127. C₁₇	#10667	semidirect

H ≅ G/N

type

1. #14528938

C₂

semidirect

2. #14527560

C₂

semidirect

4. #14526613

C₂

semidirect

127. C₁₇

#10667

semidirect

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

subgroup	index
#14528938	2
#14527560	2
#14527183	2
#14526613	2
C₁₇×#1963	2
#10667	17

subgroup

index

C₁₇×#1963

Subgroups — #16939342

Conjugacy classes of subgroups

Complements

Maximal subgroups