Group #16933719 — subgroups

Subgroups — #16933719

Order 4352.

478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 111 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	#14539212	C₂	1	1	yes	maximal, characteristic
2	#14527343	C₂	1	4	yes	maximal, complemented
3	#14527076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10587590	—	2	2	no	—
6	#11763359	C₂²	1	2	yes	—
7	#11763193	—	2	2	no	—
8	#11763189	C₄	1	1	yes	characteristic, complemented
9	#11763186	C₄	1	1	yes	characteristic, complemented
10	#11763182	C₄	1	2	yes	complemented
11	#11762862	C₂²	1	2	yes	—
12	#11762542	—	2	8	no	—
13	#11762531	—	2	2	no	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#382	C₂²	1	2	yes	—
16	C₁₇×#340	—	2	2	no	—
17	C₂×#63252	—	2	2	no	—
18	C₂²×C₈×C₁₇	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	#10587616	—	2	4	no	—
22	C₂×Dic₆₈	—	2	2	no	—
23	#10587590	—	4	8	no	—
24	C₂×D₁₃₆	—	2	2	no	—
25	C₂×#63219	—	2	4	no	—
26	#10587561	—	2	2	no	—
27	#10587560	C₂×C₄	1	2	yes	—
28	C₄×D₆₈	—	2	4	no	—
29	#10587555	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	D₄	1	2	yes	—
33	C₁₇×#95	—	2	4	no	—
34	C₂²⋊#3622	D₄	1	4	yes	—
35	C₄⋊#3622	—	2	2	no	—
36	C₄×#3622	—	2	2	no	—
37	C₂²×C₄×C₁₇	—	4	2	no	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₇	—	4	2	no	—
41	#63252	—	4	8	no	—
42	C₂×D₆₈	—	2	2	no	—
43	C₂×D₆₈	—	2	4	no	—
44	C₂×C₄×D₁₇	—	4	2	no	—
45	C₂×Dic₃₄	—	2	2	no	—
46	C₂×C₈×C₁₇	—	2	4	no	—
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
49	C₄⋊C₄×C₁₇	—	4	4	no	—
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₇	#41	1	2	yes	—
53	#63227	—	4	4	no	—
54	C₄⋊Dic₁₇	—	2	4	no	—
55	C₂×#3622	—	2	4	no	—
56	Dic₆₈	—	4	4	no	—
57	D₁₃₆	—	4	4	no	—
58	#63219	—	4	8	no	—
59	#9760	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	#127	1	1	yes	characteristic
61	C₂²×D₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	—	4	2	no	—
63	C₂×C₄×C₁₇	—	8	4	no	—
64	C₂×C₄×C₁₇	—	8	2	no	—
65	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₁₇	#121	1	2	yes	—
68	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²⋊D₁₇	—	8	4	no	—
71	C₂×Dic₁₇	—	4	2	no	—
72	D₆₈	—	4	4	no	—
73	D₆₈	—	8	4	no	—
74	C₄×D₁₇	—	8	4	no	—
75	Dic₃₄	—	4	4	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	—	2	2	no	—
78	#3622	—	4	4	no	—
79	#3147	—	17	1	no	—
80	#1835	D₁₇	1	1	yes	characteristic, complemented
81	#1517	—	17	4	no	—
82	#1452	—	17	1	no	—
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	#390	1	1	yes	characteristic
85	C₂²×C₁₇	#496	1	2	yes	—
86	C₂×D₁₇	—	8	2	no	—
87	C₂×D₁₇	—	8	4	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	16	2	no	—
91	C₄×C₁₇	#418	1	2	yes	—
92	C₄×C₁₇	#490	1	2	yes	—
93	Dic₁₇	—	8	2	no	—
94	C₂×#135	—	34	2	no	—
95	#532	—	17	2	no	—
96	#495	—	34	2	no	—
97	#494	—	17	1	no	—
98	#493	—	17	1	no	—
99	#492	—	17	2	no	—
100	#431	—	17	2	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#382	C₂×D₁₇	1	2	yes	—
103	#340	—	2	2	no	—
104	#327	—	34	8	no	—
105	#322	—	34	2	no	—
106	C₂×C₁₇	—	2	2	no	—
107	C₂×C₁₇	#1961	1	1	yes	characteristic
108	C₂×C₁₇	#1976	1	2	yes	—
109	D₁₇	—	16	2	no	—
110	C₂×#41	—	34	2	no	—
111	#135	—	68	8	no	—
112	C₂×Q₁₆	—	34	2	no	—
113	C₂×SD₁₆	—	34	4	no	—
114	C₂×D₈	—	34	2	no	—
115	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
116	C₄⋊Q₈	—	17	2	no	—
117	#127	—	17	2	no	—
118	#124	—	34	2	no	—
119	#121	—	34	4	no	—
120	C₄×D₄	—	34	4	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊C₈	—	34	2	no	—
124	C₂²⋊C₈	—	17	4	no	—
125	C₄×C₈	—	34	2	no	—
126	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
127	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
128	C₄×C₈	D₆₈	1	2	yes	—
129	#95	—	2	4	no	—
130	C₁₇	#9760	1	1	yes	characteristic, complemented, Sylow
131	#41	—	68	8	no	—
132	C₂×Q₈	—	34	2	no	—
133	C₂×D₄	—	34	4	no	—
134	C₂×D₄	—	34	2	no	—
135	C₂×D₄	—	68	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	68	2	no	—
138	C₂²×C₄	D₄×D₁₇	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	Q₁₆	—	68	4	no	—
141	SD₁₆	—	68	8	no	—
142	D₈	—	68	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	34	4	no	—
145	C₂×C₈	#63227	1	2	yes	—
146	C₂×C₈	#63227	1	2	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	34	4	no	—
149	C₂²⋊C₄	—	68	4	no	—
150	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
151	C₄²	C₂×D₆₈	1	1	yes	characteristic
152	C₄²	#63256	1	2	yes	—
153	C₂³	—	68	2	no	—
154	C₂³	#10587626	1	1	yes	characteristic
155	Q₈	—	68	4	no	—
156	D₄	—	68	4	no	—
157	D₄	—	136	4	no	—
158	D₄	—	136	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	68	2	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	#10587633	1	1	yes	characteristic
168	C₂×C₄	#10587635	1	4	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	68	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	136	2	no	—
174	C₂²	—	136	4	no	—
175	C₂²	#11762710	1	1	yes	characteristic
176	C₂²	#11763199	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	16	2	no	—
180	C₄	—	136	2	no	—
181	C₄	#11762813	1	2	yes	—
182	C₄	#11763175	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	272	2	no	—
185	C₂	#14529607	1	1	yes	characteristic
186	C₂	#14529680	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539212

C₂

yes

maximal, characteristic

#14527343

C₂

yes

maximal, complemented

#14527076

C₂

yes

maximal, characteristic, complemented

C₁₇×#1835

C₂

yes

maximal, characteristic, complemented

C₂×#10587590

—

#11763359

C₂²

yes

—

#11763193

—

#11763189

C₄

yes

characteristic, complemented

#11763186

C₄

yes

characteristic, complemented

#11763182

C₄

yes

complemented

#11762862

C₂²

yes

—

#11762542

—

#11762531

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#382

C₂²

yes

—

C₁₇×#340

—

C₂×#63252

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

#10587616

—

C₂×Dic₆₈

—

#10587590

—

C₂×D₁₃₆

—

C₂×#63219

—

#10587561

—

#10587560

C₂×C₄

yes

—

C₄×D₆₈

—

#10587555

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

C₁₇×#95

—

C₂²⋊#3622

D₄

yes

—

C₄⋊#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₇

—

#63252

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

#63227

—

C₄⋊Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

D₁₃₆

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

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

#124

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3147

—

#1835

D₁₇

yes

characteristic, complemented

#1517

—

#1452

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#496

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#418

yes

—

C₄×C₁₇

#490

yes

—

Dic₁₇

—

—

—

—

—

—

—

100

#431

—

101

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

102

#382

C₂×D₁₇

yes

—

103

#340

—

104

#327

—

105

#322

—

106

C₂×C₁₇

—

107

C₂×C₁₇

#1961

yes

characteristic

108

C₂×C₁₇

#1976

yes

—

109

D₁₇

—

110

C₂×#41

—

111

#135

—

112

C₂×Q₁₆

—

113

C₂×SD₁₆

—

114

C₂×D₈

—

115

C₂²×C₈

C₂²⋊D₁₇

yes

—

116

C₄⋊Q₈

—

117

#127

—

118

#124

—

119

#121

—

120

C₄×D₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₂²×D₁₇

yes

characteristic

123

C₄⋊C₈

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₄×D₁₇

yes

characteristic

127

C₄×C₈

C₄×D₁₇

yes

characteristic

128

C₄×C₈

D₆₈

yes

—

129

#95

—

130

C₁₇

#9760

yes

characteristic, complemented, Sylow

131

#41

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

D₄×D₁₇

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

140

Q₁₆

—

141

SD₁₆

—

142

D₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#63227

yes

—

146

C₂×C₈

#63227

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

C₂×C₄×D₁₇

yes

characteristic

151

C₄²

C₂×D₆₈

yes

characteristic

152

C₄²

#63256

yes

—

153

C₂³

—

154

C₂³

#10587626

yes

characteristic

155

Q₈

—

156

D₄

—

157

D₄

—

136

—

158

D₄

—

136

—

159

C₂×C₄

—

160

C₂×C₄

—

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

#10587633

yes

characteristic

168

C₂×C₄

#10587635

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₂²

—

173

C₂²

—

136

—

174

C₂²

—

136

—

175

C₂²

#11762710

yes

characteristic

176

C₂²

#11763199

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

136

—

181

C₄

#11762813

yes

—

182

C₄

#11763175

yes

—

183

C₂

—

184

C₂

—

272

—

185

C₂

#14529607

yes

characteristic

186

C₂

#14529680

yes

—

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1835	C₂
socle	C₂²×C₁₇	#390
prime radical	#14539212	C₂
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1835	D₁₇
hypocenter	C₁₇	#9760
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#11762710

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1835

C₂

C₂²×C₁₇

C₂

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
2. #14527343	C₂	semidirect
3. #14527076	C₂	semidirect
4. C₁₇×#1835	C₂	semidirect
8. #11763189	C₄	semidirect
9. #11763186	C₄	semidirect
10. #11763182	C₄	semidirect
80. #1835	D₁₇	semidirect
130. C₁₇	#9760	semidirect

H ≅ G/N

type

2. #14527343

C₂

semidirect

3. #14527076

C₂

semidirect

4. C₁₇×#1835

C₂

semidirect

8. #11763189

C₄

semidirect

9. #11763186

C₄

semidirect

10. #11763182

C₄

semidirect

80. #1835

D₁₇

semidirect

130. C₁₇

#9760

semidirect

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

subgroup	index
#14539212	2
#14527343	2
#14527076	2
C₁₇×#1835	2
#9760	17

subgroup

index

C₁₇×#1835

Subgroups — #16933719

Conjugacy classes of subgroups

Complements

Maximal subgroups