Group #17961047 — subgroups

Subgroups — #17961047

Order 4352.

308 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 112 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	#14545131	C₂	1	2	yes	maximal
2	#14544586	C₂	1	2	yes	maximal
3	#14543854	C₂	1	1	yes	maximal, characteristic
4	#14543852	C₂	1	1	yes	maximal, characteristic
5	#14530377	C₂	1	1	yes	maximal, characteristic
6	D₁₇⋊#108	C₂²	1	2	yes	—
7	#11763819	C₂²	1	2	yes	—
8	#11763772	—	2	2	no	—
9	#11763734	C₄	1	4	yes	complemented
10	#11763733	C₄	1	4	yes	complemented
11	#11763680	—	2	2	no	—
12	C₂×#10587469	C₂²	1	1	yes	characteristic
13	C₂×#10587469	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
15	#11762585	C₄	1	1	yes	characteristic, complemented
16	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic, complemented
17	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic, complemented
18	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
19	#10587677	—	2	2	no	—
20	D₁₇⋊M₁₆	—	2	2	no	—
21	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
23	C₄²×D₁₇	D₄	1	1	yes	characteristic
24	#10587536	—	2	2	no	—
25	#10587535	C₂×C₄	1	4	yes	—
26	#10587534	—	2	2	no	—
27	#10587533	C₂×C₄	1	4	yes	—
28	#10587532	—	4	2	no	—
29	C₈×#595	—	4	2	no	—
30	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
35	C₂×#63214	C₂×C₄	1	1	yes	characteristic
36	C₂×#63214	C₂×C₄	1	1	yes	characteristic
37	C₄×#3622	Q₈	1	1	yes	characteristic
38	#10587469	—	2	2	no	—
39	#10587469	C₂×C₄	1	2	yes	—
40	#10587469	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₇	—	2	1	no	—
42	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊#595	—	4	2	no	—
44	#63247	—	4	2	no	—
45	C₄⋊#595	—	2	4	no	—
46	C₄×#595	—	4	2	no	—
47	#63243	—	2	4	no	—
48	#63242	—	4	2	no	—
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
55	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
56	C₈×D₁₇	SD₁₆	1	2	yes	—
57	C₈×D₁₇	SD₁₆	1	2	yes	—
58	C₈×D₁₇	C₂²×C₄	1	2	yes	—
59	C₈×D₁₇	#41	1	2	yes	—
60	C₁₆×C₁₇	—	2	1	no	—
61	C₁₆×C₁₇	C₄²	1	2	yes	—
62	#63214	—	2	1	no	—
63	#63214	C₄²	1	1	yes	characteristic
64	#63214	C₄²	1	1	yes	characteristic
65	#12668	—	17	1	no	maximal, Sylow
66	C₂²×D₁₇	#135	1	1	yes	characteristic
67	C₂×#595	—	4	4	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×Dic₁₇	—	2	1	no	—
71	C₂×Dic₁₇	#135	1	1	yes	characteristic
72	C₄×D₁₇	—	4	2	no	—
73	C₄×D₁₇	C₄×D₄	1	2	yes	—
74	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
75	#3624	—	4	4	no	—
76	C₈×C₁₇	#106	1	2	yes	—
77	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
78	C₈×C₁₇	#117	1	1	yes	characteristic
79	#3622	#106	1	1	yes	characteristic
80	#3622	#106	1	1	yes	characteristic
81	#3622	C₄×Q₈	1	2	yes	—
82	#2155	—	17	2	no	—
83	C₄×M₃₂	—	17	1	no	—
84	#1772	—	17	2	no	—
85	#1385	—	17	1	no	—
86	#1384	—	17	1	no	—
87	C₂²×C₁₇	#427	1	1	yes	characteristic
88	C₂×D₁₇	—	4	1	no	—
89	C₂×D₁₇	C₄×SD₁₆	1	2	yes	—
90	#595	—	8	4	no	—
91	C₄×C₁₇	—	4	1	no	—
92	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
94	Dic₁₇	—	2	2	no	—
95	Dic₁₇	C₄×SD₁₆	1	2	yes	—
96	C₂×M₃₂	—	17	1	no	—
97	C₂×M₃₂	—	17	1	no	—
98	C₂×#108	—	17	2	no	—
99	#427	—	17	2	no	—
100	#405	—	34	2	no	—
101	C₂×C₄×C₈	—	17	1	no	—
102	#365	—	17	4	no	—
103	#365	—	17	4	no	—
104	#346	—	17	1	no	—
105	#346	#595	1	1	yes	characteristic, complemented
106	C₄×C₁₆	—	17	1	no	—
107	C₄×C₁₆	—	17	1	no	—
108	#339	—	34	2	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	C₄×#106	1	1	yes	characteristic
111	D₁₇	—	2	2	no	—
112	C₂×M₁₆	—	34	2	no	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	17	1	no	—
115	#117	—	34	2	no	—
116	C₂×C₄²	—	17	1	no	—
117	M₃₂	—	17	2	no	—
118	M₃₂	—	17	2	no	—
119	M₃₂	—	34	2	no	—
120	C₂×C₁₆	—	17	1	no	—
121	C₂×C₁₆	—	17	1	no	—
122	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
123	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
124	#108	—	17	4	no	—
125	#108	—	34	2	no	—
126	#107	—	34	2	no	—
127	#106	—	17	4	no	—
128	#97	—	68	2	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	17	1	no	—
131	C₄×C₈	—	17	1	no	—
132	C₄×C₈	—	68	2	no	—
133	C₄×C₈	C₂×#595	1	1	yes	characteristic
134	C₁₇	#12668	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	—	17	1	no	—
136	C₂²×C₄	—	34	1	no	—
137	M₁₆	—	34	4	no	—
138	M₁₆	—	68	2	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	2	no	—
144	C₂×C₈	—	17	2	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
147	C₂×C₈	C₂²×#595	1	1	yes	characteristic
148	C₄⋊C₄	—	34	4	no	—
149	C₂²⋊C₄	—	68	2	no	—
150	C₄²	—	17	1	no	—
151	C₄²	—	34	1	no	—
152	C₄²	—	68	2	no	—
153	C₄²	C₄⋊#595	1	1	yes	characteristic
154	C₁₆	—	2	1	no	—
155	C₁₆	—	17	1	no	—
156	C₁₆	—	17	1	no	—
157	C₁₆	—	34	1	no	—
158	C₁₆	C₄×#595	1	2	yes	—
159	C₂³	—	17	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	2	no	—
164	C₂×C₄	—	34	1	no	—
165	C₂×C₄	—	68	2	no	—
166	C₂×C₄	—	68	4	no	—
167	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
168	C₈	—	17	2	no	—
169	C₈	—	17	1	no	—
170	C₈	—	17	1	no	—
171	C₈	—	68	4	no	—
172	C₈	#10587533	1	2	yes	—
173	C₈	C₂×C₄×#595	1	1	yes	characteristic
174	C₈	#10587677	1	1	yes	characteristic
175	C₂²	—	17	2	no	—
176	C₂²	—	68	1	no	—
177	C₂²	#11763819	1	1	yes	characteristic
178	C₄	—	4	1	no	—
179	C₄	—	17	2	no	—
180	C₄	—	34	2	no	—
181	C₄	—	136	4	no	—
182	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
183	C₄	C₂×#10587533	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	34	2	no	—
186	C₂	C₄×#10587533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545131

C₂

yes

maximal

#14544586

C₂

yes

maximal

#14543854

C₂

yes

maximal, characteristic

#14543852

C₂

yes

maximal, characteristic

#14530377

C₂

yes

maximal, characteristic

D₁₇⋊#108

C₂²

yes

—

#11763819

C₂²

yes

—

#11763772

—

#11763734

C₄

yes

complemented

#11763733

C₄

yes

complemented

#11763680

—

C₂×#10587469

C₂²

yes

characteristic

C₂×#10587469

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic, complemented

C₁₆×Dic₁₇

C₄

yes

characteristic, complemented

C₁₆×Dic₁₇

C₄

yes

characteristic, complemented

C₁₇×#346

C₄

yes

characteristic, complemented

#10587677

—

D₁₇⋊M₁₆

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

#10587536

—

#10587535

C₂×C₄

yes

—

#10587534

—

#10587533

C₂×C₄

yes

—

#10587532

—

C₈×#595

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₄×#3622

Q₈

yes

characteristic

#10587469

—

#10587469

C₂×C₄

yes

—

#10587469

C₂×C₄

yes

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂²⋊#595

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

C₂²×C₄

yes

—

C₈×D₁₇

#41

yes

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₄²

yes

—

#63214

—

#63214

C₄²

yes

characteristic

#63214

C₄²

yes

—

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#135

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

—

C₄×D₁₇

C₂×SD₁₆

yes

—

#3624

—

C₈×C₁₇

#106

yes

—

C₈×C₁₇

C₂×C₄²

yes

C₈×C₁₇

yes

yes

yes

C₄×Q₈

yes

—

#2155

—

C₄×M₃₂

—

#1772

—

#1385

—

#1384

—

C₂²×C₁₇

#427

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₄×SD₁₆

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₂×#106

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄×SD₁₆

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#108

—

#427

—

100

#405

—

101

C₂×C₄×C₈

—

102

#365

—

103

#365

—

104

#346

—

105

#346

#595

yes

characteristic, complemented

106

C₄×C₁₆

—

107

C₄×C₁₆

—

108

#339

—

109

C₂×C₁₇

—

110

C₂×C₁₇

C₄×#106

yes

characteristic

111

D₁₇

—

112

C₂×M₁₆

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

#117

—

116

C₂×C₄²

—

117

M₃₂

—

118

M₃₂

—

119

M₃₂

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

C₂×#595

yes

characteristic

123

C₂×C₁₆

C₂×#595

yes

characteristic

124

#108

—

125

#108

—

126

#107

—

127

#106

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×#595

yes

characteristic

134

C₁₇

#12668

yes

characteristic, complemented, Sylow

135

C₂²×C₄

—

136

C₂²×C₄

—

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

C₄⋊#595

yes

characteristic

147

C₂×C₈

C₂²×#595

yes

characteristic

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

C₄⋊#595

yes

characteristic

154

C₁₆

—

155

C₁₆

—

156

C₁₆

—

157

C₁₆

—

158

C₁₆

C₄×#595

yes

—

159

C₂³

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

#10587533

yes

—

173

C₈

C₂×C₄×#595

yes

characteristic

174

C₈

#10587677

yes

characteristic

175

C₂²

—

176

C₂²

—

177

C₂²

#11763819

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

136

—

182

C₄

C₄×C₄⋊#595

yes

characteristic

183

C₄

C₂×#10587533

yes

characteristic

184

C₂

—

185

C₂

—

186

C₂

C₄×#10587533

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#10587533
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂×C₁₇	C₄×#106
prime radical	C₂²×D₁₇	#135
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12668
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

C₂×#10587533

commutator

C₈×C₁₇

C₂×C₄²

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

C₂×C₁₇

C₂²×D₁₇

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
9. #11763734	C₄	semidirect
10. #11763733	C₄	semidirect
15. #11762585	C₄	semidirect
16. C₁₆×Dic₁₇	C₄	semidirect
17. C₁₆×Dic₁₇	C₄	semidirect
18. C₁₇×#346	C₄	semidirect
105. #346	#595	semidirect
134. C₁₇	#12668	semidirect

H ≅ G/N

type

9. #11763734

C₄

semidirect

10. #11763733

C₄

semidirect

15. #11762585

C₄

semidirect

16. C₁₆×Dic₁₇

C₄

semidirect

17. C₁₆×Dic₁₇

C₄

semidirect

18. C₁₇×#346

C₄

semidirect

105. #346

#595

semidirect

134. C₁₇

#12668

semidirect

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

subgroup	index
#14545131	2
#14544586	2
#14543854	2
#14543852	2
#14530377	2
#12668	17

subgroup

index

Subgroups — #17961047

Conjugacy classes of subgroups

Complements

Maximal subgroups