Group #17948606 — subgroups

Subgroups — #17948606

Order 4352.

262 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 82 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	#14544991	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587533	C₂	1	1	yes	maximal, characteristic
3	#14527529	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587530	—	2	1	no	—
5	C₂×#10587533	C₄	1	2	yes	—
6	C₄×C₄⋊#595	—	2	1	no	—
7	#11763697	—	2	1	no	—
8	#11763695	C₄	1	2	yes	complemented
9	#11763683	—	2	1	no	—
10	#11762861	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762561	—	2	1	no	—
13	#11762560	C₄	1	1	yes	characteristic, complemented
14	C₁₇×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#595	—	2	2	no	—
16	C₂×C₄×#595	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	(C₂×#595)⋊C₄	—	4	1	no	—
21	C₂×#63216	—	2	2	no	—
22	#10587533	—	2	2	no	—
23	#10587533	C₈	1	4	yes	complemented
24	#10587530	—	2	4	no	—
25	C₁₇×C₄⋊C₈	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587491	—	4	1	no	—
28	C₈×Dic₁₇	—	4	1	no	—
29	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	C₄⋊#3622	—	2	1	no	—
32	C₄×#3622	D₄	1	1	yes	characteristic
33	C₂²×#595	—	4	2	no	—
34	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
36	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
37	C₄⋊#595	—	2	4	no	—
38	C₄⋊#595	—	4	2	no	—
39	C₄×#595	—	8	2	no	—
40	C₂×C₈×C₁₇	—	4	1	no	—
41	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
46	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
47	C₂×#3622	—	4	1	no	—
48	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₈×D₁₇	—	2	2	no	—
51	C₈×D₁₇	C₂×C₈	1	2	yes	—
52	C₈×D₁₇	M₁₆	1	2	yes	—
53	#63216	—	2	4	no	—
54	#7174	—	17	1	no	maximal, Sylow
55	C₂²×D₁₇	#102	1	1	yes	characteristic
56	C₂×#595	—	4	4	no	—
57	C₂×#595	—	8	2	no	—
58	C₂×#595	—	8	2	no	—
59	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
62	C₂×Dic₁₇	#103	1	1	yes	characteristic
63	C₂×Dic₁₇	#106	1	1	yes	characteristic
64	C₂×Dic₁₇	#107	1	1	yes	characteristic
65	C₄×D₁₇	—	2	2	no	—
66	C₄×D₁₇	—	4	1	no	—
67	C₄×D₁₇	C₂²⋊C₈	1	2	yes	—
68	C₄×D₁₇	#104	1	2	yes	—
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	8	1	no	—
71	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
72	C₈×C₁₇	#97	1	1	yes	characteristic
73	#3622	—	2	1	no	—
74	#3622	—	8	1	no	—
75	#3622	C₄⋊C₈	1	2	yes	—
76	C₄×M₃₂	—	17	1	no	—
77	C₄×#106	—	17	1	no	—
78	#1564	—	17	1	no	—
79	C₂²×C₁₇	#340	1	1	yes	characteristic
80	C₂×D₁₇	—	2	1	no	—
81	C₂×D₁₇	—	2	1	no	—
82	C₂×D₁₇	D₄⋊C₈	1	2	yes	—
83	#595	—	8	4	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	#336	1	1	yes	characteristic
87	C₄×C₁₇	#339	1	1	yes	characteristic
88	Dic₁₇	—	2	1	no	—
89	Dic₁₇	Q₈⋊C₈	1	2	yes	—
90	Dic₁₇	#365	1	2	yes	—
91	C₂×M₃₂	—	34	1	no	—
92	#431	—	34	1	no	—
93	C₂×#106	—	17	2	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	C₄×C₄⋊C₄	—	34	1	no	—
96	#346	—	34	1	no	—
97	C₄×C₁₆	—	17	2	no	—
98	#340	—	34	1	no	—
99	#335	—	34	1	no	—
100	#334	—	17	1	no	—
101	#334	#595	1	1	yes	characteristic, complemented
102	C₂×C₁₇	#1279	1	1	yes	characteristic
103	C₂×C₁₇	#1383	1	1	yes	characteristic
104	C₂×C₁₇	#1384	1	1	yes	characteristic
105	D₁₇	—	2	2	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂×C₄⋊C₄	—	34	2	no	—
109	C₂×C₄²	—	17	1	no	—
110	C₂×C₄²	—	68	1	no	—
111	M₃₂	—	34	4	no	—
112	C₂×C₁₆	—	34	2	no	—
113	#106	—	17	4	no	—
114	#106	—	34	2	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	34	1	no	—
117	C₂²⋊C₈	—	68	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	68	1	no	—
122	C₄×C₈	C₂×#595	1	1	yes	characteristic
123	#95	—	68	1	no	—
124	C₁₇	#7174	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	—	17	1	no	—
127	C₂²×C₄	—	17	1	no	—
128	C₂²×C₄	—	68	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	2	no	—
132	C₂×C₈	—	17	2	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	34	2	no	—
135	C₂×C₈	—	68	1	no	—
136	C₂×C₈	C₄×#595	1	1	yes	characteristic
137	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
138	C₄⋊C₄	—	34	4	no	—
139	C₄⋊C₄	—	68	2	no	—
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₄²	—	17	1	no	—
143	C₄²	—	136	2	no	—
144	C₄²	C₂²⋊#595	1	1	yes	characteristic
145	C₁₆	—	34	4	no	—
146	C₂³	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	2	no	—
150	C₂×C₄	—	17	2	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	34	2	no	—
153	C₂×C₄	—	68	1	no	—
154	C₂×C₄	—	68	4	no	—
155	C₂×C₄	—	136	2	no	—
156	C₂×C₄	—	136	2	no	—
157	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
158	C₂×C₄	#10587547	1	1	yes	characteristic
159	C₂×C₄	#10587549	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	8	1	no	—
162	C₈	—	17	2	no	—
163	C₈	—	34	1	no	—
164	C₈	—	136	1	no	—
165	C₈	C₈×#595	1	1	yes	characteristic
166	C₈	#10587532	1	1	yes	characteristic
167	C₂²	—	17	2	no	—
168	C₂²	—	34	1	no	—
169	C₂²	—	34	1	no	—
170	C₂²	#11763682	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	17	2	no	—
174	C₄	—	17	2	no	—
175	C₄	—	34	1	no	—
176	C₄	—	136	4	no	—
177	C₄	#11763674	1	1	yes	characteristic
178	C₄	#11763681	1	1	yes	characteristic
179	C₂	—	34	2	no	—
180	C₂	#14543618	1	1	yes	characteristic
181	C₂	#14543850	1	1	yes	characteristic
182	C₂	#14543851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544991

C₂

yes

maximal, characteristic

C₄×#10587533

C₂

yes

maximal, characteristic

#14527529

C₂

yes

maximal, characteristic

C₂×#10587530

—

C₂×#10587533

C₄

yes

—

C₄×C₄⋊#595

—

#11763697

—

#11763695

C₄

yes

complemented

#11763683

—

#11762861

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762561

—

#11762560

C₄

yes

characteristic, complemented

C₁₇×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄×#595

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

(C₂×#595)⋊C₄

—

C₂×#63216

—

#10587533

—

#10587533

C₈

yes

complemented

#10587530

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

Q₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

C₈×D₁₇

M₁₆

yes

—

—

—

C₂²×D₁₇

yes

—

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂²⋊C₈

yes

—

C₄×D₁₇

#104

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

—

—

C₄⋊C₈

yes

—

C₄×M₃₂

—

C₄×#106

—

#1564

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

D₄⋊C₈

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

Q₈⋊C₈

yes

—

Dic₁₇

#365

yes

—

C₂×M₃₂

—

#431

—

C₂×#106

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#346

—

C₄×C₁₆

—

#340

—

#335

—

100

#334

—

101

#334

#595

yes

characteristic, complemented

102

C₂×C₁₇

#1279

yes

characteristic

103

C₂×C₁₇

#1383

yes

characteristic

104

C₂×C₁₇

#1384

yes

characteristic

105

D₁₇

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

M₃₂

—

112

C₂×C₁₆

—

113

#106

—

114

#106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₂²⋊C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#595

yes

characteristic

123

#95

—

124

C₁₇

#7174

yes

characteristic, complemented, Sylow

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₄×#595

yes

characteristic

137

C₂×C₈

C₄⋊#595

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

136

—

144

C₄²

C₂²⋊#595

yes

characteristic

145

C₁₆

—

146

C₂³

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

136

—

156

C₂×C₄

—

136

—

157

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

158

C₂×C₄

#10587547

yes

characteristic

159

C₂×C₄

#10587549

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

136

—

165

C₈

C₈×#595

yes

characteristic

166

C₈

#10587532

yes

characteristic

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11763682

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

136

—

177

C₄

#11763674

yes

characteristic

178

C₄

#11763681

yes

characteristic

179

C₂

—

180

C₂

#14543618

yes

characteristic

181

C₂

#14543850

yes

characteristic

182

C₂

#14543851

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#334	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#102
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#334	#595
hypocenter	C₁₇	#7174
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#11763682

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#334

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
8. #11763695	C₄	semidirect
13. #11762560	C₄	semidirect
14. C₁₇×#334	C₄	semidirect
23. #10587533	C₈	semidirect
101. #334	#595	semidirect
124. C₁₇	#7174	semidirect

H ≅ G/N

type

8. #11763695

C₄

semidirect

13. #11762560

C₄

semidirect

14. C₁₇×#334

C₄

semidirect

23. #10587533

C₈

semidirect

101. #334

#595

semidirect

124. C₁₇

#7174

semidirect

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

subgroup	index
#14544991	2
C₄×#10587533	2
#14527529	2
#7174	17

subgroup

index

Subgroups — #17948606

Conjugacy classes of subgroups

Complements

Maximal subgroups