Group #17948545 — subgroups

Subgroups — #17948545

Order 4352.

234 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 88 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	#14544985	C₂	1	1	yes	maximal, characteristic
2	#14544326	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587529	C₄	1	2	yes	—
5	#11763810	—	2	1	no	—
6	#11763698	—	2	1	no	—
7	#11763697	—	2	1	no	—
8	#11763671	—	2	1	no	—
9	#11763670	C₄	1	2	yes	—
10	C₄⋊C₈×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762560	C₄	1	1	yes	characteristic
13	#11762559	—	2	1	no	—
14	C₁₇×#334	C₄	1	1	yes	characteristic
15	C₂×#63242	—	4	1	no	—
16	C₂×C₈×D₁₇	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	#10587544	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₂×#63216	—	2	2	no	—
23	#10587529	—	2	2	no	—
24	#10587529	C₈	1	4	yes	complemented
25	C₁₇×C₄⋊C₈	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587485	—	4	1	no	—
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₄⋊#3622	—	2	1	no	—
31	C₄×#3622	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	#63242	—	8	2	no	—
37	C₂×C₈×C₁₇	—	4	1	no	—
38	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	D₈	1	1	yes	characteristic
43	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
44	C₂×#3622	—	4	1	no	—
45	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₁₇	—	2	2	no	—
48	C₈×D₁₇	—	8	2	no	—
49	C₈×D₁₇	C₂×C₈	1	2	yes	—
50	C₈×D₁₇	M₁₆	1	2	yes	—
51	#63216	—	2	4	no	—
52	#7156	—	17	1	no	maximal, Sylow
53	C₂²×D₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
57	C₂×Dic₁₇	#102	1	1	yes	characteristic
58	C₂×Dic₁₇	#103	1	1	yes	characteristic
59	C₂×Dic₁₇	#107	1	1	yes	characteristic
60	C₄×D₁₇	—	2	2	no	—
61	C₄×D₁₇	—	2	2	no	—
62	C₄×D₁₇	C₄⋊C₈	1	2	yes	—
63	C₄×D₁₇	#108	1	2	yes	—
64	#3624	—	8	2	no	—
65	C₈×C₁₇	—	2	1	no	—
66	C₈×C₁₇	—	8	1	no	—
67	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
68	C₈×C₁₇	#97	1	1	yes	characteristic
69	#3622	—	2	1	no	—
70	#3622	—	8	1	no	—
71	#3622	C₂²⋊C₈	1	2	yes	—
72	C₂×#346	—	17	1	no	—
73	C₂×#334	—	17	1	no	—
74	C₂×#334	—	17	1	no	—
75	C₂²×C₁₇	#340	1	1	yes	characteristic
76	C₂×D₁₇	#334	1	2	yes	—
77	C₂×D₁₇	#365	1	2	yes	—
78	C₂×D₁₇	#365	1	2	yes	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#336	1	1	yes	characteristic
82	C₄×C₁₇	#341	1	1	yes	characteristic
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	—	2	1	no	—
85	Dic₁₇	#335	1	2	yes	—
86	C₂²×C₁₆	—	17	2	no	—
87	C₂×C₄⋊C₈	—	34	1	no	—
88	C₂×C₄⋊C₈	—	34	1	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	#346	—	34	1	no	—
91	#346	—	34	1	no	—
92	#334	—	17	2	no	—
93	#334	—	17	1	no	—
94	#334	—	34	1	no	—
95	#334	—	34	1	no	—
96	#334	#595	1	1	yes	characteristic
97	C₂×C₁₇	#1277	1	1	yes	characteristic
98	C₂×C₁₇	#1383	1	1	yes	characteristic
99	C₂×C₁₇	#1383	1	1	yes	characteristic
100	D₁₇	#1368	1	4	yes	—
101	C₂²×C₈	—	17	1	no	—
102	C₂²×C₈	—	17	1	no	—
103	C₂²×C₈	—	68	1	no	—
104	C₂²×C₈	—	68	1	no	—
105	C₂×C₄²	—	17	1	no	—
106	C₂×C₁₆	—	17	4	no	—
107	C₂×C₁₆	—	34	2	no	—
108	C₂×C₁₆	—	34	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	34	2	no	—
111	C₄⋊C₈	—	34	1	no	—
112	C₄⋊C₈	—	68	1	no	—
113	C₄⋊C₈	—	68	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	C₂×#595	1	1	yes	characteristic
118	C₁₇	#7156	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	17	1	no	—
120	C₂²×C₄	—	17	1	no	—
121	C₂²×C₄	—	17	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	—	17	2	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	34	2	no	—
128	C₂×C₈	—	68	2	no	—
129	C₂×C₈	—	68	1	no	—
130	C₂×C₈	—	136	2	no	—
131	C₂×C₈	—	136	2	no	—
132	C₂×C₈	C₄×#595	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
134	C₄²	—	17	1	no	—
135	C₄²	—	17	1	no	—
136	C₄²	—	17	1	no	—
137	C₄²	C₂²⋊#595	1	1	yes	characteristic
138	C₁₆	—	34	4	no	—
139	C₂³	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	17	2	no	—
142	C₂×C₄	—	17	2	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	34	2	no	—
146	C₂×C₄	—	34	2	no	—
147	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#10587547	1	1	yes	characteristic
149	C₂×C₄	#10587549	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	8	1	no	—
152	C₈	—	17	2	no	—
153	C₈	—	34	1	no	—
154	C₈	—	136	1	no	—
155	C₈	—	136	2	no	—
156	C₈	C₈×#595	1	1	yes	characteristic
157	C₈	#10587532	1	1	yes	characteristic
158	C₂²	—	17	2	no	—
159	C₂²	—	17	2	no	—
160	C₂²	—	17	2	no	—
161	C₂²	#11763682	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	17	2	no	—
165	C₄	—	34	1	no	—
166	C₄	—	34	1	no	—
167	C₄	#11763674	1	1	yes	characteristic
168	C₄	#11763684	1	1	yes	characteristic
169	C₂	—	17	4	no	—
170	C₂	#14543613	1	1	yes	characteristic
171	C₂	#14543848	1	1	yes	characteristic
172	C₂	#14543850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544985

C₂

yes

maximal, characteristic

#14544326

C₂

yes

maximal, characteristic

D₁₇×#334

C₂

yes

maximal, characteristic

C₂×#10587529

C₄

yes

—

—

—

—

—

C₄

yes

—

C₄⋊C₈×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762560

C₄

yes

characteristic

#11762559

—

C₁₇×#334

C₄

yes

characteristic

C₂×#63242

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

C₈

yes

complemented

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×#3624

—

#63242

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

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

—

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

C₄⋊C₈

yes

—

C₄×D₁₇

#108

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

—

—

C₂²⋊C₈

yes

—

C₂×#346

—

C₂×#334

—

C₂×#334

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

#334

yes

—

C₂×D₁₇

#365

yes

—

C₂×D₁₇

#365

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#335

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

—

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

100

D₁₇

#1368

yes

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₂×#595

yes

characteristic

118

C₁₇

#7156

yes

characteristic, complemented, Sylow

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

136

—

131

C₂×C₈

—

136

—

132

C₂×C₈

C₄×#595

yes

characteristic

133

C₂×C₈

C₄⋊#595

yes

characteristic

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²⋊#595

yes

characteristic

138

C₁₆

—

139

C₂³

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

148

C₂×C₄

#10587547

yes

characteristic

149

C₂×C₄

#10587549

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

136

—

155

C₈

—

136

—

156

C₈

C₈×#595

yes

characteristic

157

C₈

#10587532

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#11763682

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#11763674

yes

characteristic

168

C₄

#11763684

yes

characteristic

169

C₂

—

170

C₂

#14543613

yes

characteristic

171

C₂

#14543848

yes

characteristic

172

C₂

#14543850

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	#106
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#334	#595
hypocenter	C₁₇	#7156
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
24. #10587529	C₈	semidirect
118. C₁₇	#7156	semidirect

H ≅ G/N

type

24. #10587529

C₈

semidirect

118. C₁₇

#7156

semidirect

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

subgroup	index
#14544985	2
#14544326	2
D₁₇×#334	2
#7156	17

subgroup

index

#14544985

#14544326

D₁₇×#334

#7156

Subgroups — #17948545

Conjugacy classes of subgroups

Complements

Maximal subgroups