Group #17948561 — subgroups

Subgroups — #17948561

Order 4352.

200 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 76 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	#14545123	C₂	1	1	yes	maximal, characteristic
2	#14544991	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587530	C₄	1	1	yes	characteristic
5	C₂×#10587530	C₄	1	1	yes	characteristic
6	#11763865	—	2	1	no	—
7	#11763732	—	2	1	no	—
8	#11763731	C₄	1	2	yes	—
9	#11763697	—	2	1	no	—
10	#11763695	—	2	1	no	—
11	M₃₂×D₁₇	—	2	1	no	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762630	C₄	1	1	yes	characteristic
14	#11762629	—	2	1	no	—
15	C₁₇×#364	C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₄²×D₁₇	Q₈	1	1	yes	characteristic
19	#10587538	—	2	2	no	—
20	C₂×#63216	—	2	1	no	—
21	C₂×#63216	—	2	1	no	—
22	#10587530	—	2	1	no	—
23	#10587530	—	2	1	no	—
24	#10587530	—	4	1	no	—
25	#10587530	C₈	1	2	yes	—
26	#10587530	C₈	1	2	yes	—
27	#10587529	—	4	1	no	—
28	C₁₇×M₃₂	—	2	1	no	—
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₇	D₄	1	1	yes	characteristic
32	#10587483	—	2	1	no	—
33	C₄×#3622	D₄	1	1	yes	characteristic
34	#10587469	—	4	1	no	—
35	C₁₆×D₁₇	—	4	1	no	—
36	C₂×C₄×D₁₇	—	2	1	no	—
37	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	—	2	1	no	—
42	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
46	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
47	C₈×D₁₇	M₁₆	1	1	yes	characteristic
48	C₈×D₁₇	M₁₆	1	1	yes	characteristic
49	C₈×D₁₇	D₈	1	1	yes	characteristic
50	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
51	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
52	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
53	#63216	—	2	2	no	—
54	#63216	—	2	2	no	—
55	#63216	—	4	2	no	—
56	C₁₆×C₁₇	—	4	1	no	—
57	#63214	—	4	1	no	—
58	#7162	—	17	1	no	maximal, Sylow
59	C₂²×D₁₇	#108	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
62	C₂×Dic₁₇	—	2	1	no	—
63	C₂×Dic₁₇	#108	1	1	yes	characteristic
64	C₄×D₁₇	—	4	2	no	—
65	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
67	C₄×D₁₇	#106	1	1	yes	characteristic
68	C₄×D₁₇	#107	1	1	yes	characteristic
69	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
70	C₈×C₁₇	#97	1	1	yes	characteristic
71	C₈×C₁₇	#102	1	1	yes	characteristic
72	C₈×C₁₇	#103	1	1	yes	characteristic
73	#3622	C₂²⋊C₈	1	1	yes	characteristic
74	#3622	C₂²⋊C₈	1	1	yes	characteristic
75	#3622	#102	1	1	yes	characteristic
76	#3622	#103	1	1	yes	characteristic
77	C₂×#364	—	17	1	no	—
78	C₂×#364	—	17	1	no	—
79	C₄×M₃₂	—	17	1	no	—
80	C₂²×C₁₇	#341	1	1	yes	characteristic
81	C₂×D₁₇	—	4	1	no	—
82	C₂×D₁₇	#334	1	1	yes	characteristic
83	C₂×D₁₇	#335	1	1	yes	characteristic
84	C₄×C₁₇	—	4	1	no	—
85	C₄×C₁₇	#336	1	1	yes	characteristic
86	C₄×C₁₇	#340	1	1	yes	characteristic
87	Dic₁₇	—	2	2	no	—
88	Dic₁₇	#334	1	1	yes	characteristic
89	Dic₁₇	#335	1	1	yes	characteristic
90	C₂×M₃₂	—	17	1	no	—
91	C₂×M₃₂	—	17	1	no	—
92	C₂×M₃₂	—	34	1	no	—
93	C₂×M₃₂	—	34	1	no	—
94	C₂×C₄×C₈	—	17	1	no	—
95	#364	—	17	2	no	—
96	#364	—	17	1	no	—
97	#364	—	34	1	no	—
98	#364	—	34	1	no	—
99	#364	#595	1	1	yes	characteristic
100	#346	—	34	1	no	—
101	C₄×C₁₆	—	34	1	no	—
102	C₂×C₁₇	—	2	1	no	—
103	C₂×C₁₇	#1277	1	1	yes	characteristic
104	D₁₇	—	2	1	no	—
105	D₁₇	—	2	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂×C₄²	—	17	1	no	—
109	M₃₂	—	2	1	no	—
110	M₃₂	—	17	2	no	—
111	M₃₂	—	17	2	no	—
112	M₃₂	—	34	1	no	—
113	M₃₂	—	34	1	no	—
114	M₃₂	—	34	2	no	—
115	M₃₂	—	34	1	no	—
116	M₃₂	—	68	1	no	—
117	M₃₂	—	68	1	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	—	34	1	no	—
120	C₂×C₁₆	—	68	1	no	—
121	C₂×C₁₆	—	68	1	no	—
122	C₄×C₈	—	17	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	C₂×#595	1	1	yes	characteristic
126	C₁₇	#7162	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₄	—	17	1	no	—
128	C₂²×C₄	—	34	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	1	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	C₄×#595	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
141	C₄²	—	17	1	no	—
142	C₄²	—	34	1	no	—
143	C₄²	C₄⋊#595	1	1	yes	characteristic
144	C₁₆	—	4	1	no	—
145	C₁₆	—	34	2	no	—
146	C₁₆	—	34	2	no	—
147	C₁₆	—	68	2	no	—
148	C₁₆	—	68	1	no	—
149	C₂³	—	17	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	1	no	—
154	C₂×C₄	—	17	1	no	—
155	C₂×C₄	—	17	1	no	—
156	C₂×C₄	—	34	1	no	—
157	C₂×C₄	—	68	2	no	—
158	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
159	C₈	—	17	1	no	—
160	C₈	—	17	1	no	—
161	C₈	—	17	1	no	—
162	C₈	—	17	1	no	—
163	C₈	C₈×#595	1	1	yes	characteristic
164	C₈	#10587532	1	1	yes	characteristic
165	C₈	#10587547	1	1	yes	characteristic
166	C₈	#10587549	1	1	yes	characteristic
167	C₂²	—	17	1	no	—
168	C₂²	—	17	1	no	—
169	C₂²	—	68	1	no	—
170	C₂²	#11763684	1	1	yes	characteristic
171	C₄	—	4	1	no	—
172	C₄	—	17	1	no	—
173	C₄	—	17	1	no	—
174	C₄	—	34	2	no	—
175	C₄	#11763674	1	1	yes	characteristic
176	C₄	#11763682	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	34	1	no	—
179	C₂	—	34	1	no	—
180	C₂	#14543613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545123

C₂

yes

maximal, characteristic

#14544991

C₂

yes

maximal, characteristic

D₁₇×#364

C₂

yes

maximal, characteristic

C₂×#10587530

C₄

yes

characteristic

C₂×#10587530

C₄

yes

—

—

C₄

yes

—

#11763697

—

#11763695

—

M₃₂×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762630

C₄

yes

characteristic

#11762629

—

C₁₇×#364

C₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

Q₈

yes

—

—

—

—

—

—

C₈

yes

—

#10587530

C₈

yes

—

#10587529

—

C₁₇×M₃₂

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

#10587483

—

C₄×#3622

D₄

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄²

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

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

—

—

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#108

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

C₄⋊C₈

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

C₂²⋊C₈

yes

characteristic

#3622

C₂²⋊C₈

yes

yes

yes

—

—

C₄×M₃₂

—

C₂²×C₁₇

#341

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

yes

Dic₁₇

yes

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

—

—

—

—

yes

100

—

101

C₄×C₁₆

—

102

C₂×C₁₇

—

103

C₂×C₁₇

#1277

yes

characteristic

104

D₁₇

—

105

D₁₇

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

M₃₂

—

110

M₃₂

—

111

M₃₂

—

112

M₃₂

—

113

M₃₂

—

114

M₃₂

—

115

M₃₂

—

116

M₃₂

—

117

M₃₂

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

C₂×#595

yes

characteristic

126

C₁₇

#7162

yes

characteristic, complemented, Sylow

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

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₄×#595

yes

characteristic

140

C₂×C₈

C₂²⋊#595

yes

characteristic

141

C₄²

—

142

C₄²

—

143

C₄²

C₄⋊#595

yes

characteristic

144

C₁₆

—

145

C₁₆

—

146

C₁₆

—

147

C₁₆

—

148

C₁₆

—

149

C₂³

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

C₈×#595

yes

characteristic

164

C₈

#10587532

yes

characteristic

165

C₈

#10587547

yes

characteristic

166

C₈

#10587549

yes

characteristic

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11763684

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#11763674

yes

characteristic

176

C₄

#11763682

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

—

180

C₂

#14543613

yes

characteristic

	subgroup	G/H
center	C₈	#10587547
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#364	C₄
socle	C₂×C₁₇	#1277
prime radical	C₂²×D₁₇	#108
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#364	#595
hypocenter	C₁₇	#7162
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₈

#10587547

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#364

C₄

C₂×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
126. C₁₇	#7162	semidirect

H ≅ G/N

type

126. C₁₇

#7162

semidirect

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

subgroup	index
#14545123	2
#14544991	2
D₁₇×#364	2
#7162	17

subgroup

index

#14545123

#14544991

D₁₇×#364

#7162

Subgroups — #17948561

Conjugacy classes of subgroups

Complements

Maximal subgroups