Group #17948601 — subgroups

Subgroups — #17948601

Order 4352.

246 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₄×#10587536	C₂	1	1	yes	maximal, characteristic
2	#14544334	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
4	#11763828	—	2	1	no	—
5	C₂×#10587536	C₄	1	2	yes	—
6	C₄×#63243	—	2	1	no	—
7	#11763685	—	2	1	no	—
8	#11763672	C₄	1	2	yes	—
9	#11763671	—	2	1	no	—
10	C₂×C₁₆×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	C₄×#63214	C₄	1	1	yes	characteristic
13	C₁₆×Dic₁₇	—	2	1	no	—
14	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂×#63243	—	2	2	no	—
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₄²×D₁₇	D₄	1	1	yes	characteristic
19	#10587543	—	4	1	no	—
20	#10587542	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₄×#3624	—	4	1	no	—
23	C₄×#3624	—	4	1	no	—
24	#10587536	—	2	2	no	—
25	#10587536	C₈	1	4	yes	complemented
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₁₆×D₁₇	—	2	4	no	—
33	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
36	C₂×#3624	—	4	2	no	—
37	C₂×#3624	—	4	2	no	—
38	#63243	—	2	4	no	—
39	#63243	—	4	2	no	—
40	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	D₈	1	1	yes	characteristic
45	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₁₇	—	2	2	no	—
49	C₈×D₁₇	C₂×C₈	1	2	yes	—
50	C₈×D₁₇	M₁₆	1	2	yes	—
51	C₁₆×C₁₇	—	2	2	no	—
52	#63214	—	2	2	no	—
53	#7172	—	17	1	no	maximal, Sylow
54	C₂²×D₁₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
58	C₂×Dic₁₇	#102	1	1	yes	characteristic
59	C₂×Dic₁₇	#102	1	1	yes	characteristic
60	C₂×Dic₁₇	#103	1	1	yes	characteristic
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	2	no	—
64	C₄×D₁₇	C₂²⋊C₈	1	2	yes	—
65	C₄×D₁₇	#104	1	2	yes	—
66	#3624	—	4	4	no	—
67	#3624	—	8	2	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
70	C₈×C₁₇	#97	1	1	yes	characteristic
71	#3622	—	2	1	no	—
72	#3622	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₁₆	—	17	1	no	—
74	C₄×#108	—	17	1	no	—
75	#1564	—	17	1	no	—
76	C₂²×C₁₇	#340	1	1	yes	characteristic
77	C₂×D₁₇	—	2	1	no	—
78	C₂×D₁₇	—	2	1	no	—
79	C₂×D₁₇	Q₈⋊C₈	1	2	yes	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#336	1	1	yes	characteristic
82	C₄×C₁₇	#339	1	1	yes	characteristic
83	C₄×C₁₇	#368	1	1	yes	characteristic
84	C₄×C₁₇	#368	1	1	yes	characteristic
85	Dic₁₇	—	2	1	no	—
86	Dic₁₇	D₄⋊C₈	1	2	yes	—
87	Dic₁₇	#359	1	2	yes	—
88	C₂²×C₁₆	—	34	1	no	—
89	#431	—	34	1	no	—
90	C₂×#108	—	17	2	no	—
91	C₄×M₁₆	—	34	1	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₄×C₁₆	—	17	1	no	—
94	C₄×C₁₆	—	34	1	no	—
95	C₄×C₁₆	#595	1	1	yes	characteristic
96	#341	—	34	1	no	—
97	#335	—	17	2	no	—
98	#334	—	34	1	no	—
99	C₂×C₁₇	#1279	1	1	yes	characteristic
100	C₂×C₁₇	#1381	1	1	yes	characteristic
101	C₂×C₁₇	#1382	1	1	yes	characteristic
102	D₁₇	—	2	2	no	—
103	C₂×M₁₆	—	34	2	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂×C₄²	—	17	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	34	4	no	—
109	C₂×C₁₆	—	34	1	no	—
110	#108	—	17	4	no	—
111	#108	—	34	2	no	—
112	C₄⋊C₈	—	34	2	no	—
113	C₂²⋊C₈	—	68	1	no	—
114	#97	—	68	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	68	1	no	—
119	C₄×C₈	—	68	1	no	—
120	C₄×C₈	C₂×#595	1	1	yes	characteristic
121	C₁₇	#7172	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	—	17	1	no	—
123	C₂²×C₄	—	17	1	no	—
124	C₂²×C₄	—	17	1	no	—
125	M₁₆	—	34	4	no	—
126	M₁₆	—	68	2	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	17	2	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	34	2	no	—
132	C₂×C₈	—	68	2	no	—
133	C₂×C₈	—	68	2	no	—
134	C₂×C₈	C₄×#595	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
136	C₄²	—	17	1	no	—
137	C₄²	—	17	1	no	—
138	C₄²	—	17	1	no	—
139	C₄²	C₄⋊#595	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	34	2	no	—
142	C₂³	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	2	no	—
146	C₂×C₄	—	17	2	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	34	2	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	34	1	no	—
151	C₂×C₄	#10587533	1	1	yes	characteristic
152	C₂×C₄	#10587534	1	1	yes	characteristic
153	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	17	2	no	—
156	C₈	—	34	1	no	—
157	C₈	—	68	4	no	—
158	C₈	—	136	2	no	—
159	C₈	C₈×#595	1	1	yes	characteristic
160	C₈	#10587532	1	1	yes	characteristic
161	C₂²	—	17	2	no	—
162	C₂²	—	34	1	no	—
163	C₂²	—	34	1	no	—
164	C₂²	#11763683	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	17	2	no	—
167	C₄	—	17	2	no	—
168	C₄	—	34	1	no	—
169	C₄	#11763674	1	1	yes	characteristic
170	C₄	#11763680	1	1	yes	characteristic
171	C₄	#11763737	1	1	yes	characteristic
172	C₄	#11763738	1	1	yes	characteristic
173	C₂	—	34	2	no	—
174	C₂	#14543619	1	1	yes	characteristic
175	C₂	#14543844	1	1	yes	characteristic
176	C₂	#14543846	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587536

C₂

yes

maximal, characteristic

#14544334

C₂

yes

maximal, characteristic

C₄×C₁₆×D₁₇

C₂

yes

maximal, characteristic

#11763828

—

C₂×#10587536

C₄

yes

—

C₄×#63243

—

#11763685

—

#11763672

C₄

yes

—

#11763671

—

C₂×C₁₆×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×#63214

C₄

yes

characteristic

C₁₆×Dic₁₇

—

C₄×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

—

C₈

yes

complemented

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₂×#63214

—

C₄×#3622

Q₈

yes

characteristic

C₁₆×D₁₇

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

D₈

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

C₈×D₁₇

M₁₆

yes

—

C₁₆×C₁₇

—

—

—

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×D₁₇

C₂²⋊C₈

yes

—

C₄×D₁₇

#104

yes

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

—

C₄⋊C₈

yes

—

C₂×C₄×C₁₆

—

C₄×#108

—

#1564

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

Q₈⋊C₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

D₄⋊C₈

yes

—

Dic₁₇

#359

yes

—

C₂²×C₁₆

—

#431

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

—

C₂×C₁₇

#1279

yes

characteristic

100

C₂×C₁₇

#1381

yes

characteristic

101

C₂×C₁₇

#1382

yes

characteristic

102

D₁₇

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

#108

—

111

#108

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂×#595

yes

characteristic

121

C₁₇

#7172

yes

characteristic, complemented, Sylow

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

M₁₆

—

126

M₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₄×#595

yes

characteristic

135

C₂×C₈

C₂²⋊#595

yes

characteristic

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₄⋊#595

yes

characteristic

140

C₁₆

—

141

C₁₆

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#10587533

yes

characteristic

152

C₂×C₄

#10587534

yes

characteristic

153

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

136

—

159

C₈

C₈×#595

yes

characteristic

160

C₈

#10587532

yes

characteristic

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

#11763683

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#11763674

yes

characteristic

170

C₄

#11763680

yes

characteristic

171

C₄

#11763737

yes

characteristic

172

C₄

#11763738

yes

characteristic

173

C₂

—

174

C₂

#14543619

yes

characteristic

175

C₂

#14543844

yes

characteristic

176

C₂

#14543846

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587534
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#103
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#7172
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₄

#10587534

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₄×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₄×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
25. #10587536	C₈	semidirect
121. C₁₇	#7172	semidirect

H ≅ G/N

type

25. #10587536

C₈

semidirect

121. C₁₇

#7172

semidirect

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

subgroup	index
C₄×#10587536	2
#14544334	2
C₄×C₁₆×D₁₇	2
#7172	17

subgroup

index

C₄×#10587536

#14544334

C₄×C₁₆×D₁₇

#7172

Subgroups — #17948601

Conjugacy classes of subgroups

Complements

Maximal subgroups