Group #17948551 — subgroups

Subgroups — #17948551

Order 4352.

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

normal subgroups only

This group has 222 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	#14544334	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587530	C₄	1	2	yes	—
5	#11763828	—	2	1	no	—
6	#11763697	—	2	1	no	—
7	#11763695	—	2	1	no	—
8	#11763672	C₄	1	2	yes	—
9	#11763671	—	2	1	no	—
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×C₈×D₁₇	—	4	1	no	—
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
18	C₄²×D₁₇	D₄	1	1	yes	characteristic
19	#10587543	—	4	1	no	—
20	C₄⋊#3624	—	2	2	no	—
21	C₄×#3624	—	4	1	no	—
22	C₂×#63216	—	2	2	no	—
23	#10587530	—	2	2	no	—
24	#10587530	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	D₈	1	1	yes	characteristic
34	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	C₂×C₈×C₁₇	—	4	1	no	—
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
43	C₂×#3622	—	4	1	no	—
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₁₇	—	2	2	no	—
47	C₈×D₁₇	—	8	2	no	—
48	C₈×D₁₇	C₂×C₈	1	2	yes	—
49	C₈×D₁₇	M₁₆	1	2	yes	—
50	#63216	—	2	4	no	—
51	#7158	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
56	C₂×Dic₁₇	#102	1	1	yes	characteristic
57	C₂×Dic₁₇	#103	1	1	yes	characteristic
58	C₂×Dic₁₇	#106	1	1	yes	characteristic
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	—	4	1	no	—
61	C₄×D₁₇	C₄⋊C₈	1	2	yes	—
62	C₄×D₁₇	#108	1	2	yes	—
63	#3624	—	8	2	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	8	1	no	—
66	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
67	C₈×C₁₇	#97	1	1	yes	characteristic
68	#3622	—	2	1	no	—
69	#3622	—	8	1	no	—
70	#3622	C₂²⋊C₈	1	2	yes	—
71	C₄×M₃₂	—	17	1	no	—
72	#1564	—	17	1	no	—
73	C₂×#334	—	17	1	no	—
74	C₂²×C₁₇	#340	1	1	yes	characteristic
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	—	2	1	no	—
77	C₂×D₁₇	#335	1	2	yes	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	#336	1	1	yes	characteristic
81	C₄×C₁₇	#341	1	1	yes	characteristic
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	#334	1	2	yes	—
84	Dic₁₇	#360	1	2	yes	—
85	C₂×M₃₂	—	17	2	no	—
86	#431	—	34	1	no	—
87	C₂×C₄⋊C₈	—	34	1	no	—
88	C₂×C₄×C₈	—	17	1	no	—
89	#346	—	34	1	no	—
90	C₄×C₁₆	—	34	1	no	—
91	#335	—	17	2	no	—
92	#334	—	17	1	no	—
93	#334	—	34	1	no	—
94	#334	—	34	1	no	—
95	#334	#595	1	1	yes	characteristic
96	C₂×C₁₇	#1277	1	1	yes	characteristic
97	C₂×C₁₇	#1386	1	1	yes	characteristic
98	C₂×C₁₇	#1388	1	1	yes	characteristic
99	D₁₇	—	2	2	no	—
100	C₂²×C₈	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂²×C₈	—	68	1	no	—
103	C₂×C₄²	—	17	1	no	—
104	M₃₂	—	17	4	no	—
105	M₃₂	—	34	2	no	—
106	C₂×C₁₆	—	34	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	34	2	no	—
109	C₄⋊C₈	—	34	1	no	—
110	C₄⋊C₈	—	68	1	no	—
111	C₂²⋊C₈	—	68	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	68	1	no	—
116	C₄×C₈	C₂×#595	1	1	yes	characteristic
117	C₁₇	#7158	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	—	17	1	no	—
119	C₂²×C₄	—	17	1	no	—
120	C₂²×C₄	—	17	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	2	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	34	2	no	—
127	C₂×C₈	—	68	2	no	—
128	C₂×C₈	—	68	1	no	—
129	C₂×C₈	—	136	2	no	—
130	C₂×C₈	C₄×#595	1	1	yes	characteristic
131	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
132	C₄²	—	17	1	no	—
133	C₄²	—	17	1	no	—
134	C₄²	—	17	1	no	—
135	C₄²	C₂²⋊#595	1	1	yes	characteristic
136	C₁₆	—	34	4	no	—
137	C₂³	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	—	17	2	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	34	2	no	—
144	C₂×C₄	—	68	1	no	—
145	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#10587547	1	1	yes	characteristic
147	C₂×C₄	#10587549	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	17	2	no	—
151	C₈	—	34	1	no	—
152	C₈	—	136	1	no	—
153	C₈	—	136	2	no	—
154	C₈	C₈×#595	1	1	yes	characteristic
155	C₈	#10587532	1	1	yes	characteristic
156	C₂²	—	17	2	no	—
157	C₂²	—	34	1	no	—
158	C₂²	—	34	1	no	—
159	C₂²	#11763682	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	17	2	no	—
163	C₄	—	17	2	no	—
164	C₄	—	34	1	no	—
165	C₄	#11763674	1	1	yes	characteristic
166	C₄	#11763684	1	1	yes	characteristic
167	C₂	—	34	2	no	—
168	C₂	#14543613	1	1	yes	characteristic
169	C₂	#14543857	1	1	yes	characteristic
170	C₂	#14543863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544991

C₂

yes

maximal, characteristic

#14544334

C₂

yes

maximal, characteristic

D₁₇×#334

C₂

yes

maximal, characteristic

C₂×#10587530

C₄

yes

—

—

—

—

C₄

yes

—

#11763671

—

C₄⋊C₈×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762560

C₄

yes

characteristic

#11762559

—

C₁₇×#334

C₄

yes

characteristic

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

D₈

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

characteristic

C₂×#3624

—

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

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

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

—

C₄×M₃₂

—

#1564

—

C₂×#334

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#335

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

#334

yes

—

Dic₁₇

#360

yes

—

C₂×M₃₂

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₂²⋊C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×#595

yes

characteristic

117

C₁₇

#7158

yes

characteristic, complemented, Sylow

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

136

—

130

C₂×C₈

C₄×#595

yes

characteristic

131

C₂×C₈

C₄⋊#595

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#595

yes

characteristic

136

C₁₆

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

146

C₂×C₄

#10587547

yes

characteristic

147

C₂×C₄

#10587549

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

136

—

153

C₈

—

136

—

154

C₈

C₈×#595

yes

characteristic

155

C₈

#10587532

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#11763682

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#11763674

yes

characteristic

166

C₄

#11763684

yes

characteristic

167

C₂

—

168

C₂

#14543613

yes

characteristic

169

C₂

#14543857

yes

characteristic

170

C₂

#14543863

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	#107
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#334	#595
hypocenter	C₁₇	#7158
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. #10587530	C₈	semidirect
117. C₁₇	#7158	semidirect

H ≅ G/N

type

24. #10587530

C₈

semidirect

117. C₁₇

#7158

semidirect

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

subgroup	index
#14544991	2
#14544334	2
D₁₇×#334	2
#7158	17

subgroup

index

#14544991

#14544334

D₁₇×#334

#7158

Subgroups — #17948551

Conjugacy classes of subgroups

Complements

Maximal subgroups