Group #17948837 — subgroups

Subgroups — #17948837

Order 4352.

198 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 64 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	#14545114	C₂	1	1	yes	maximal, characteristic
2	#14544347	C₂	1	1	yes	maximal, characteristic
3	#14527538	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587529	C₄	1	2	yes	—
5	#11763831	—	2	1	no	—
6	#11763730	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	#11763673	C₄	1	2	yes	—
9	#11763671	—	2	1	no	—
10	#11762831	—	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×#63218	—	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	#10587542	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₂×#63216	—	2	2	no	—
23	#10587529	—	4	2	no	—
24	C₁₇×C₄⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	#10587485	—	4	1	no	—
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₄⋊#3622	—	2	1	no	—
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
34	C₂×#3624	—	4	2	no	—
35	C₂×C₈×C₁₇	—	4	1	no	—
36	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	D₈	1	1	yes	characteristic
41	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
42	C₂×#3622	—	4	1	no	—
43	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
45	#63218	—	8	2	no	—
46	C₈×D₁₇	—	2	1	no	—
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	4	1	no	—
49	#63216	—	4	2	no	—
50	#7243	—	17	1	no	maximal, Sylow
51	C₂²×D₁₇	#106	1	1	yes	characteristic
52	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
55	C₂×Dic₁₇	#102	1	1	yes	characteristic
56	C₂×Dic₁₇	#103	1	1	yes	characteristic
57	C₂×Dic₁₇	#107	1	1	yes	characteristic
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	2	2	no	—
62	#3624	—	8	2	no	—
63	C₈×C₁₇	—	2	1	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	8	1	no	—
66	#3622	—	2	1	no	—
67	#3622	—	8	1	no	—
68	#3622	#100	1	1	yes	characteristic
69	#3622	#101	1	1	yes	characteristic
70	C₂×C₄⋊C₁₆	—	17	1	no	—
71	#1568	—	17	1	no	—
72	#1566	—	17	1	no	—
73	C₂²×C₁₇	#340	1	1	yes	characteristic
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	#365	1	2	yes	—
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	#338	1	1	yes	characteristic
80	C₄×C₁₇	#343	1	1	yes	characteristic
81	Dic₁₇	—	2	1	no	—
82	Dic₁₇	—	2	1	no	—
83	Dic₁₇	#359	1	2	yes	—
84	C₂²×C₁₆	—	17	2	no	—
85	#432	—	34	1	no	—
86	#423	—	34	1	no	—
87	C₂×C₄×C₈	—	17	1	no	—
88	C₄⋊C₁₆	—	34	1	no	—
89	C₄⋊C₁₆	—	34	1	no	—
90	#335	—	17	2	no	—
91	#334	—	17	1	no	—
92	#334	—	34	1	no	—
93	#334	—	34	1	no	—
94	#334	#595	1	1	yes	characteristic
95	C₂×C₁₇	#1292	1	1	yes	characteristic
96	C₂×C₁₇	#1382	1	1	yes	characteristic
97	C₂×C₁₇	#1383	1	1	yes	characteristic
98	D₁₇	—	2	2	no	—
99	C₂×M₁₆	—	68	1	no	—
100	C₂²×C₈	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂×C₄²	—	17	1	no	—
103	C₂×C₁₆	—	34	2	no	—
104	C₂×C₁₆	—	68	2	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	34	2	no	—
107	C₄⋊C₈	—	34	1	no	—
108	C₄⋊C₈	—	68	1	no	—
109	C₂²⋊C₈	—	68	1	no	—
110	#97	—	68	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	C₂×#595	1	1	yes	characteristic
115	C₁₇	#7243	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	—	17	1	no	—
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	1	no	—
119	M₁₆	—	136	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	34	1	no	—
124	C₂×C₈	—	34	1	no	—
125	C₂×C₈	—	68	1	no	—
126	C₂×C₈	—	68	2	no	—
127	C₂×C₈	—	68	1	no	—
128	C₂×C₈	C₄×#595	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
130	C₄²	—	17	1	no	—
131	C₄²	—	17	1	no	—
132	C₄²	—	17	1	no	—
133	C₄²	C₂²⋊#595	1	1	yes	characteristic
134	C₁₆	—	68	2	no	—
135	C₂³	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	34	2	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#10587547	1	1	yes	characteristic
145	C₂×C₄	#10587549	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	17	1	no	—
150	C₈	—	17	1	no	—
151	C₈	—	34	1	no	—
152	C₈	—	136	1	no	—
153	C₈	—	136	2	no	—
154	C₂²	—	17	2	no	—
155	C₂²	—	34	1	no	—
156	C₂²	—	34	1	no	—
157	C₂²	#11763682	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	17	2	no	—
161	C₄	—	34	1	no	—
162	C₄	—	34	1	no	—
163	C₄	#11763678	1	1	yes	characteristic
164	C₄	#11763688	1	1	yes	characteristic
165	C₂	—	34	2	no	—
166	C₂	#14543653	1	1	yes	characteristic
167	C₂	#14543847	1	1	yes	characteristic
168	C₂	#14543848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545114

C₂

yes

maximal, characteristic

#14544347

C₂

yes

maximal, characteristic

#14527538

C₂

yes

maximal, characteristic

C₂×#10587529

C₄

yes

—

—

—

—

C₄

yes

—

#11763671

—

#11762831

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762560

C₄

yes

characteristic

#11762559

—

C₁₇×#334

C₄

yes

characteristic

C₂×#63218

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

—

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

—

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

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

—

—

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

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

yes

C₂×C₄⋊C₁₆

—

#1568

—

#1566

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#365

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#359

yes

—

C₂²×C₁₆

—

#432

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#595

yes

characteristic

115

C₁₇

#7243

yes

characteristic, complemented, Sylow

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

136

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₄×#595

yes

characteristic

129

C₂×C₈

C₄⋊#595

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²⋊#595

yes

characteristic

134

C₁₆

—

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

144

C₂×C₄

#10587547

yes

characteristic

145

C₂×C₄

#10587549

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

136

—

153

C₈

—

136

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#11763682

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11763678

yes

characteristic

164

C₄

#11763688

yes

characteristic

165

C₂

—

166

C₂

#14543653

yes

characteristic

167

C₂

#14543847

yes

characteristic

168

C₂

#14543848

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
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	#7243
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#334

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
115. C₁₇	#7243	semidirect

H ≅ G/N

type

115. C₁₇

#7243

semidirect

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

subgroup	index
#14545114	2
#14544347	2
#14527538	2
#7243	17

subgroup

index

Subgroups — #17948837

Conjugacy classes of subgroups

Complements

Maximal subgroups