Group #17949000 — subgroups

Subgroups — #17949000

Order 4352.

188 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14545148	C₂	1	1	yes	maximal, characteristic
2	#14544339	C₂	1	1	yes	maximal, characteristic
3	#14527531	C₂	1	1	yes	maximal, characteristic
4	#11763813	—	2	1	no	—
5	C₄⋊#63216	—	2	1	no	—
6	#11763704	C₄	1	2	yes	—
7	#11763698	—	2	1	no	—
8	#11763670	—	2	1	no	—
9	#11763670	C₄	1	2	yes	—
10	#11762861	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762559	—	2	1	no	—
14	C₁₇×#335	C₄	1	1	yes	characteristic
15	C₂×#63243	—	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	C₄⋊#3624	—	2	2	no	—
20	C₄⋊#3624	—	4	1	no	—
21	C₂×#63216	—	2	2	no	—
22	C₁₇×C₄⋊C₈	—	2	1	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587491	—	4	1	no	—
25	C₈×Dic₁₇	—	4	1	no	—
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₄⋊#3622	—	2	1	no	—
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
32	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
33	C₂×#3624	—	4	2	no	—
34	#63243	—	8	1	no	—
35	#63243	—	8	1	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	1	no	—
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	4	1	no	—
49	#63216	—	4	2	no	—
50	#7295	—	17	1	no	maximal, Sylow
51	C₂²×D₁₇	#107	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	#106	1	1	yes	characteristic
58	C₄×D₁₇	—	2	1	no	—
59	C₄×D₁₇	—	2	1	no	—
60	C₄×D₁₇	—	2	1	no	—
61	C₄×D₁₇	—	2	1	no	—
62	C₄×D₁₇	—	4	1	no	—
63	#3624	—	8	2	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	8	1	no	—
66	C₈×C₁₇	#108	1	1	yes	characteristic
67	C₈×C₁₇	#108	1	1	yes	characteristic
68	#3622	—	2	1	no	—
69	#3622	—	2	1	no	—
70	#3622	—	8	1	no	—
71	#2163	—	17	1	no	—
72	#1566	—	17	1	no	—
73	#1564	—	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	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	#341	1	1	yes	characteristic
80	C₄×C₁₇	#344	1	1	yes	characteristic
81	C₄×C₁₇	#360	1	1	yes	characteristic
82	C₄×C₁₇	#360	1	1	yes	characteristic
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	—	2	1	no	—
85	Dic₁₇	—	2	1	no	—
86	#431	—	34	1	no	—
87	#423	—	34	1	no	—
88	C₂×C₄×C₈	—	17	1	no	—
89	C₄⋊C₁₆	—	34	1	no	—
90	C₂²⋊C₁₆	—	17	2	no	—
91	#346	—	34	1	no	—
92	#335	—	17	1	no	—
93	#335	#595	1	1	yes	characteristic
94	#334	—	17	2	no	—
95	#334	—	34	1	no	—
96	#334	—	34	1	no	—
97	C₂×C₁₇	#1301	1	1	yes	characteristic
98	C₂×C₁₇	#1385	1	1	yes	characteristic
99	C₂×C₁₇	#1386	1	1	yes	characteristic
100	D₁₇	—	4	1	no	—
101	C₂×M₁₆	—	68	1	no	—
102	C₂²×C₈	—	17	1	no	—
103	C₂²×C₈	—	17	1	no	—
104	C₂×C₄²	—	17	1	no	—
105	C₂×C₁₆	—	34	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	34	2	no	—
108	C₄⋊C₈	—	34	1	no	—
109	C₄⋊C₈	—	68	1	no	—
110	C₂²⋊C₈	—	68	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	68	1	no	—
115	C₄×C₈	C₂×#595	1	1	yes	characteristic
116	C₁₇	#7295	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	1	no	—
119	C₂²×C₄	—	17	1	no	—
120	M₁₆	—	136	1	no	—
121	M₁₆	—	136	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	34	1	no	—
126	C₂×C₈	—	34	1	no	—
127	C₂×C₈	—	68	1	no	—
128	C₂×C₈	—	68	2	no	—
129	C₂×C₈	—	68	1	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	—	68	2	no	—
137	C₂³	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	34	1	no	—
142	C₂×C₄	—	34	1	no	—
143	C₂×C₄	—	34	1	no	—
144	C₂×C₄	—	34	1	no	—
145	C₂×C₄	—	68	1	no	—
146	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#10587547	1	1	yes	characteristic
148	C₂×C₄	#10587549	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	34	1	no	—
152	C₈	—	34	1	no	—
153	C₈	—	136	2	no	—
154	C₈	—	136	1	no	—
155	C₈	#10587535	1	1	yes	characteristic
156	C₈	#10587536	1	1	yes	characteristic
157	C₂²	—	34	1	no	—
158	C₂²	—	34	1	no	—
159	C₂²	—	34	1	no	—
160	C₂²	#11763682	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	34	1	no	—
163	C₄	—	34	1	no	—
164	C₄	—	34	1	no	—
165	C₄	#11763685	1	1	yes	characteristic
166	C₄	#11763692	1	1	yes	characteristic
167	C₄	#11763725	1	1	yes	characteristic
168	C₄	#11763726	1	1	yes	characteristic
169	C₂	—	68	1	no	—
170	C₂	#14543676	1	1	yes	characteristic
171	C₂	#14543853	1	1	yes	characteristic
172	C₂	#14543855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545148

C₂

yes

maximal, characteristic

#14544339

C₂

yes

maximal, characteristic

#14527531

C₂

yes

maximal, characteristic

#11763813

—

C₄⋊#63216

—

#11763704

C₄

yes

—

#11763698

—

#11763670

—

#11763670

C₄

yes

—

#11762861

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762559

—

C₁₇×#335

C₄

yes

characteristic

C₂×#63243

—

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

#10587491

—

C₈×Dic₁₇

—

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

—

—

—

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

—

—

—

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

—

C₄×D₁₇

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#431

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

yes

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

100

D₁₇

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₂²⋊C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#595

yes

characteristic

116

C₁₇

#7295

yes

characteristic, complemented, Sylow

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

M₁₆

—

136

—

121

M₁₆

—

136

—

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

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

—

146

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

147

C₂×C₄

#10587547

yes

characteristic

148

C₂×C₄

#10587549

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

136

—

154

C₈

—

136

—

155

C₈

#10587535

yes

characteristic

156

C₈

#10587536

yes

characteristic

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#11763682

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#11763685

yes

characteristic

166

C₄

#11763692

yes

characteristic

167

C₄

#11763725

yes

characteristic

168

C₄

#11763726

yes

characteristic

169

C₂

—

170

C₂

#14543676

yes

characteristic

171

C₂

#14543853

yes

characteristic

172

C₂

#14543855

yes

characteristic

	subgroup	G/H
center	C₂²	#11763682
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#107
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#335	#595
hypocenter	C₁₇	#7295
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#11763682

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#335

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
116. C₁₇	#7295	semidirect

H ≅ G/N

type

116. C₁₇

#7295

semidirect

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

subgroup	index
#14545148	2
#14544339	2
#14527531	2
#7295	17

subgroup

index

Subgroups — #17949000

Conjugacy classes of subgroups

Complements

Maximal subgroups