Group #17949016 — subgroups

Subgroups — #17949016

Order 4352.

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14545147	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587534	C₂	1	1	yes	maximal, characteristic
3	#14527544	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587534	—	2	1	no	—
5	C₄×C₄⋊#595	—	2	1	no	—
6	#11763730	C₄	1	2	yes	complemented
7	#11763704	—	2	1	no	—
8	#11763697	—	2	1	no	—
9	#11763683	C₄	1	2	yes	—
10	#11762865	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762561	—	2	1	no	—
13	#11762560	C₄	1	1	yes	characteristic, complemented
14	C₁₇×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#595	—	2	2	no	—
16	C₂×C₄×#595	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	(C₂×#595)⋊C₄	—	4	1	no	—
21	C₂×#63216	—	2	2	no	—
22	#10587534	—	2	4	no	—
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	#10587491	—	4	1	no	—
26	#10587486	—	4	1	no	—
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
29	C₄⋊#3622	—	2	1	no	—
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₂²×#595	—	4	2	no	—
32	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
34	C₂×C₄×D₁₇	SD₁₆	1	1	yes	characteristic
35	C₄⋊#595	—	2	4	no	—
36	C₄⋊#595	—	4	2	no	—
37	C₄×#595	—	8	2	no	—
38	C₂×C₈×C₁₇	—	4	1	no	—
39	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
45	C₂×#3622	—	4	1	no	—
46	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₁₇	—	2	2	no	—
49	C₈×D₁₇	—	4	1	no	—
50	#63216	—	4	2	no	—
51	#7301	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	#102	1	1	yes	characteristic
53	C₂×#595	—	4	4	no	—
54	C₂×#595	—	8	2	no	—
55	C₂×#595	—	8	2	no	—
56	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
59	C₂×Dic₁₇	#103	1	1	yes	characteristic
60	C₂×Dic₁₇	#106	1	1	yes	characteristic
61	C₂×Dic₁₇	#107	1	1	yes	characteristic
62	C₄×D₁₇	—	2	1	no	—
63	C₄×D₁₇	—	2	1	no	—
64	C₄×D₁₇	—	2	1	no	—
65	C₄×D₁₇	—	2	1	no	—
66	C₄×D₁₇	—	4	1	no	—
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	—	8	1	no	—
70	#3622	—	2	1	no	—
71	#3622	—	8	1	no	—
72	#3622	#104	1	2	yes	—
73	#2163	—	17	1	no	—
74	C₄×#107	—	17	1	no	—
75	#1568	—	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	#361	1	1	yes	characteristic
80	C₂×D₁₇	#362	1	1	yes	characteristic
81	#595	—	8	4	no	—
82	C₄×C₁₇	—	2	1	no	—
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	#339	1	1	yes	characteristic
85	C₄×C₁₇	#344	1	1	yes	characteristic
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	—	2	1	no	—
89	#432	—	34	1	no	—
90	C₂×#107	—	34	1	no	—
91	C₂×C₄×C₈	—	17	1	no	—
92	C₄×C₄⋊C₄	—	34	1	no	—
93	C₄⋊C₁₆	—	17	2	no	—
94	C₂²⋊C₁₆	—	34	1	no	—
95	#346	—	34	1	no	—
96	#340	—	17	2	no	—
97	#335	—	34	1	no	—
98	#334	—	17	1	no	—
99	#334	#595	1	1	yes	characteristic, complemented
100	C₂×C₁₇	#1303	1	1	yes	characteristic
101	C₂×C₁₇	#1384	1	1	yes	characteristic
102	C₂×C₁₇	#1387	1	1	yes	characteristic
103	D₁₇	—	2	1	no	—
104	D₁₇	—	2	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂×C₄⋊C₄	—	34	2	no	—
108	C₂×C₄²	—	17	1	no	—
109	C₂×C₄²	—	68	1	no	—
110	C₂×C₁₆	—	34	2	no	—
111	#107	—	34	4	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	34	1	no	—
114	C₂²⋊C₈	—	68	1	no	—
115	#97	—	68	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	C₂×#595	1	1	yes	characteristic
120	#95	—	68	1	no	—
121	C₁₇	#7301	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	C₂²×C₄	—	68	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	34	2	no	—
130	C₂×C₈	—	68	1	no	—
131	C₂×C₈	—	68	1	no	—
132	C₂×C₈	C₄×#595	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
134	C₄⋊C₄	—	34	4	no	—
135	C₄⋊C₄	—	68	2	no	—
136	C₄²	—	17	1	no	—
137	C₄²	—	17	1	no	—
138	C₄²	—	17	1	no	—
139	C₄²	—	136	2	no	—
140	C₄²	C₂²⋊#595	1	1	yes	characteristic
141	C₁₆	—	68	2	no	—
142	C₂³	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	34	1	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	34	1	no	—
150	C₂×C₄	—	68	1	no	—
151	C₂×C₄	—	68	4	no	—
152	C₂×C₄	—	136	2	no	—
153	C₂×C₄	—	136	2	no	—
154	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#10587547	1	1	yes	characteristic
156	C₂×C₄	#10587549	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	17	2	no	—
161	C₈	—	34	1	no	—
162	C₈	—	136	1	no	—
163	C₂²	—	17	1	no	—
164	C₂²	—	17	1	no	—
165	C₂²	—	34	1	no	—
166	C₂²	—	34	1	no	—
167	C₂²	#11763682	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	34	1	no	—
171	C₄	—	34	1	no	—
172	C₄	—	34	1	no	—
173	C₄	—	136	4	no	—
174	C₄	#11763681	1	1	yes	characteristic
175	C₄	#11763691	1	1	yes	characteristic
176	C₂	—	34	1	no	—
177	C₂	—	34	1	no	—
178	C₂	#14543680	1	1	yes	characteristic
179	C₂	#14543851	1	1	yes	characteristic
180	C₂	#14543860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545147

C₂

yes

maximal, characteristic

C₄×#10587534

C₂

yes

maximal, characteristic

#14527544

C₂

yes

maximal, characteristic

C₂×#10587534

—

C₄×C₄⋊#595

—

#11763730

C₄

yes

—

—

C₄

yes

—

#11762865

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762561

—

#11762560

C₄

yes

characteristic, complemented

C₁₇×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄×#595

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

(C₂×#595)⋊C₄

—

C₂×#63216

—

#10587534

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

#10587486

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

SD₁₆

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

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

Q₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

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

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

yes

—

#2163

—

C₄×#107

—

#1568

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#432

—

C₂×#107

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

yes

characteristic, complemented

100

C₂×C₁₇

#1303

yes

characteristic

101

C₂×C₁₇

#1384

yes

characteristic

102

C₂×C₁₇

#1387

yes

characteristic

103

D₁₇

—

104

D₁₇

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

111

#107

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

C₂×#595

yes

characteristic

120

#95

—

121

C₁₇

#7301

yes

characteristic, complemented, Sylow

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₄×#595

yes

characteristic

133

C₂×C₈

C₄⋊#595

yes

characteristic

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

136

—

140

C₄²

C₂²⋊#595

yes

characteristic

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

—

152

C₂×C₄

—

136

—

153

C₂×C₄

—

136

—

154

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

155

C₂×C₄

#10587547

yes

characteristic

156

C₂×C₄

#10587549

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

136

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#11763682

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

136

—

174

C₄

#11763681

yes

characteristic

175

C₄

#11763691

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14543680

yes

characteristic

179

C₂

#14543851

yes

characteristic

180

C₂

#14543860

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	#102
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#334	#595
hypocenter	C₁₇	#7301
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×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 ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
6. #11763730	C₄	semidirect
13. #11762560	C₄	semidirect
14. C₁₇×#334	C₄	semidirect
99. #334	#595	semidirect
121. C₁₇	#7301	semidirect

H ≅ G/N

type

6. #11763730

C₄

semidirect

13. #11762560

C₄

semidirect

14. C₁₇×#334

C₄

semidirect

99. #334

#595

semidirect

121. C₁₇

#7301

semidirect

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

subgroup	index
#14545147	2
C₄×#10587534	2
#14527544	2
#7301	17

subgroup

index

Subgroups — #17949016

Conjugacy classes of subgroups

Complements

Maximal subgroups