Group #17962211 — subgroups

Subgroups — #17962211

Order 4352.

298 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₂×#11763731	C₂	1	1	yes	maximal, characteristic
2	#14545121	C₂	1	2	yes	maximal
3	C₂×#11763695	C₂	1	1	yes	maximal, characteristic
4	#14543839	C₂	1	2	yes	maximal
5	#14528666	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂²×#63216	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	#11763731	—	2	2	no	—
10	#11763730	—	2	4	no	—
11	#11763695	—	2	2	no	—
12	#11762847	C₄	1	1	yes	characteristic
13	#11762767	C₂²	1	2	yes	—
14	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	#11762570	C₄	1	2	yes	—
16	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	#10587612	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂³	1	1	yes	characteristic
22	#10587538	—	2	2	no	—
23	#10587538	—	2	2	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	2	1	no	—
27	C₂×#63216	—	2	1	no	—
28	C₂×#63216	—	2	2	no	—
29	C₂×#63216	D₄	1	2	yes	—
30	C₂×#63216	Q₈	1	2	yes	—
31	C₁₇×#108	—	2	2	no	—
32	#10587489	C₈	1	2	yes	—
33	#10587488	C₈	1	2	yes	—
34	#10587486	—	2	2	no	—
35	C₈×Dic₁₇	—	2	2	no	—
36	C₈×Dic₁₇	D₄	1	2	yes	—
37	C₈×Dic₁₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	2	1	no	—
40	M₁₆×C₁₇	—	2	2	no	—
41	M₁₆×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
47	C₂²⋊Dic₁₇	—	2	2	no	—
48	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
49	C₄×Dic₁₇	—	2	2	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
52	C₂×#3622	C₂×D₄	1	1	yes	characteristic
53	C₂×#3622	C₂×D₄	1	1	yes	characteristic
54	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
55	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
56	C₂×#3622	#41	1	1	yes	characteristic
57	C₂×#3622	#41	1	1	yes	characteristic
58	#63216	—	2	2	no	—
59	#63216	—	2	2	no	—
60	#63216	—	4	2	no	—
61	#13161	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	#131	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
69	C₂×Dic₁₇	—	2	4	no	—
70	C₂×Dic₁₇	—	2	2	no	—
71	C₂×Dic₁₇	—	2	4	no	—
72	C₈×C₁₇	—	4	2	no	—
73	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
74	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
75	#3622	#121	1	1	yes	characteristic
76	#3622	C₂²⋊Q₈	1	1	yes	characteristic
77	#3622	#125	1	1	yes	characteristic
78	#3622	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#364	—	17	1	no	—
80	#2151	—	17	2	no	—
81	C₂×C₄×C₁₆	—	17	1	no	—
82	#1772	—	17	1	no	—
83	#1379	—	17	2	no	—
84	C₂²×C₁₇	—	2	2	no	—
85	C₂²×C₁₇	#424	1	1	yes	characteristic
86	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
87	C₂²×C₁₇	#446	1	1	yes	characteristic
88	C₄×C₁₇	#389	1	1	yes	characteristic
89	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₁₇	#436	1	1	yes	characteristic
91	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
92	Dic₁₇	—	2	4	no	—
93	Dic₁₇	—	4	4	no	—
94	C₂×M₃₂	—	17	2	no	—
95	C₂²×C₁₆	—	17	1	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂×#108	#595	1	1	yes	characteristic
98	#427	—	17	1	no	—
99	#405	—	17	2	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	#364	—	34	2	no	—
102	C₄⋊C₁₆	—	34	4	no	—
103	C₄×C₁₆	—	34	2	no	—
104	#339	—	17	2	no	—
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	—	2	1	no	—
107	C₂×C₁₇	#1939	1	1	yes	characteristic
108	C₂×C₁₇	#2170	1	2	yes	—
109	C₂×M₁₆	C₂×#595	1	2	yes	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	C₂×#595	1	1	yes	characteristic
112	#117	—	17	2	no	—
113	C₂×C₄²	—	17	1	no	—
114	M₃₂	—	34	2	no	—
115	M₃₂	—	34	2	no	—
116	C₂×C₁₆	—	17	2	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	—	34	1	no	—
120	C₂×C₁₆	—	34	1	no	—
121	C₂×C₁₆	—	34	1	no	—
122	C₂×C₁₆	—	34	2	no	—
123	#108	—	2	2	no	—
124	#107	—	17	2	no	—
125	#106	—	17	2	no	—
126	#97	—	34	2	no	—
127	C₄×C₈	—	17	2	no	—
128	C₄×C₈	—	17	2	no	—
129	C₄×C₈	—	34	2	no	—
130	C₁₇	#13161	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	—	34	1	no	—
132	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	#63242	1	1	yes	characteristic
143	C₂×C₈	#63242	1	1	yes	characteristic
144	C₂×C₈	C₄⋊#595	1	2	yes	—
145	C₂×C₈	#63247	1	2	yes	—
146	C₄⋊C₄	—	17	4	no	—
147	C₂²⋊C₄	—	34	2	no	—
148	C₄²	—	17	2	no	—
149	C₄²	—	34	1	no	—
150	C₄²	—	34	2	no	—
151	C₁₆	—	34	2	no	—
152	C₁₆	—	34	2	no	—
153	C₁₆	—	68	2	no	—
154	C₂³	#10587680	1	1	yes	characteristic
155	C₂×C₄	—	34	4	no	—
156	C₂×C₄	—	34	2	no	—
157	C₂×C₄	—	34	4	no	—
158	C₂×C₄	C₂×#63242	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
160	C₂×C₄	#10587678	1	1	yes	characteristic
161	C₂×C₄	D₄×#595	1	1	yes	characteristic
162	C₂×C₄	Q₈×#595	1	1	yes	characteristic
163	C₂×C₄	C₂×#63247	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	—	17	1	no	—
166	C₈	—	17	1	no	—
167	C₈	—	17	1	no	—
168	C₈	—	17	1	no	—
169	C₈	#10587544	1	2	yes	—
170	C₈	#10587544	1	2	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	#11763816	1	1	yes	characteristic
173	C₂²	#11763837	1	1	yes	characteristic
174	C₂²	#11763860	1	1	yes	characteristic
175	C₄	—	34	4	no	—
176	C₄	—	68	4	no	—
177	C₄	#11763760	1	1	yes	characteristic
178	C₄	C₂×#10587544	1	1	yes	characteristic
179	C₄	#11763844	1	1	yes	characteristic
180	C₄	#11763858	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#14544882	1	1	yes	characteristic
184	C₂	#14545173	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763731

C₂

yes

maximal, characteristic

#14545121

C₂

yes

maximal

C₂×#11763695

C₂

yes

maximal, characteristic

#14543839

C₂

yes

maximal

#14528666

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

C₂²×#63216

C₂²

yes

characteristic

C₂²×#63216

C₂²

yes

—

—

—

C₄

yes

characteristic

#11762767

C₂²

yes

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762570

C₄

yes

—

C₂×C₁₇×#108

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587612

C₂×C₄

yes

—

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#63216

Q₈

yes

—

C₁₇×#108

—

#10587489

C₈

yes

—

#10587488

C₈

yes

—

#10587486

—

C₈×Dic₁₇

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₈

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₂×#3622

C₂×Q₈

yes

yes

yes

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₄⋊C₈

yes

—

yes

C₂²⋊Q₈

yes

yes

C₄⋊Q₈

yes

characteristic

C₂×#364

—

#2151

—

C₂×C₄×C₁₆

—

#1772

—

#1379

—

C₂²×C₁₇

—

C₂²×C₁₇

#424

yes

characteristic

C₂²×C₁₇

C₈×D₄

yes

C₂²×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₇

#436

yes

characteristic

C₄×C₁₇

C₈×Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

yes

—

—

100

C₂×C₄×C₈

—

101

#364

—

102

C₄⋊C₁₆

—

103

C₄×C₁₆

—

104

#339

—

105

C₂×C₁₇

—

106

C₂×C₁₇

—

107

C₂×C₁₇

#1939

yes

characteristic

108

C₂×C₁₇

#2170

yes

—

109

C₂×M₁₆

C₂×#595

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×#595

yes

characteristic

112

#117

—

113

C₂×C₄²

—

114

M₃₂

—

115

M₃₂

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

123

#108

—

124

#107

—

125

#106

—

126

#97

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₁₇

#13161

yes

characteristic, complemented, Sylow

131

C₂²×C₄

—

132

C₂²×C₄

C₂²×#595

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#63242

yes

characteristic

143

C₂×C₈

#63242

yes

characteristic

144

C₂×C₈

C₄⋊#595

yes

—

145

C₂×C₈

#63247

yes

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₁₆

—

152

C₁₆

—

153

C₁₆

—

154

C₂³

#10587680

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₂×#63242

yes

characteristic

159

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

160

C₂×C₄

#10587678

yes

characteristic

161

C₂×C₄

D₄×#595

yes

characteristic

162

C₂×C₄

Q₈×#595

yes

characteristic

163

C₂×C₄

C₂×#63247

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

#10587544

yes

—

170

C₈

#10587544

yes

—

171

C₂²

—

172

C₂²

#11763816

yes

characteristic

173

C₂²

#11763837

yes

characteristic

174

C₂²

#11763860

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

#11763760

yes

characteristic

178

C₄

C₂×#10587544

yes

characteristic

179

C₄

#11763844

yes

characteristic

180

C₄

#11763858

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

#14544882

yes

characteristic

184

C₂

#14545173

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#108	C₄
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×#108	#595
hypocenter	C₁₇	#13161
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₂×C₁₇×#108

C₄

socle

C₂²×C₁₇

C₈×D₄

prime radical

C₂³×C₁₇

#131

cosocle

C₂²×#3622

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
130. C₁₇	#13161	semidirect

H ≅ G/N

type

130. C₁₇

#13161

semidirect

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

subgroup	index
C₂×#11763731	2
#14545121	2
C₂×#11763695	2
#14543839	2
#14528666	2
#13161	17

subgroup

index

Subgroups — #17962211

Conjugacy classes of subgroups

Complements

Maximal subgroups