Group #16954202 — subgroups

Subgroups — #16954202

Order 4352.

284 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 91 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×#11762637	C₂	1	1	yes	maximal, characteristic, complemented
2	#14528848	C₂	1	2	yes	maximal
3	#14526893	C₂	1	2	yes	maximal, complemented
4	#14526878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	C₂²	1	2	yes	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762810	—	2	2	no	—
9	#11762637	—	2	2	no	—
10	#11762577	—	2	2	no	—
11	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×C₂²⋊C₁₆	—	2	2	no	—
15	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
16	C₁₇×#131	—	2	2	no	—
17	C₁₇×#131	—	2	2	no	—
18	C₂×M₁₆×C₁₇	—	2	1	no	—
19	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	—	2	2	no	—
23	C₂×#63223	—	2	2	no	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	—	2	1	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
28	#10587506	—	4	2	no	—
29	#10587503	—	4	2	no	—
30	#10587490	—	2	2	no	—
31	#10587490	—	2	2	no	—
32	#10587489	—	2	2	no	—
33	#10587489	—	2	2	no	—
34	#41×C₁₇	—	2	2	no	—
35	#41×C₁₇	—	2	2	no	—
36	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
37	C₂×D₄×C₁₇	—	2	1	no	—
38	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	—	2	1	no	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	4	2	no	—
42	M₁₆×C₁₇	—	2	2	no	—
43	M₁₆×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	4	1	no	—
46	C₂×C₈×C₁₇	—	4	1	no	—
47	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	2	yes	—
50	C₂×C₈×C₁₇	#41	1	2	yes	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	#63223	—	4	2	no	—
54	#63223	—	4	2	no	—
55	C₂×#3622	—	4	2	no	—
56	C₁₆×C₁₇	—	2	2	no	—
57	C₁₆×C₁₇	—	4	1	no	—
58	#13496	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	—	2	1	no	—
60	C₂³×C₁₇	#135	1	1	yes	characteristic
61	Q₈×C₁₇	—	2	2	no	—
62	D₄×C₁₇	—	2	2	no	—
63	D₄×C₁₇	—	4	2	no	—
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	—	4	1	no	—
66	C₂×C₄×C₁₇	—	4	1	no	—
67	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#123	1	2	yes	—
69	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#137	1	2	yes	—
71	C₂×Dic₁₇	—	8	2	no	—
72	C₂×Dic₁₇	—	16	2	no	—
73	C₈×C₁₇	—	4	1	no	—
74	C₈×C₁₇	#123	1	1	yes	characteristic
75	C₈×C₁₇	#124	1	1	yes	characteristic
76	C₈×C₁₇	#126	1	2	yes	—
77	#3622	—	8	2	no	—
78	C₂×#367	—	17	1	no	—
79	#2113	D₁₇	1	1	yes	characteristic
80	#1816	—	17	2	no	—
81	#1387	—	17	2	no	—
82	#1381	—	17	1	no	—
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	—	4	1	no	—
85	C₂²×C₁₇	—	4	2	no	—
86	C₂²×C₁₇	#466	1	1	yes	characteristic
87	C₂²×C₁₇	#485	1	2	yes	—
88	C₄×C₁₇	—	4	1	no	—
89	C₄×C₁₇	#397	1	1	yes	characteristic
90	C₄×C₁₇	#470	1	1	yes	characteristic
91	C₄×C₁₇	#480	1	2	yes	—
92	Dic₁₇	—	16	2	no	—
93	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
94	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
95	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
96	C₂×#108	—	17	2	no	—
97	C₂×#107	—	17	2	no	—
98	#417	—	34	2	no	—
99	#367	—	34	2	no	—
100	C₂²⋊C₁₆	—	2	2	no	—
101	#343	—	34	2	no	—
102	C₂×C₁₇	—	2	2	no	—
103	C₂×C₁₇	—	8	1	no	—
104	C₂×C₁₇	#2075	1	1	yes	characteristic
105	C₂×C₁₇	#2223	1	2	yes	—
106	C₂×#41	D₆₈	1	1	yes	characteristic
107	#131	—	2	2	no	—
108	#131	—	2	2	no	—
109	C₂×M₁₆	—	2	1	no	—
110	C₂×M₁₆	—	34	2	no	—
111	C₂×M₁₆	D₆₈	1	1	yes	characteristic
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	34	2	no	—
115	M₃₂	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
119	#108	—	34	2	no	—
120	#108	—	34	2	no	—
121	#107	—	34	2	no	—
122	#107	—	34	2	no	—
123	#103	—	68	2	no	—
124	#102	—	68	2	no	—
125	C₁₇	#13496	1	1	yes	characteristic, complemented, Sylow
126	#41	—	2	2	no	—
127	#41	—	2	2	no	—
128	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	D₁₃₆	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	68	2	no	—
133	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	4	2	no	—
136	M₁₆	—	68	2	no	—
137	M₁₆	—	68	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	68	2	no	—
142	C₂×C₈	#63252	1	1	yes	characteristic
143	C₂×C₈	#63254	1	2	yes	—
144	C₂×C₈	#63254	1	2	yes	—
145	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄⋊C₄	—	68	2	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	—	2	1	no	—
151	C₂³	#10587590	1	1	yes	characteristic
152	Q₈	—	2	2	no	—
153	D₄	—	2	2	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	272	2	no	—
159	C₂×C₄	—	136	2	no	—
160	C₂×C₄	#10587572	1	2	yes	—
161	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
162	C₂×C₄	#10587595	1	2	yes	—
163	C₂×C₄	#10587616	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	136	2	no	—
166	C₈	#10587565	1	2	yes	—
167	C₈	#10587621	1	1	yes	characteristic
168	C₈	#10587622	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	#11763045	1	1	yes	characteristic
173	C₂²	#11763151	1	2	yes	—
174	C₄	—	4	1	no	—
175	C₄	—	272	2	no	—
176	C₄	#11762734	1	1	yes	characteristic
177	C₄	#11763067	1	1	yes	characteristic
178	C₄	#11763123	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	8	1	no	—
181	C₂	#14530207	1	1	yes	characteristic
182	C₂	#14530914	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762637

C₂

yes

maximal, characteristic, complemented

#14528848

C₂

yes

maximal

#14526893

C₂

yes

maximal, complemented

#14526878

C₂

yes

maximal, characteristic, complemented

C₁₇×#2113

C₂

yes

maximal, characteristic

C₂×#10587490

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

#11762810

—

#11762637

—

#11762577

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×C₂²⋊C₁₆

—

C₂×#41×C₁₇

D₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×#63223

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

D₄

yes

—

—

—

—

—

—

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

D₈

yes

characteristic

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

D₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

C₂×C₈×C₁₇

yes

C₂×C₈×C₁₇

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

#63223

—

C₂×#3622

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

#13496

—

maximal, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

#135

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

#3622

—

C₂×#367

—

#2113

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#466

yes

characteristic

C₂²×C₁₇

#485

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×#131

C₂×D₁₇

yes

characteristic

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

C₂×D₁₇

yes

—

—

—

—

100

C₂²⋊C₁₆

—

101

#343

—

102

C₂×C₁₇

—

103

C₂×C₁₇

—

104

C₂×C₁₇

#2075

yes

characteristic

105

C₂×C₁₇

#2223

yes

—

106

C₂×#41

D₆₈

yes

characteristic

107

#131

—

108

#131

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

D₆₈

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₁₇

yes

characteristic

114

C₂×C₄⋊C₄

—

115

M₃₂

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₂×C₁₆

C₂²⋊D₁₇

yes

—

119

#108

—

120

#108

—

121

#107

—

122

#107

—

123

#103

—

124

#102

—

125

C₁₇

#13496

yes

characteristic, complemented, Sylow

126

#41

—

127

#41

—

128

C₂×Q₈

D₁₃₆

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

D₁₃₆

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂×D₆₈

yes

characteristic

134

M₁₆

—

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#63252

yes

characteristic

143

C₂×C₈

#63254

yes

—

144

C₂×C₈

#63254

yes

—

145

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

#10587590

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

272

—

159

C₂×C₄

—

136

—

160

C₂×C₄

#10587572

yes

—

161

C₂×C₄

C₂×D₁₃₆

yes

characteristic

162

C₂×C₄

#10587595

yes

—

163

C₂×C₄

#10587616

yes

characteristic

164

C₈

—

165

C₈

—

136

—

166

C₈

#10587565

yes

—

167

C₈

#10587621

yes

characteristic

168

C₈

#10587622

yes

characteristic

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#11763045

yes

characteristic

173

C₂²

#11763151

yes

—

174

C₄

—

175

C₄

—

272

—

176

C₄

#11762734

yes

characteristic

177

C₄

#11763067

yes

characteristic

178

C₄

#11763123

yes

—

179

C₂

—

180

C₂

—

181

C₂

#14530207

yes

characteristic

182

C₂

#14530914

yes

—

	subgroup	G/H
center	C₂²	#11763045
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2113	C₂
socle	C₂²×C₁₇	#466
prime radical	C₂×#41×C₁₇	D₄
hypercenter	#2113	D₁₇
hypocenter	C₁₇	#13496
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₂²

#11763045

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2113

C₂

socle

C₂²×C₁₇

#466

prime radical

C₂×#41×C₁₇

D₄

D₁₇

C₁₇

G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
1. C₂×#11762637	C₂	semidirect
3. #14526893	C₂	semidirect
4. #14526878	C₂	semidirect
125. C₁₇	#13496	semidirect

H ≅ G/N

type

1. C₂×#11762637

C₂

semidirect

3. #14526893

C₂

semidirect

4. #14526878

C₂

semidirect

125. C₁₇

#13496

semidirect

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

subgroup	index
C₂×#11762637	2
#14528848	2
#14526893	2
#14526878	2
C₁₇×#2113	2
#13496	17

subgroup

index

C₁₇×#2113

Subgroups — #16954202

Conjugacy classes of subgroups

Complements

Maximal subgroups