Group #16954204 — subgroups

Subgroups — #16954204

Order 4352.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14530397	C₂	1	1	yes	maximal, characteristic
3	#14528845	C₂	1	2	yes	maximal
4	#14526894	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587587	C₂²	1	1	yes	characteristic
7	C₂×#10587483	C₂²	1	1	yes	characteristic
8	C₂×#10587489	C₂²	1	2	yes	—
9	#11762811	—	2	2	no	—
10	#11762637	—	2	2	no	—
11	#11762588	—	2	2	no	—
12	#11762577	—	2	2	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#108	C₂²	1	2	yes	—
15	C₂×#63252	D₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₂×#63223	D₄	1	1	yes	characteristic
20	#10587587	—	2	2	no	—
21	#10587587	—	2	2	no	—
22	C₂×#63218	—	2	1	no	—
23	C₂×C₈×D₁₇	—	2	1	no	—
24	C₂×C₁₆×C₁₇	—	2	2	no	—
25	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
26	C₁₇×#108	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	#10587492	—	4	2	no	—
29	#10587489	—	2	2	no	—
30	#10587489	—	2	2	no	—
31	#10587487	—	4	2	no	—
32	#10587483	—	2	2	no	—
33	C₂×#63214	—	2	1	no	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₇	—	2	1	no	—
36	C₂²×Dic₁₇	—	4	2	no	—
37	#63252	—	2	2	no	—
38	#63252	—	2	2	no	—
39	C₂×D₆₈	D₈	1	1	yes	characteristic
40	C₂×C₄×D₁₇	—	2	1	no	—
41	C₂×Dic₃₄	D₈	1	1	yes	characteristic
42	M₁₆×C₁₇	—	4	2	no	—
43	M₁₆×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	#41	1	2	yes	—
47	C₂×C₈×C₁₇	#41	1	2	yes	—
48	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
49	C₄⋊Dic₁₇	—	4	2	no	—
50	C₄⋊Dic₁₇	—	4	2	no	—
51	#63223	—	2	2	no	—
52	C₂×#3622	—	2	1	no	—
53	#63218	—	4	2	no	—
54	C₈×D₁₇	—	4	1	no	—
55	C₈×D₁₇	—	4	1	no	—
56	C₁₆×C₁₇	—	2	2	no	—
57	#63214	—	4	1	no	—
58	#13496	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	#135	1	1	yes	characteristic
60	C₂²×D₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#123	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#137	1	2	yes	—
65	C₂²⋊D₁₇	—	4	2	no	—
66	C₂×Dic₁₇	—	2	1	no	—
67	C₂×Dic₁₇	—	8	2	no	—
68	C₂×Dic₁₇	—	16	2	no	—
69	D₆₈	—	2	2	no	—
70	C₄×D₁₇	—	4	1	no	—
71	C₄×D₁₇	—	4	1	no	—
72	Dic₃₄	—	2	2	no	—
73	C₈×C₁₇	—	8	2	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	—	4	1	no	—
78	C₂×#367	—	17	1	no	—
79	#2113	—	17	1	no	—
80	#1816	—	17	2	no	—
81	#1387	—	17	2	no	—
82	#1381	D₁₇	1	1	yes	characteristic, complemented
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	#466	1	1	yes	characteristic
85	C₂²×C₁₇	#485	1	2	yes	—
86	C₂×D₁₇	—	4	1	no	—
87	C₂×D₁₇	—	4	2	no	—
88	C₄×C₁₇	#397	1	1	yes	characteristic
89	C₄×C₁₇	#470	1	1	yes	characteristic
90	C₄×C₁₇	#480	1	2	yes	—
91	Dic₁₇	—	4	1	no	—
92	Dic₁₇	—	16	2	no	—
93	C₂×#131	—	17	1	no	—
94	C₂×M₃₂	—	17	1	no	—
95	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
96	C₂×#108	C₂×D₁₇	1	2	yes	—
97	C₂×#107	—	17	2	no	—
98	#417	—	34	2	no	—
99	#367	—	34	2	no	—
100	C₂²⋊C₁₆	—	34	2	no	—
101	#343	—	34	2	no	—
102	C₂×C₁₇	—	2	2	no	—
103	C₂×C₁₇	#2075	1	1	yes	characteristic
104	C₂×C₁₇	#2223	1	2	yes	—
105	D₁₇	—	8	1	no	—
106	C₂×#41	—	17	1	no	—
107	#131	—	34	2	no	—
108	#131	—	34	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	17	1	no	—
111	C₂×M₁₆	—	34	1	no	—
112	C₂²×C₈	—	34	1	no	—
113	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	34	2	no	—
115	M₃₂	—	34	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	34	1	no	—
118	C₂×C₁₆	D₆₈	1	2	yes	—
119	#108	—	2	2	no	—
120	#108	—	2	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	—	34	2	no	—
127	#41	—	34	2	no	—
128	C₂×Q₈	—	17	1	no	—
129	C₂×D₄	—	17	1	no	—
130	C₂×D₄	—	34	1	no	—
131	C₂²×C₄	—	34	1	no	—
132	C₂²×C₄	—	68	2	no	—
133	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
134	M₁₆	—	4	2	no	—
135	M₁₆	—	4	2	no	—
136	M₁₆	—	34	2	no	—
137	M₁₆	—	68	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	34	1	no	—
140	C₂×C₈	—	68	1	no	—
141	C₂×C₈	—	68	1	no	—
142	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
143	C₂×C₈	#63252	1	2	yes	—
144	C₂×C₈	#63254	1	2	yes	—
145	C₂×C₈	#63256	1	1	yes	characteristic
146	C₄⋊C₄	—	68	2	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	68	1	no	—
150	C₂³	—	34	1	no	—
151	C₂³	#10587605	1	1	yes	characteristic
152	Q₈	—	34	2	no	—
153	D₄	—	34	2	no	—
154	D₄	—	68	2	no	—
155	C₂×C₄	—	272	2	no	—
156	C₂×C₄	—	34	1	no	—
157	C₂×C₄	—	68	1	no	—
158	C₂×C₄	—	68	1	no	—
159	C₂×C₄	—	136	2	no	—
160	C₂×C₄	#10587569	1	2	yes	—
161	C₂×C₄	#10587581	1	1	yes	characteristic
162	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
163	C₂×C₄	#10587601	1	2	yes	—
164	C₈	—	8	2	no	—
165	C₈	—	68	1	no	—
166	C₈	#10587561	1	1	yes	characteristic
167	C₈	#10587572	1	1	yes	characteristic
168	C₈	#10587584	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	68	2	no	—
171	C₂²	—	68	1	no	—
172	C₂²	#11763046	1	1	yes	characteristic
173	C₂²	#11763149	1	2	yes	—
174	C₄	—	272	2	no	—
175	C₄	—	68	1	no	—
176	C₄	#11762736	1	1	yes	characteristic
177	C₄	#11763068	1	1	yes	characteristic
178	C₄	#11763121	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	136	1	no	—
181	C₂	#14530209	1	1	yes	characteristic
182	C₂	#14530915	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762637

C₂

yes

maximal, characteristic, complemented

#14530397

C₂

yes

maximal, characteristic

#14528845

C₂

yes

maximal

#14526894

C₂

yes

maximal, complemented

C₁₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#10587587

C₂²

yes

characteristic

C₂×#10587483

C₂²

yes

characteristic

C₂×#10587489

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₂×#63252

D₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×#63223

D₄

yes

—

—

—

C₂×C₈×D₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

—

—

—

—

—

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₇

—

C₂²×Dic₁₇

—

#63252

—

#63252

—

C₂×D₆₈

D₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

D₈

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

C₂×#3622

—

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

#466

yes

characteristic

C₂²×C₁₇

#485

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

C₂×#108

C₂×D₁₇

yes

—

C₂×#107

—

#417

—

#367

—

100

C₂²⋊C₁₆

—

101

#343

—

102

C₂×C₁₇

—

103

C₂×C₁₇

#2075

yes

characteristic

104

C₂×C₁₇

#2223

yes

—

105

D₁₇

—

106

C₂×#41

—

107

#131

—

108

#131

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

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

D₆₈

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

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

D₄×D₁₇

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

C₂×D₆₈

yes

characteristic

143

C₂×C₈

#63252

yes

—

144

C₂×C₈

#63254

yes

—

145

C₂×C₈

#63256

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

#10587605

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

272

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

136

—

160

C₂×C₄

#10587569

yes

—

161

C₂×C₄

#10587581

yes

characteristic

162

C₂×C₄

D₈×D₁₇

yes

characteristic

163

C₂×C₄

#10587601

yes

—

164

C₈

—

165

C₈

—

166

C₈

#10587561

yes

characteristic

167

C₈

#10587572

yes

characteristic

168

C₈

#10587584

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#11763046

yes

characteristic

173

C₂²

#11763149

yes

—

174

C₄

—

272

—

175

C₄

—

176

C₄

#11762736

yes

characteristic

177

C₄

#11763068

yes

characteristic

178

C₄

#11763121

yes

—

179

C₂

—

180

C₂

—

136

—

181

C₂

#14530209

yes

characteristic

182

C₂

#14530915

yes

—

	subgroup	G/H
center	C₂²	#11763046
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1381	C₂
socle	C₂²×C₁₇	#466
prime radical	C₂×#63252	D₄
hypercenter	#1381	D₁₇
hypocenter	C₁₇	#13496
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#11763046

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1381

C₂

C₂²×C₁₇

D₄

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂×#11762637	C₂	semidirect
4. #14526894	C₂	semidirect
5. C₁₇×#1381	C₂	semidirect
82. #1381	D₁₇	semidirect
125. C₁₇	#13496	semidirect

H ≅ G/N

type

1. C₂×#11762637

C₂

semidirect

4. #14526894

C₂

semidirect

5. C₁₇×#1381

C₂

semidirect

82. #1381

D₁₇

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
#14530397	2
#14528845	2
#14526894	2
C₁₇×#1381	2
#13496	17

subgroup

index

C₁₇×#1381

Subgroups — #16954204

Conjugacy classes of subgroups

Complements

Maximal subgroups