Group #16954220 — subgroups

Subgroups — #16954220

Order 4352.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	#14530399	C₂	1	1	yes	maximal, characteristic
2	#14528839	C₂	1	2	yes	maximal
3	#14526886	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526881	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587587	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	#11762847	C₂²	1	2	yes	—
9	#11762825	—	2	2	no	—
10	#11762588	—	2	2	no	—
11	#11762580	—	2	2	no	—
12	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#108	C₂²	1	2	yes	—
14	C₁₇×#368	—	2	2	no	—
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	#10587612	—	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	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
26	C₁₇×#108	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	#10587496	—	4	4	no	—
29	#10587489	—	2	2	no	—
30	#10587488	—	2	2	no	—
31	C₂×#63214	—	2	1	no	—
32	C₂×#63214	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₇	—	2	1	no	—
35	#63252	—	2	2	no	—
36	#63252	—	2	2	no	—
37	C₂×D₆₈	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₁₇	—	2	1	no	—
39	C₂×Dic₃₄	Q₁₆	1	1	yes	characteristic
40	M₁₆×C₁₇	—	4	2	no	—
41	M₁₆×C₁₇	—	4	2	no	—
42	C₂×C₈×C₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₇	#41	1	2	yes	—
47	C₂²⋊Dic₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄×Dic₁₇	—	4	2	no	—
50	#63223	—	2	2	no	—
51	C₂×#3622	—	2	1	no	—
52	#63218	—	4	2	no	—
53	C₈×D₁₇	—	4	1	no	—
54	C₈×D₁₇	—	4	1	no	—
55	C₁₆×C₁₇	—	2	2	no	—
56	#63214	—	4	1	no	—
57	#13499	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	#135	1	1	yes	characteristic
59	C₂²×D₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#123	1	2	yes	—
62	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₂²⋊D₁₇	—	4	2	no	—
65	C₂×Dic₁₇	—	2	1	no	—
66	C₂×Dic₁₇	—	8	2	no	—
67	D₆₈	—	2	2	no	—
68	C₄×D₁₇	—	4	1	no	—
69	C₄×D₁₇	—	4	1	no	—
70	Dic₃₄	—	2	2	no	—
71	C₈×C₁₇	—	8	2	no	—
72	C₈×C₁₇	#123	1	1	yes	characteristic
73	C₈×C₁₇	#124	1	1	yes	characteristic
74	C₈×C₁₇	#126	1	2	yes	—
75	#3622	—	4	1	no	—
76	#2161	D₁₇	1	1	yes	characteristic, complemented
77	#2113	—	17	1	no	—
78	#1814	—	17	2	no	—
79	#1384	—	17	1	no	—
80	#1382	—	17	2	no	—
81	C₂²×C₁₇	—	4	1	no	—
82	C₂²×C₁₇	#467	1	1	yes	characteristic
83	C₂²×C₁₇	#482	1	2	yes	—
84	C₂×D₁₇	—	4	2	no	—
85	C₂×D₁₇	—	4	1	no	—
86	C₄×C₁₇	#397	1	1	yes	characteristic
87	C₄×C₁₇	#469	1	1	yes	characteristic
88	C₄×C₁₇	#484	1	2	yes	—
89	Dic₁₇	—	4	1	no	—
90	Dic₁₇	—	16	2	no	—
91	C₂×#131	—	17	1	no	—
92	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
93	C₂²×C₁₆	—	17	1	no	—
94	C₂×#108	C₂×D₁₇	1	2	yes	—
95	#427	—	17	2	no	—
96	#421	—	34	2	no	—
97	#368	—	2	2	no	—
98	C₂²⋊C₁₆	—	34	2	no	—
99	#344	—	34	2	no	—
100	C₂×C₁₇	—	2	2	no	—
101	C₂×C₁₇	—	2	1	no	—
102	C₂×C₁₇	#2076	1	1	yes	characteristic
103	D₁₇	—	8	1	no	—
104	C₂×#41	—	17	1	no	—
105	#131	—	34	2	no	—
106	#131	—	34	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	17	1	no	—
109	C₂×M₁₆	—	34	1	no	—
110	C₂²×C₈	—	34	1	no	—
111	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
112	#117	—	34	2	no	—
113	M₃₂	—	2	2	no	—
114	C₂×C₁₆	—	34	2	no	—
115	C₂×C₁₆	—	34	1	no	—
116	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#107	—	34	2	no	—
120	#106	—	34	2	no	—
121	#104	—	68	4	no	—
122	C₁₇	#13499	1	1	yes	characteristic, complemented, Sylow
123	#41	—	34	2	no	—
124	#41	—	34	2	no	—
125	C₂×Q₈	—	17	1	no	—
126	C₂×D₄	—	17	1	no	—
127	C₂×D₄	—	34	1	no	—
128	C₂²×C₄	—	34	1	no	—
129	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	34	2	no	—
133	M₁₆	—	68	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	34	1	no	—
136	C₂×C₈	—	68	1	no	—
137	C₂×C₈	—	68	1	no	—
138	C₂×C₈	#63252	1	2	yes	—
139	C₂×C₈	#63254	1	1	yes	characteristic
140	C₂×C₈	#63256	1	2	yes	—
141	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
142	C₄⋊C₄	—	68	2	no	—
143	C₂²⋊C₄	—	68	2	no	—
144	C₄²	—	68	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	68	1	no	—
147	C₂³	—	34	1	no	—
148	C₂³	#10587605	1	1	yes	characteristic
149	Q₈	—	34	2	no	—
150	D₄	—	34	2	no	—
151	D₄	—	68	2	no	—
152	C₂×C₄	—	34	1	no	—
153	C₂×C₄	—	68	1	no	—
154	C₂×C₄	—	68	1	no	—
155	C₂×C₄	—	136	2	no	—
156	C₂×C₄	#10587580	1	2	yes	—
157	C₂×C₄	#10587597	1	2	yes	—
158	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
159	C₂×C₄	#10587624	1	1	yes	characteristic
160	C₈	—	8	2	no	—
161	C₈	—	68	1	no	—
162	C₈	#10587584	1	2	yes	—
163	C₈	#10587615	1	1	yes	characteristic
164	C₈	#10587633	1	1	yes	characteristic
165	C₂²	—	4	1	no	—
166	C₂²	—	68	2	no	—
167	C₂²	—	68	1	no	—
168	C₂²	#11763047	1	1	yes	characteristic
169	C₂²	#11763136	1	2	yes	—
170	C₄	—	272	2	no	—
171	C₄	—	68	1	no	—
172	C₄	#11762735	1	1	yes	characteristic
173	C₄	#11763059	1	1	yes	characteristic
174	C₄	#11763146	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	136	1	no	—
178	C₂	#14530213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530399

C₂

yes

maximal, characteristic

#14528839

C₂

yes

maximal

#14526886

C₂

yes

maximal, characteristic, complemented

#14526881

C₂

yes

maximal, complemented

C₁₇×#2161

C₂

yes

maximal, characteristic, complemented

C₂×#10587587

C₂²

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

#11762847

C₂²

yes

—

#11762825

—

#11762588

—

#11762580

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₁₇×#368

—

C₂×#63252

D₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

#10587612

—

C₂×#63223

D₄

yes

—

—

—

C₂×C₈×D₁₇

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#108

—

C₁₇×#108

—

—

—

—

—

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₇

—

#63252

—

#63252

—

C₂×D₆₈

Q₁₆

yes

characteristic

C₂×C₄×D₁₇

—

C₂×Dic₃₄

Q₁₆

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

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂²⋊Dic₁₇

—

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×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

#3622

—

#2161

D₁₇

yes

characteristic, complemented

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#467

yes

characteristic

C₂²×C₁₇

#482

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×#131

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₁₇

yes

—

#427

—

#421

—

#368

—

C₂²⋊C₁₆

—

#344

—

100

C₂×C₁₇

—

101

C₂×C₁₇

—

102

C₂×C₁₇

#2076

yes

characteristic

103

D₁₇

—

104

C₂×#41

—

105

#131

—

106

#131

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂²×D₁₇

yes

characteristic

112

#117

—

113

M₃₂

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

C₂²⋊D₁₇

yes

—

117

#108

—

118

#108

—

119

#107

—

120

#106

—

121

#104

—

122

C₁₇

#13499

yes

characteristic, complemented, Sylow

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

C₂²×C₄

D₄×D₁₇

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

#63252

yes

—

139

C₂×C₈

#63254

yes

characteristic

140

C₂×C₈

#63256

yes

—

141

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₁₆

—

146

C₁₆

—

147

C₂³

—

148

C₂³

#10587605

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

136

—

156

C₂×C₄

#10587580

yes

—

157

C₂×C₄

#10587597

yes

—

158

C₂×C₄

Q₁₆×D₁₇

yes

characteristic

159

C₂×C₄

#10587624

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

#10587584

yes

—

163

C₈

#10587615

yes

characteristic

164

C₈

#10587633

yes

characteristic

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#11763047

yes

characteristic

169

C₂²

#11763136

yes

—

170

C₄

—

272

—

171

C₄

—

172

C₄

#11762735

yes

characteristic

173

C₄

#11763059

yes

characteristic

174

C₄

#11763146

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

136

—

178

C₂

#14530213

yes

characteristic

	subgroup	G/H
center	C₄	#11763059
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2161	C₂
socle	C₂×C₁₇	#2076
prime radical	C₂×#63252	D₄
hypercenter	#2161	D₁₇
hypocenter	C₁₇	#13499
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#11763059

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2161

C₂

C₂×C₁₇

D₄

D₁₇

C₁₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
3. #14526886	C₂	semidirect
4. #14526881	C₂	semidirect
5. C₁₇×#2161	C₂	semidirect
76. #2161	D₁₇	semidirect
122. C₁₇	#13499	semidirect

H ≅ G/N

type

3. #14526886

C₂

semidirect

4. #14526881

C₂

semidirect

5. C₁₇×#2161

C₂

semidirect

76. #2161

D₁₇

semidirect

122. C₁₇

#13499

semidirect

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

subgroup	index
#14530399	2
#14528839	2
#14526886	2
#14526881	2
C₁₇×#2161	2
#13499	17

subgroup

index

C₁₇×#2161

Subgroups — #16954220

Conjugacy classes of subgroups

Complements

Maximal subgroups