Group #16949076 — subgroups

Subgroups — #16949076

Order 4352.

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

normal subgroups only

This group has 340 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	#14531034	C₂	1	2	yes	maximal, complemented
2	#14531008	C₂	1	2	yes	maximal, complemented
3	#14528399	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₇₂	—	2	4	no	—
7	#11763214	C₂²	1	1	yes	characteristic
8	#11763186	—	2	1	no	—
9	#11763186	—	2	1	no	—
10	#11763186	C₂²	1	1	yes	characteristic
11	#11763186	C₂²	1	2	yes	—
12	#11763182	—	2	1	no	—
13	#11763156	—	2	1	no	—
14	#11762611	—	2	4	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×D₁₃₆	—	2	2	no	—
18	C₂×D₁₃₆	—	2	2	no	—
19	C₂×D₁₃₆	—	2	2	no	—
20	C₂×D₁₃₆	—	4	2	no	—
21	C₂×D₁₃₆	—	4	2	no	—
22	C₂×#63219	—	4	2	no	—
23	#10587560	—	2	1	no	—
24	#10587560	—	2	2	no	—
25	#10587555	—	2	1	no	—
26	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	#10587492	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	D₂₇₂	—	4	8	no	—
33	C₂×D₆₈	—	4	1	no	—
34	C₂×D₆₈	—	4	2	no	—
35	C₂×D₆₈	—	4	1	no	—
36	C₂×D₆₈	—	4	1	no	—
37	C₂×D₆₈	—	4	2	no	—
38	C₂×D₆₈	—	4	2	no	—
39	C₂×Dic₃₄	—	4	1	no	—
40	C₂×C₈×C₁₇	D₈	1	2	yes	—
41	C₂×C₈×C₁₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
43	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₁₇	—	4	1	no	—
45	C₄⋊Dic₁₇	—	4	1	no	—
46	D₁₃₆	—	4	4	no	—
47	D₁₃₆	—	4	4	no	—
48	D₁₃₆	—	4	4	no	—
49	D₁₃₆	—	8	4	no	—
50	D₁₃₆	—	8	4	no	—
51	#63219	—	8	4	no	—
52	C₁₆×C₁₇	—	2	4	no	—
53	#12525	—	17	1	no	maximal, Sylow
54	C₂²×D₁₇	—	8	1	no	—
55	C₂²×D₁₇	—	8	2	no	—
56	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₇	—	8	1	no	—
60	D₆₈	—	8	2	no	—
61	D₆₈	—	8	2	no	—
62	D₆₈	—	8	2	no	—
63	D₆₈	—	8	2	no	—
64	D₆₈	—	8	4	no	—
65	D₆₈	—	8	2	no	—
66	D₆₈	—	8	4	no	—
67	Dic₃₄	—	8	2	no	—
68	C₈×C₁₇	—	2	2	no	—
69	C₈×C₁₇	D₁₆	1	4	yes	complemented
70	C₈×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
71	#2246	—	17	2	no	—
72	#2240	—	17	2	no	—
73	#1712	—	17	1	no	—
74	#1706	—	17	1	no	—
75	#1312	D₁₇	1	1	yes	characteristic, complemented
76	C₂²×C₁₇	#493	1	1	yes	characteristic
77	C₂×D₁₇	—	16	1	no	—
78	C₂×D₁₇	—	16	2	no	—
79	C₂×D₁₇	—	16	1	no	—
80	C₂×D₁₇	—	16	2	no	—
81	C₂×D₁₇	—	16	1	no	—
82	C₂×D₁₇	—	16	2	no	—
83	C₄×C₁₇	#496	1	2	yes	—
84	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
85	C₄×C₁₇	#509	1	2	yes	—
86	Dic₁₇	—	16	1	no	—
87	C₂×D₁₆	—	34	4	no	—
88	#500	—	17	1	no	—
89	#493	—	17	1	no	—
90	#493	—	17	2	no	—
91	#493	—	34	1	no	—
92	#493	—	34	1	no	—
93	#492	—	34	1	no	—
94	#486	—	34	1	no	—
95	#357	—	34	4	no	—
96	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
97	C₈²	C₂×D₁₇	1	1	yes	characteristic
98	C₂×C₁₇	#1721	1	1	yes	characteristic
99	C₂×C₁₇	#2246	1	1	yes	characteristic
100	C₂×C₁₇	#2250	1	1	yes	characteristic
101	D₁₇	—	32	1	no	—
102	D₁₇	—	32	2	no	—
103	C₂×SD₁₆	—	68	2	no	—
104	C₂×D₈	—	34	2	no	—
105	C₂×D₈	—	34	2	no	—
106	C₂×D₈	—	34	2	no	—
107	C₂×D₈	—	68	2	no	—
108	C₂×D₈	—	68	2	no	—
109	C₄⋊Q₈	—	34	1	no	—
110	#127	—	34	1	no	—
111	#127	—	34	2	no	—
112	D₁₆	—	68	8	no	—
113	C₂×C₁₆	D₆₈	1	4	yes	—
114	#107	—	34	2	no	—
115	#102	—	68	2	no	—
116	C₄×C₈	D₆₈	1	1	yes	characteristic
117	C₄×C₈	D₆₈	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
119	C₁₇	#12525	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₈	—	68	1	no	—
121	C₂×D₄	—	68	1	no	—
122	C₂×D₄	—	68	2	no	—
123	C₂×D₄	—	68	1	no	—
124	C₂×D₄	—	68	2	no	—
125	C₂×D₄	—	68	2	no	—
126	C₂×D₄	—	68	1	no	—
127	SD₁₆	—	136	4	no	—
128	D₈	—	68	4	no	—
129	D₈	—	68	4	no	—
130	D₈	—	68	4	no	—
131	D₈	—	136	4	no	—
132	D₈	—	136	4	no	—
133	C₂×C₈	D₁₃₆	1	2	yes	—
134	C₂×C₈	D₁₃₆	1	2	yes	—
135	C₂×C₈	C₂×D₆₈	1	2	yes	—
136	C₄⋊C₄	—	68	1	no	—
137	C₄⋊C₄	—	68	1	no	—
138	C₄²	C₂×D₆₈	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	136	1	no	—
141	C₂³	—	136	2	no	—
142	Q₈	—	136	2	no	—
143	D₄	—	136	4	no	—
144	D₄	—	136	2	no	—
145	D₄	—	136	4	no	—
146	D₄	—	136	2	no	—
147	D₄	—	136	2	no	—
148	D₄	—	136	2	no	—
149	D₄	—	136	2	no	—
150	C₂×C₄	—	136	1	no	—
151	C₂×C₄	#10587560	1	1	yes	characteristic
152	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
153	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	D₂₇₂	1	4	yes	complemented
156	C₈	#10587594	1	4	yes	—
157	C₂²	—	272	1	no	—
158	C₂²	—	272	2	no	—
159	C₂²	—	272	2	no	—
160	C₂²	—	272	1	no	—
161	C₂²	—	272	2	no	—
162	C₂²	—	272	1	no	—
163	C₂²	#11763186	1	1	yes	characteristic
164	C₄	—	272	1	no	—
165	C₄	#11763198	1	2	yes	—
166	C₄	C₂×D₂₇₂	1	2	yes	—
167	C₄	#11763247	1	2	yes	—
168	C₂	—	544	2	no	—
169	C₂	—	544	1	no	—
170	C₂	#14528450	1	1	yes	characteristic
171	C₂	#14531034	1	1	yes	characteristic
172	C₂	#14531050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531034

C₂

yes

maximal, complemented

#14531008

C₂

yes

maximal, complemented

#14528399

C₂

yes

maximal, characteristic, complemented

#14528376

C₂

yes

maximal, characteristic, complemented

C₁₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₂₇₂

—

#11763214

C₂²

yes

—

—

C₂²

yes

characteristic

#11763186

C₂²

yes

—

#11763182

—

#11763156

—

#11762611

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

—

—

—

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587492

—

#10587489

—

D₂₇₂

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

D₁₃₆

—

#63219

—

C₁₆×C₁₇

—

#12525

—

maximal, Sylow

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

D₁₆

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#496

yes

—

C₄×C₁₇

C₂×D₁₆

yes

—

C₄×C₁₇

#509

yes

—

Dic₁₇

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

100

C₂×C₁₇

#2250

yes

characteristic

101

D₁₇

—

102

D₁₇

—

103

C₂×SD₁₆

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

—

108

C₂×D₈

—

109

C₄⋊Q₈

—

110

#127

—

111

#127

—

112

D₁₆

—

113

C₂×C₁₆

D₆₈

yes

—

114

#107

—

115

#102

—

116

C₄×C₈

D₆₈

yes

characteristic

117

C₄×C₈

D₆₈

yes

characteristic

118

C₄×C₈

C₂²×D₁₇

yes

characteristic

119

C₁₇

#12525

yes

characteristic, complemented, Sylow

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

SD₁₆

—

136

—

128

D₈

—

129

D₈

—

130

D₈

—

131

D₈

—

136

—

132

D₈

—

136

—

133

C₂×C₈

D₁₃₆

yes

—

134

C₂×C₈

D₁₃₆

yes

—

135

C₂×C₈

C₂×D₆₈

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×D₆₈

yes

characteristic

139

C₁₆

—

140

C₂³

—

136

—

141

C₂³

—

136

—

142

Q₈

—

136

—

143

D₄

—

136

—

144

D₄

—

136

—

145

D₄

—

136

—

146

D₄

—

136

—

147

D₄

—

136

—

148

D₄

—

136

—

149

D₄

—

136

—

150

C₂×C₄

—

136

—

151

C₂×C₄

#10587560

yes

characteristic

152

C₂×C₄

C₂×D₁₃₆

yes

characteristic

153

C₂×C₄

C₂×D₁₃₆

yes

characteristic

154

C₈

—

155

C₈

D₂₇₂

yes

complemented

156

C₈

#10587594

yes

—

157

C₂²

—

272

—

158

C₂²

—

272

—

159

C₂²

—

272

—

160

C₂²

—

272

—

161

C₂²

—

272

—

162

C₂²

—

272

—

163

C₂²

#11763186

yes

characteristic

164

C₄

—

272

—

165

C₄

#11763198

yes

—

166

C₄

C₂×D₂₇₂

yes

—

167

C₄

#11763247

yes

—

168

C₂

—

544

—

169

C₂

—

544

—

170

C₂

#14528450

yes

characteristic

171

C₂

#14531034

yes

characteristic

172

C₂

#14531050

yes

characteristic

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1312	C₂
socle	C₂²×C₁₇	#493
hypercenter	#1312	D₁₇
hypocenter	C₁₇	#12525
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#11763186

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1312

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #14531034	C₂	semidirect
2. #14531008	C₂	semidirect
3. #14528399	C₂	semidirect
4. #14528376	C₂	semidirect
5. C₁₇×#1312	C₂	semidirect
69. C₈×C₁₇	D₁₆	semidirect
75. #1312	D₁₇	semidirect
119. C₁₇	#12525	semidirect
155. C₈	D₂₇₂	semidirect

H ≅ G/N

type

1. #14531034

C₂

semidirect

2. #14531008

C₂

semidirect

3. #14528399

C₂

semidirect

4. #14528376

C₂

semidirect

5. C₁₇×#1312

C₂

semidirect

69. C₈×C₁₇

D₁₆

semidirect

75. #1312

D₁₇

semidirect

119. C₁₇

#12525

semidirect

155. C₈

D₂₇₂

semidirect

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

subgroup	index
#14531034	2
#14531008	2
#14528399	2
#14528376	2
C₁₇×#1312	2
#12525	17

subgroup

index

C₁₇×#1312

Subgroups — #16949076

Conjugacy classes of subgroups

Complements

Maximal subgroups