Group #16949077 — subgroups

Subgroups — #16949077

Order 4352.

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

normal subgroups only

This group has 340 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	#14531033	C₂	1	2	yes	maximal, complemented
2	#14531007	C₂	1	2	yes	maximal, complemented
3	#14528376	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526702	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1712	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587497	—	2	4	no	—
7	#11763214	C₂²	1	1	yes	characteristic
8	#11763186	C₂²	1	1	yes	characteristic
9	#11763182	—	2	1	no	—
10	#11763156	—	2	1	no	—
11	#11762610	—	2	4	no	—
12	C₄×#63214	C₂²	1	2	yes	—
13	C₁₇×#493	—	2	1	no	—
14	C₁₇×#493	—	2	1	no	—
15	C₁₇×#493	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₇	—	2	2	no	—
18	C₂×D₈×C₁₇	—	2	2	no	—
19	C₂×D₈×C₁₇	—	4	2	no	—
20	C₂×D₈×C₁₇	—	4	2	no	—
21	C₁₇×#127	—	2	2	no	—
22	C₂×D₁₃₆	—	2	2	no	—
23	C₂×#63219	—	4	2	no	—
24	#10587560	—	2	1	no	—
25	#10587555	—	2	1	no	—
26	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	#10587497	—	4	8	no	—
30	#10587492	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	C₂×#63214	D₄	1	4	yes	—
33	C₂×D₄×C₁₇	—	4	2	no	—
34	C₂×D₄×C₁₇	—	4	2	no	—
35	C₂×D₄×C₁₇	—	4	2	no	—
36	C₂×D₆₈	—	4	1	no	—
37	C₂×D₆₈	—	4	1	no	—
38	C₂×D₆₈	—	4	1	no	—
39	C₂×Dic₃₄	—	4	1	no	—
40	D₈×C₁₇	—	4	4	no	—
41	D₈×C₁₇	—	4	4	no	—
42	D₈×C₁₇	—	8	4	no	—
43	D₈×C₁₇	—	8	4	no	—
44	C₂×C₈×C₁₇	D₈	1	2	yes	—
45	C₂×C₈×C₁₇	D₈	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₁₇	—	4	1	no	—
49	C₄⋊Dic₁₇	—	4	1	no	—
50	D₁₃₆	—	4	4	no	—
51	#63219	—	8	4	no	—
52	#63214	—	2	4	no	—
53	#12525	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	—	8	2	no	—
55	C₂²×D₁₇	—	8	1	no	—
56	D₄×C₁₇	—	8	2	no	—
57	D₄×C₁₇	—	8	2	no	—
58	D₄×C₁₇	—	8	4	no	—
59	D₄×C₁₇	—	8	4	no	—
60	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
63	C₂×Dic₁₇	—	8	1	no	—
64	D₆₈	—	8	2	no	—
65	D₆₈	—	8	2	no	—
66	D₆₈	—	8	2	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	D₁₇	1	1	yes	characteristic, complemented
74	#1706	—	17	1	no	—
75	#1312	—	17	1	no	—
76	C₂²×C₁₇	—	16	2	no	—
77	C₂²×C₁₇	—	16	2	no	—
78	C₂²×C₁₇	—	16	2	no	—
79	C₂²×C₁₇	#493	1	1	yes	characteristic
80	C₂×D₁₇	—	16	1	no	—
81	C₂×D₁₇	—	16	1	no	—
82	C₂×D₁₇	—	16	1	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	—	2	1	no	—
90	#493	—	2	1	no	—
91	#493	—	17	1	no	—
92	#493	C₂×D₁₇	1	2	yes	—
93	#492	—	34	1	no	—
94	#486	—	34	1	no	—
95	#357	—	34	4	no	—
96	C₄×C₁₆	—	17	2	no	—
97	C₈²	C₂×D₁₇	1	1	yes	characteristic
98	C₂×C₁₇	—	32	2	no	—
99	C₂×C₁₇	#1721	1	1	yes	characteristic
100	C₂×C₁₇	#2246	1	1	yes	characteristic
101	C₂×C₁₇	#2250	1	1	yes	characteristic
102	D₁₇	—	32	1	no	—
103	C₂×SD₁₆	—	68	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	4	2	no	—
107	C₂×D₈	—	4	2	no	—
108	C₂×D₈	—	34	2	no	—
109	C₄⋊Q₈	—	34	1	no	—
110	#127	—	2	2	no	—
111	#127	—	34	1	no	—
112	D₁₆	—	68	8	no	—
113	C₂×C₁₆	—	17	4	no	—
114	#107	—	34	2	no	—
115	#102	—	68	2	no	—
116	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₁₇	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	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	C₂×D₄	—	4	2	no	—
124	C₂×D₄	—	68	1	no	—
125	C₂×D₄	—	68	1	no	—
126	C₂×D₄	—	68	1	no	—
127	SD₁₆	—	136	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	4	4	no	—
130	D₈	—	8	4	no	—
131	D₈	—	8	4	no	—
132	D₈	—	68	4	no	—
133	C₂×C₈	#63228	1	2	yes	—
134	C₂×C₈	#63228	1	2	yes	—
135	C₂×C₈	D₄×D₁₇	1	2	yes	—
136	C₄⋊C₄	—	68	1	no	—
137	C₄⋊C₄	—	68	1	no	—
138	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
139	C₁₆	—	34	4	no	—
140	C₂³	—	8	2	no	—
141	C₂³	—	136	1	no	—
142	Q₈	—	136	2	no	—
143	D₄	—	8	4	no	—
144	D₄	—	8	4	no	—
145	D₄	—	8	2	no	—
146	D₄	—	8	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	C₂×#63228	1	1	yes	characteristic
152	C₂×C₄	C₂×#63228	1	1	yes	characteristic
153	C₂×C₄	#10587626	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	#10587497	1	4	yes	complemented
156	C₈	#10587597	1	4	yes	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	2	no	—
160	C₂²	—	272	1	no	—
161	C₂²	—	272	1	no	—
162	C₂²	—	272	1	no	—
163	C₂²	#11763185	1	1	yes	characteristic
164	C₄	—	272	1	no	—
165	C₄	#11763197	1	2	yes	—
166	C₄	C₂×#10587497	1	2	yes	—
167	C₄	#11763245	1	2	yes	—
168	C₂	—	32	2	no	—
169	C₂	—	544	1	no	—
170	C₂	#14528451	1	1	yes	characteristic
171	C₂	#14531033	1	1	yes	characteristic
172	C₂	#14531048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531033

C₂

yes

maximal, complemented

#14531007

C₂

yes

maximal, complemented

#14528376

C₂

yes

maximal, characteristic, complemented

#14526702

C₂

yes

maximal, characteristic, complemented

C₁₇×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#10587497

—

#11763214

C₂²

yes

characteristic

#11763186

C₂²

yes

—

—

—

C₂²

yes

—

C₁₇×#493

—

C₁₇×#493

—

C₁₇×#493

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₁₇×#127

—

C₂×D₁₃₆

—

C₂×#63219

—

#10587560

—

#10587555

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

D₈×C₁₇

—

D₈×C₁₇

—

D₈×C₁₇

—

D₈×C₁₇

—

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

—

—

—

—

C₂³×C₁₇

—

C₂²×D₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

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

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

D₁₆

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

#2246

—

#2240

—

#1712

D₁₇

yes

characteristic, complemented

#1706

—

#1312

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

#496

yes

—

C₄×C₁₇

C₂×D₁₆

yes

—

C₄×C₁₇

#509

yes

—

Dic₁₇

—

C₂×D₁₆

—

—

—

—

—

C₂×D₁₇

yes

—

#492

—

#486

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#1721

yes

characteristic

100

C₂×C₁₇

#2246

yes

characteristic

101

C₂×C₁₇

#2250

yes

characteristic

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

—

114

#107

—

115

#102

—

116

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

117

C₄×C₈

C₂²⋊D₁₇

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

—

132

D₈

—

133

C₂×C₈

#63228

yes

—

134

C₂×C₈

#63228

yes

—

135

C₂×C₈

D₄×D₁₇

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

139

C₁₆

—

140

C₂³

—

141

C₂³

—

136

—

142

Q₈

—

136

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

136

—

148

D₄

—

136

—

149

D₄

—

136

—

150

C₂×C₄

—

136

—

151

C₂×C₄

C₂×#63228

yes

characteristic

152

C₂×C₄

C₂×#63228

yes

characteristic

153

C₂×C₄

#10587626

yes

characteristic

154

C₈

—

155

C₈

#10587497

yes

complemented

156

C₈

#10587597

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

272

—

161

C₂²

—

272

—

162

C₂²

—

272

—

163

C₂²

#11763185

yes

characteristic

164

C₄

—

272

—

165

C₄

#11763197

yes

—

166

C₄

C₂×#10587497

yes

—

167

C₄

#11763245

yes

—

168

C₂

—

169

C₂

—

544

—

170

C₂

#14528451

yes

characteristic

171

C₂

#14531033

yes

characteristic

172

C₂

#14531048

yes

characteristic

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1712	C₂
socle	C₂²×C₁₇	#493
hypercenter	#1712	D₁₇
hypocenter	C₁₇	#12525
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1712

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #14531033	C₂	semidirect
2. #14531007	C₂	semidirect
3. #14528376	C₂	semidirect
4. #14526702	C₂	semidirect
5. C₁₇×#1712	C₂	semidirect
69. C₈×C₁₇	D₁₆	semidirect
73. #1712	D₁₇	semidirect
119. C₁₇	#12525	semidirect
155. C₈	#10587497	semidirect

H ≅ G/N

type

1. #14531033

C₂

semidirect

2. #14531007

C₂

semidirect

3. #14528376

C₂

semidirect

4. #14526702

C₂

semidirect

5. C₁₇×#1712

C₂

semidirect

69. C₈×C₁₇

D₁₆

semidirect

73. #1712

D₁₇

semidirect

119. C₁₇

#12525

semidirect

155. C₈

#10587497

semidirect

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

subgroup	index
#14531033	2
#14531007	2
#14528376	2
#14526702	2
C₁₇×#1712	2
#12525	17

subgroup

index

C₁₇×#1712

Subgroups — #16949077

Conjugacy classes of subgroups

Complements

Maximal subgroups