Group #35323781 — subgroups

Subgroups — #35323781

Order 21248.

540 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19127514	C₂	1	2	yes	maximal, complemented
2	#24909179	C₂	1	2	yes	maximal, complemented
3	#24908211	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905545	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂×#14581728	C₂²	1	1	yes	characteristic
8	#19127739	C₂²	1	1	yes	characteristic
9	#19127727	—	2	1	no	—
10	#19127514	—	2	4	no	—
11	#19127502	—	2	4	no	—
12	#19127461	—	2	1	no	—
13	C₂²×D₈×C₈₃	—	2	1	no	—
14	C₂²×D₈×C₈₃	C₂²	1	2	yes	—
15	C₂×#127×C₈₃	—	2	1	no	—
16	#493×C₈₃	—	2	4	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050574	—	2	1	no	—
19	#14581751	—	2	1	no	—
20	#14581732	—	2	2	no	—
21	#14581732	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	2	no	—
23	#14581728	—	2	2	no	—
24	#14581726	D₄	1	2	yes	complemented
25	#14581724	—	2	1	no	—
26	Dic₈₃⋊C₈	—	4	2	no	—
27	C₂²×D₄×C₈₃	—	2	2	no	—
28	C₂²×D₄×C₈₃	—	4	2	no	—
29	C₂×D₈×C₈₃	—	2	4	no	—
30	C₂×D₈×C₈₃	—	2	2	no	—
31	C₂×D₈×C₈₃	—	2	2	no	—
32	C₂×D₈×C₈₃	—	2	2	no	—
33	C₂×D₈×C₈₃	—	2	2	no	—
34	C₂×D₈×C₈₃	—	2	2	no	—
35	C₂×D₈×C₈₃	—	4	2	no	—
36	C₂×D₈×C₈₃	—	4	2	no	—
37	C₂×D₈×C₈₃	D₄	1	4	yes	—
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	#127×C₈₃	—	2	4	no	—
41	#127×C₈₃	—	4	2	no	—
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	—	2	2	no	—
44	#13050574	—	4	1	no	—
45	#13050574	—	4	1	no	—
46	C₂×C₈₃⋊C₈	—	4	1	no	—
47	C₄⋊Dic₈₃	—	2	2	no	—
48	C₂²⋊Dic₈₃	—	4	1	no	—
49	C₄×Dic₈₃	—	4	1	no	—
50	C₈₃⋊C₁₆	—	4	2	no	—
51	C₂⁴×C₈₃	—	4	2	no	—
52	C₂×D₄×C₈₃	—	2	4	no	—
53	C₂×D₄×C₈₃	—	4	8	no	—
54	C₂×D₄×C₈₃	—	4	2	no	—
55	C₂×D₄×C₈₃	—	4	2	no	—
56	C₂×D₄×C₈₃	—	4	2	no	—
57	C₂×D₄×C₈₃	—	4	2	no	—
58	C₂×D₄×C₈₃	—	4	2	no	—
59	C₂×D₄×C₈₃	—	8	8	no	—
60	C₂²×C₄×C₈₃	—	2	1	no	—
61	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	D₈×C₈₃	—	2	4	no	—
63	D₈×C₈₃	—	2	4	no	—
64	D₈×C₈₃	—	4	8	no	—
65	D₈×C₈₃	—	4	2	no	—
66	D₈×C₈₃	—	4	2	no	—
67	C₂×C₈×C₈₃	—	2	1	no	—
68	C₂×C₈×C₈₃	—	2	1	no	—
69	C₂×C₈×C₈₃	D₈	1	2	yes	—
70	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₈₃	#41	1	2	yes	—
73	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
74	C₄²×C₈₃	—	2	1	no	—
75	C₄²×C₈₃	SD₁₆	1	2	yes	—
76	C₂³×C₈₃	—	4	4	no	—
77	C₂³×C₈₃	—	8	8	no	—
78	C₂³×C₈₃	—	8	2	no	—
79	C₂³×C₈₃	—	8	2	no	—
80	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	D₄×C₈₃	—	4	4	no	—
82	D₄×C₈₃	—	4	4	no	—
83	D₄×C₈₃	—	8	16	no	—
84	D₄×C₈₃	—	8	2	no	—
85	D₄×C₈₃	—	8	2	no	—
86	C₂×C₄×C₈₃	—	2	4	no	—
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₈₃	—	4	2	no	—
95	C₈×C₈₃	—	2	2	no	—
96	C₈×C₈₃	#121	1	2	yes	—
97	C₈×C₈₃	#123	1	2	yes	—
98	C₈₃⋊C₈	—	8	1	no	—
99	C₂²×C₈₃	—	2	2	no	—
100	C₂²×C₈₃	—	8	8	no	—
101	C₂²×C₈₃	—	8	4	no	—
102	C₂²×C₈₃	—	16	4	no	—
103	C₂²×C₈₃	—	16	4	no	—
104	C₂²×C₈₃	#450	1	1	yes	characteristic
105	C₂²×C₈₃	#459	1	1	yes	characteristic
106	C₂²×C₈₃	#490	1	1	yes	characteristic
107	C₄×C₈₃	—	2	4	no	—
108	C₄×C₈₃	#394	1	1	yes	characteristic
109	C₄×C₈₃	#447	1	1	yes	characteristic
110	C₄×C₈₃	#460	1	1	yes	characteristic
111	C₄×C₈₃	#486	1	1	yes	characteristic
112	Dic₈₃	—	8	2	no	—
113	#13413	—	83	1	no	maximal, Sylow
114	C₂×C₈₃	—	2	1	no	—
115	C₂×C₈₃	—	2	1	no	—
116	C₂×C₈₃	—	16	4	no	—
117	C₂×C₈₃	#2041	1	1	yes	characteristic
118	C₂×C₈₃	#2213	1	2	yes	—
119	C₂×#493	D₈₃	1	1	yes	characteristic
120	C₂×#361	—	83	2	no	—
121	#2140	—	83	2	no	—
122	#1945	—	83	1	no	—
123	#1376	—	83	1	no	—
124	C₈₃	#13413	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₈	—	2	1	no	—
126	C₂²×D₈	C₂×D₈₃	1	2	yes	—
127	C₂×#127	—	2	1	no	—
128	C₂×M₃₂	—	83	2	no	—
129	#493	—	2	4	no	—
130	C₂×#108	—	83	1	no	—
131	#427	—	83	1	no	—
132	#424	—	166	1	no	—
133	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
134	#361	—	166	4	no	—
135	#357	—	166	4	no	—
136	#339	—	166	1	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	2	no	—
146	C₂×D₈	—	4	2	no	—
147	C₂×D₈	C₂²⋊D₈₃	1	4	yes	—
148	C₂×M₁₆	—	166	1	no	—
149	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	#117	—	166	1	no	—
154	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
155	M₃₂	—	166	2	no	—
156	M₃₂	—	166	2	no	—
157	C₂×C₁₆	—	166	2	no	—
158	#108	—	166	2	no	—
159	#107	—	83	2	no	—
160	#106	—	166	1	no	—
161	C₄⋊C₈	—	332	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂⁴	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	8	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	8	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
174	D₈	—	2	4	no	—
175	D₈	—	2	4	no	—
176	D₈	—	4	8	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	M₁₆	—	332	1	no	—
180	M₁₆	—	332	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	332	1	no	—
184	C₂×C₈	#13050575	1	2	yes	—
185	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
187	C₂×C₈	#13050589	1	2	yes	—
188	C₂×C₈	#13050589	1	1	yes	characteristic
189	C₄⋊C₄	—	166	2	no	—
190	C₂²⋊C₄	—	332	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	332	1	no	—
193	C₄²	#13050578	1	2	yes	—
194	C₁₆	—	332	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	8	8	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	2	no	—
199	C₂³	#14581818	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	16	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	332	2	no	—
208	C₂×C₄	#14581761	1	1	yes	characteristic
209	C₂×C₄	#14581763	1	1	yes	characteristic
210	C₂×C₄	#14581779	1	1	yes	characteristic
211	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
212	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
213	C₂×C₄	#14581818	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	664	1	no	—
216	C₈	#14581764	1	2	yes	—
217	C₈	#14581773	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	8	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	—	16	4	no	—
223	C₂²	#19127852	1	1	yes	characteristic
224	C₂²	#19127893	1	1	yes	characteristic
225	C₂²	#19128065	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	664	2	no	—
228	C₄	#19127615	1	1	yes	characteristic
229	C₄	#19127835	1	1	yes	characteristic
230	C₄	#19127898	1	1	yes	characteristic
231	C₄	#19128047	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	16	4	no	—
235	C₂	#24908694	1	1	yes	characteristic
236	C₂	#24909536	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127514

C₂

yes

maximal, complemented

#24909179

C₂

yes

maximal, complemented

#24908211

C₂

yes

maximal, characteristic, complemented

#24905545

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×#14581728

C₂²

yes

characteristic

#19127739

C₂²

yes

—

—

—

—

C₂²×D₈×C₈₃

—

C₂²×D₈×C₈₃

C₂²

yes

—

C₂×#127×C₈₃

—

#493×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

—

—

—

C₂×C₈₃⋊C₁₆

—

#14581728

—

#14581726

D₄

yes

complemented

#14581724

—

Dic₈₃⋊C₈

—

C₂²×D₄×C₈₃

—

C₂²×D₄×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#127×C₈₃

—

#127×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂⁴×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

D₈×C₈₃

—

D₈×C₈₃

—

D₈×C₈₃

—

D₈×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

AGL(1,ℤ/8)

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#121

yes

—

C₈×C₈₃

#123

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

100

C₂²×C₈₃

—

101

C₂²×C₈₃

—

102

C₂²×C₈₃

—

103

C₂²×C₈₃

—

104

C₂²×C₈₃

#450

yes

characteristic

105

C₂²×C₈₃

#459

yes

characteristic

106

C₂²×C₈₃

#490

yes

characteristic

107

C₄×C₈₃

—

108

C₄×C₈₃

#394

yes

characteristic

109

C₄×C₈₃

#447

yes

characteristic

110

C₄×C₈₃

#460

yes

characteristic

111

C₄×C₈₃

#486

yes

characteristic

112

Dic₈₃

—

113

#13413

—

maximal, Sylow

114

C₂×C₈₃

—

115

C₂×C₈₃

—

116

C₂×C₈₃

—

117

C₂×C₈₃

#2041

yes

characteristic

118

C₂×C₈₃

#2213

yes

—

119

C₂×#493

D₈₃

yes

characteristic

120

C₂×#361

—

121

#2140

—

122

#1945

—

123

#1376

—

124

C₈₃

#13413

yes

characteristic, complemented, Sylow

125

C₂²×D₈

—

126

C₂²×D₈

C₂×D₈₃

yes

—

127

C₂×#127

—

128

C₂×M₃₂

—

129

#493

—

130

C₂×#108

—

131

#427

—

132

#424

—

166

—

133

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

134

#361

—

166

—

135

#357

—

166

—

136

#339

—

166

—

137

C₂²×D₄

—

138

C₂²×D₄

—

139

C₂×D₈

—

140

C₂×D₈

—

141

C₂×D₈

—

142

C₂×D₈

—

143

C₂×D₈

—

144

C₂×D₈

—

145

C₂×D₈

—

146

C₂×D₈

—

147

C₂×D₈

C₂²⋊D₈₃

yes

—

148

C₂×M₁₆

—

166

—

149

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

150

C₂²×C₈

C₂²×D₈₃

yes

characteristic

151

#127

—

152

#127

—

153

#117

—

166

—

154

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

155

M₃₂

—

166

—

156

M₃₂

—

166

—

157

C₂×C₁₆

—

166

—

158

#108

—

166

—

159

#107

—

160

#106

—

166

—

161

C₄⋊C₈

—

332

—

162

C₄×C₈

—

163

C₂⁴

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

174

D₈

—

175

D₈

—

176

D₈

—

177

D₈

—

178

D₈

—

179

M₁₆

—

332

—

180

M₁₆

—

332

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

332

—

184

C₂×C₈

#13050575

yes

—

185

C₂×C₈

D₄×D₈₃

yes

characteristic

186

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

187

C₂×C₈

#13050589

yes

—

188

C₂×C₈

#13050589

yes

characteristic

189

C₄⋊C₄

—

166

—

190

C₂²⋊C₄

—

332

—

191

C₄²

—

192

C₄²

—

332

—

193

C₄²

#13050578

yes

—

194

C₁₆

—

332

—

195

C₂³

—

196

C₂³

—

197

C₂³

—

198

C₂³

—

199

C₂³

#14581818

yes

characteristic

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

D₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

332

—

208

C₂×C₄

#14581761

yes

characteristic

209

C₂×C₄

#14581763

yes

characteristic

210

C₂×C₄

#14581779

yes

characteristic

211

C₂×C₄

C₂×#13050575

yes

characteristic

212

C₂×C₄

C₂×#13050578

yes

characteristic

213

C₂×C₄

#14581818

yes

characteristic

214

C₈

—

215

C₈

—

664

—

216

C₈

#14581764

yes

—

217

C₈

#14581773

yes

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#19127852

yes

characteristic

224

C₂²

#19127893

yes

characteristic

225

C₂²

#19128065

yes

characteristic

226

C₄

—

227

C₄

—

664

—

228

C₄

#19127615

yes

characteristic

229

C₄

#19127835

yes

characteristic

230

C₄

#19127898

yes

characteristic

231

C₄

#19128047

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

235

C₂

#24908694

yes

characteristic

236

C₂

#24909536

yes

—

	subgroup	G/H
center	C₂²	#19127893
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting, prime radical	C₂×#493×C₈₃	C₂
socle	C₂²×C₈₃	#459
hypercenter	C₂×#493	D₈₃
hypocenter	C₈₃	#13413
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

#19127893

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting, prime radical

C₂×#493×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. C₂×#19127514	C₂	semidirect
2. #24909179	C₂	semidirect
3. #24908211	C₂	semidirect
4. #24905545	C₂	semidirect
24. #14581726	D₄	semidirect
124. C₈₃	#13413	semidirect

H ≅ G/N

type

1. C₂×#19127514

C₂

semidirect

2. #24909179

C₂

semidirect

3. #24908211

C₂

semidirect

4. #24905545

C₂

semidirect

24. #14581726

D₄

semidirect

124. C₈₃

#13413

semidirect

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

subgroup	index
C₂×#19127514	2
#24909179	2
#24908211	2
#24905545	2
C₂×#493×C₈₃	2
#13413	83

subgroup

index

C₂×#493×C₈₃

Subgroups — #35323781

Conjugacy classes of subgroups

Complements

Maximal subgroups