Group #19145731 — subgroups

Subgroups — #19145731

Order 5504.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	#14583302	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583295	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583247	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583247	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583223	C₂	1	4	yes	maximal, complemented
7	#14583196	C₂	1	4	yes	maximal, complemented
8	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	—	2	1	no	—
11	#13063325	C₂²	1	1	yes	characteristic
12	#13063322	C₂²	1	1	yes	characteristic
13	#13063313	—	2	1	no	—
14	#13063299	—	2	1	no	—
15	#13063299	C₂²	1	2	yes	—
16	C₄×D₁₇₂	—	2	1	no	—
17	C₄×D₁₇₂	C₂²	1	2	yes	—
18	#13063258	C₂²	1	4	yes	—
19	#13063256	C₂²	1	4	yes	—
20	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
21	#13063245	C₂²	1	4	yes	—
22	#13063245	C₂²	1	4	yes	—
23	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
24	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	#125×C₄₃	C₂²	1	2	yes	—
29	C₂²⋊Q₈×C₄₃	—	2	1	no	—
30	C₂²⋊Q₈×C₄₃	—	2	1	no	—
31	C₄×Q₈×C₄₃	—	2	1	no	—
32	C₄×Q₈×C₄₃	—	2	1	no	—
33	#117×C₄₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₃	—	2	1	no	—
36	C₂×Q₈×C₄₃	—	2	1	no	—
37	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₄₃	—	2	2	no	—
41	#10621898	—	4	4	no	—
42	C₂×D₁₇₂	—	2	1	no	—
43	C₂×D₁₇₂	—	2	1	no	—
44	C₂×D₁₇₂	C₂³	1	2	yes	—
45	C₂×C₄×D₄₃	—	2	2	no	—
46	C₂×Dic₈₆	—	2	1	no	—
47	C₄⋊C₄×C₄₃	—	2	2	no	—
48	C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
53	C₂²⋊C₄×C₄₃	—	2	2	no	—
54	C₄²×C₄₃	—	2	2	no	—
55	C₄²×C₄₃	D₄	1	1	yes	characteristic
56	C₄²×C₄₃	D₄	1	1	yes	characteristic
57	C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	(C₂×D₄₃)⋊C₄	—	2	2	no	—
60	(C₂×D₄₃)⋊C₄	—	2	2	no	—
61	C₄⋊Dic₄₃	—	2	1	no	—
62	C₄⋊Dic₄₃	C₂³	1	2	yes	—
63	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
64	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
65	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×D₄₃	—	2	2	no	—
67	Q₈×C₄₃	—	4	2	no	—
68	Q₈×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	2	2	no	—
82	D₁₇₂	—	2	4	no	—
83	D₁₇₂	—	4	2	no	—
84	D₁₇₂	—	4	2	no	—
85	C₄×D₄₃	—	4	4	no	—
86	Dic₈₆	—	4	2	no	—
87	C₄₃⋊C₈	—	2	2	no	—
88	C₄₃⋊C₈	—	2	2	no	—
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	—	2	1	no	—
91	C₂²×C₄₃	—	2	1	no	—
92	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×D₄₃	—	4	2	no	—
94	C₂×D₄₃	—	4	2	no	—
95	C₂×D₄₃	—	4	2	no	—
96	C₄×C₄₃	—	2	1	no	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	2	1	no	—
99	C₄×C₄₃	—	4	2	no	—
100	C₄×C₄₃	—	4	2	no	—
101	C₄×C₄₃	#135	1	2	yes	—
102	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
104	Dic₄₃	—	4	2	no	—
105	#3228	—	43	1	no	maximal, Sylow
106	C₂×C₄₃	—	4	1	no	—
107	C₂×C₄₃	#536	1	1	yes	characteristic
108	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
109	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	D₄₃	—	8	2	no	—
111	#533	D₄₃	1	1	yes	characteristic, complemented
112	#532	—	43	1	no	—
113	#500	—	43	1	no	—
114	#498	—	43	1	no	—
115	#486	—	43	1	no	—
116	#486	—	43	1	no	—
117	#482	—	43	4	no	—
118	#478	—	43	4	no	—
119	#431	—	43	1	no	—
120	C₄₃	#3228	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	86	1	no	—
122	C₄⋊Q₈	—	43	1	no	—
123	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
125	#127	—	43	1	no	—
126	#125	C₂×D₄₃	1	2	yes	—
127	#124	—	86	1	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	—	2	1	no	—
130	#121	—	43	2	no	—
131	#121	—	86	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×D₄	—	43	2	no	—
135	C₄×D₄	—	86	1	no	—
136	#117	C₂×D₄₃	1	2	yes	—
137	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
138	#107	—	43	4	no	—
139	#106	—	43	4	no	—
140	C₄⋊C₈	—	43	2	no	—
141	#102	—	43	4	no	—
142	#102	—	43	4	no	—
143	C₂²⋊C₈	—	43	2	no	—
144	C₄×C₈	—	43	1	no	—
145	C₄×C₈	—	43	1	no	—
146	#41	—	172	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	86	1	no	—
150	C₂×D₄	—	43	2	no	—
151	C₂×D₄	—	86	2	no	—
152	C₂×D₄	—	86	1	no	—
153	C₂×D₄	—	86	1	no	—
154	C₂²×C₄	—	86	2	no	—
155	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
158	C₂×C₈	—	43	2	no	—
159	C₂×C₈	—	43	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	43	2	no	—
165	C₄⋊C₄	—	86	1	no	—
166	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
167	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	86	2	no	—
170	C₂²⋊C₄	—	86	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
173	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
174	C₄²	C₂²×D₄₃	1	1	yes	characteristic
175	C₄²	C₂²×D₄₃	1	1	yes	characteristic
176	C₂³	—	86	2	no	—
177	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	Q₈	—	4	2	no	—
180	Q₈	—	172	2	no	—
181	D₄	—	86	4	no	—
182	D₄	—	172	4	no	—
183	D₄	—	172	2	no	—
184	D₄	—	172	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	86	2	no	—
191	C₂×C₄	—	172	4	no	—
192	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
198	C₈	—	86	2	no	—
199	C₈	—	86	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	172	2	no	—
204	C₂²	—	172	2	no	—
205	C₂²	—	172	2	no	—
206	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	172	2	no	—
213	C₄	#13063345	1	2	yes	—
214	C₄	#13063350	1	2	yes	—
215	C₄	#13063373	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	344	2	no	—
218	C₂	#14583473	1	1	yes	characteristic
219	C₂	C₂×#13063345	1	1	yes	characteristic
220	C₂	C₂×#13063350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14583302

C₂

yes

maximal, characteristic, complemented

#14583295

C₂

yes

maximal, characteristic, complemented

#14583247

C₂

yes

maximal, characteristic, complemented

#14583247

C₂

yes

maximal, characteristic, complemented

#14583223

C₂

yes

maximal, complemented

#14583196

C₂

yes

maximal, complemented

#14582952

C₂

yes

maximal, characteristic, complemented

C₄₃×#533

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

—

—

C₂²

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

—

#13063258

C₂²

yes

—

#13063256

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063245

C₂²

yes

—

#13063245

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#125×C₄₃

C₂²

yes

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

#135

yes

—

102

C₄×C₄₃

AGL(1,ℤ/8)

yes

—

103

C₄×C₄₃

2₋¹⁺⁴

yes

—

104

Dic₄₃

—

105

#3228

—

maximal, Sylow

106

C₂×C₄₃

—

107

C₂×C₄₃

#536

yes

characteristic

108

C₂×C₄₃

C₂×#135

yes

characteristic

109

C₂×C₄₃

C₂×AGL(1,ℤ/8)

yes

characteristic

110

D₄₃

—

111

#533

D₄₃

yes

characteristic, complemented

112

#532

—

113

#500

—

114

#498

—

115

#486

—

116

#486

—

117

#482

—

118

#478

—

119

#431

—

120

C₄₃

#3228

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

124

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

125

#127

—

126

#125

C₂×D₄₃

yes

—

127

#124

—

128

C₂²⋊Q₈

—

129

C₂²⋊Q₈

—

130

#121

—

131

#121

—

132

C₄×Q₈

—

133

C₄×Q₈

—

134

C₄×D₄

—

135

C₄×D₄

—

136

#117

C₂×D₄₃

yes

—

137

C₂×C₄²

C₂×D₄₃

yes

characteristic

138

#107

—

139

#106

—

140

C₄⋊C₈

—

141

#102

—

142

#102

—

143

C₂²⋊C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

#41

—

172

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

156

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

157

C₂²×C₄

C₂²×D₄₃

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

C₂²×D₄₃

yes

—

167

C₄⋊C₄

C₂²×D₄₃

yes

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

C₂²⋊D₄₃

yes

characteristic

173

C₄²

C₂²⋊D₄₃

yes

characteristic

174

C₄²

C₂²×D₄₃

yes

characteristic

175

C₄²

C₂²×D₄₃

yes

characteristic

176

C₂³

—

177

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

178

Q₈

—

179

Q₈

—

180

Q₈

—

172

—

181

D₄

—

182

D₄

—

172

—

183

D₄

—

172

—

184

D₄

—

172

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

172

—

192

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

195

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

197

C₂×C₄

C₂³×D₄₃

yes

characteristic

198

C₈

—

199

C₈

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

172

—

204

C₂²

—

172

—

205

C₂²

—

172

—

206

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

172

—

213

C₄

#13063345

yes

—

214

C₄

#13063350

yes

—

215

C₄

#13063373

yes

—

216

C₂

—

217

C₂

—

344

—

218

C₂

#14583473

yes

characteristic

219

C₂

C₂×#13063345

yes

characteristic

220

C₂

C₂×#13063350

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#533	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#533	D₄₃
hypocenter	C₄₃	#3228
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#533

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14583302	C₂	semidirect
3. #14583295	C₂	semidirect
4. #14583247	C₂	semidirect
5. #14583247	C₂	semidirect
6. #14583223	C₂	semidirect
7. #14583196	C₂	semidirect
8. #14582952	C₂	semidirect
9. C₄₃×#533	C₂	semidirect
111. #533	D₄₃	semidirect
120. C₄₃	#3228	semidirect

H ≅ G/N

type

2. #14583302

C₂

semidirect

3. #14583295

C₂

semidirect

4. #14583247

C₂

semidirect

5. #14583247

C₂

semidirect

6. #14583223

C₂

semidirect

7. #14583196

C₂

semidirect

8. #14582952

C₂

semidirect

9. C₄₃×#533

C₂

semidirect

111. #533

D₄₃

semidirect

120. C₄₃

#3228

semidirect

Subgroups — #19145731

Conjugacy classes of subgroups

Complements