Group #22581635 — subgroups

Subgroups — #22581635

Order 8896.

214 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 95 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	#17984837	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984837	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984832	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984832	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984827	C₂	1	1	yes	maximal, characteristic
6	#17984827	C₂	1	1	yes	maximal, characteristic
7	#17984820	C₂	1	1	yes	maximal, characteristic, complemented
8	#17984820	C₂	1	1	yes	maximal, characteristic, complemented
9	#17984807	C₂	1	1	yes	maximal, characteristic, complemented
10	#17984804	C₂	1	1	yes	maximal, characteristic
11	#17984800	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984799	C₂	1	1	yes	maximal, characteristic, complemented
13	#17984799	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₄×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
17	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
24	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
25	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
26	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
27	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
28	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
32	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
33	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
34	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
35	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
36	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
37	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
38	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
39	(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
40	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
41	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
42	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
43	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
44	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
45	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
48	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
50	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
51	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
52	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
53	Dic₂₇₈	—	2	2	no	—
54	D₅₅₆	—	2	2	no	—
55	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
61	C₄×D₁₃₉	—	2	2	no	—
62	C₄×D₁₃₉	—	2	2	no	—
63	C₄×D₁₃₉	—	2	2	no	—
64	C₄×D₁₃₉	—	2	2	no	—
65	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
72	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
73	C₂×D₁₃₉	—	2	1	no	—
74	C₂×D₁₃₉	—	2	1	no	—
75	C₂×D₁₃₉	—	2	1	no	—
76	C₂×D₁₃₉	—	2	1	no	—
77	C₂×D₁₃₉	—	2	1	no	—
78	C₂×D₁₃₉	#41	1	2	yes	—
79	C₄×C₁₃₉	—	2	1	no	—
80	C₄×C₁₃₉	—	2	1	no	—
81	C₄×C₁₃₉	—	2	1	no	—
82	C₄×C₁₃₉	—	2	1	no	—
83	C₄×C₁₃₉	—	2	1	no	—
84	C₄×C₁₃₉	—	2	1	no	—
85	C₄×C₁₃₉	—	2	1	no	—
86	Dic₁₃₉	—	2	1	no	—
87	Dic₁₃₉	—	2	1	no	—
88	Dic₁₃₉	—	2	1	no	—
89	Dic₁₃₉	—	2	1	no	—
90	Dic₁₃₉	—	2	1	no	—
91	Dic₁₃₉	#41	1	2	yes	—
92	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
93	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
94	C₂×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
95	D₁₃₉	—	2	2	no	—
96	D₁₃₉	—	4	1	no	—
97	C₁₃₉	#552	1	1	yes	characteristic, complemented, Sylow
98	#552	—	139	1	no	maximal, Sylow
99	#126	—	139	1	no	—
100	#126	—	139	1	no	—
101	#125	—	139	1	no	—
102	#125	—	139	1	no	—
103	#125	D₁₃₉	1	1	yes	characteristic, complemented
104	#123	—	139	1	no	—
105	#123	—	139	1	no	—
106	C₂²⋊Q₈	—	139	1	no	—
107	C₂²⋊Q₈	—	139	1	no	—
108	C₄×Q₈	—	139	1	no	—
109	C₄×D₄	—	139	1	no	—
110	#117	—	139	1	no	—
111	#117	—	139	1	no	—
112	#117	—	139	1	no	—
113	C₂×C₄⋊C₄	—	139	1	no	—
114	C₂×Q₈	—	139	1	no	—
115	C₂×D₄	—	139	1	no	—
116	C₂²×C₄	—	139	1	no	—
117	C₂²×C₄	—	139	1	no	—
118	C₂²×C₄	—	139	1	no	—
119	C₂²×C₄	—	139	1	no	—
120	C₄⋊C₄	—	139	1	no	—
121	C₄⋊C₄	—	139	1	no	—
122	C₄⋊C₄	—	139	1	no	—
123	C₄⋊C₄	—	139	1	no	—
124	C₄⋊C₄	—	139	1	no	—
125	C₄⋊C₄	—	139	1	no	—
126	C₄⋊C₄	—	139	1	no	—
127	C₄⋊C₄	—	139	1	no	—
128	C₄⋊C₄	—	139	1	no	—
129	C₄⋊C₄	—	139	1	no	—
130	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
131	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
132	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
133	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
134	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
135	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
136	C₂²⋊C₄	—	139	1	no	—
137	C₂²⋊C₄	—	139	1	no	—
138	C₂²⋊C₄	—	139	1	no	—
139	C₂²⋊C₄	—	139	1	no	—
140	C₂²⋊C₄	—	139	1	no	—
141	C₂²⋊C₄	—	139	1	no	—
142	C₂²⋊C₄	—	139	1	no	—
143	C₂²⋊C₄	—	139	1	no	—
144	C₄²	—	139	1	no	—
145	C₄²	—	139	1	no	—
146	C₄²	—	139	1	no	—
147	C₄²	—	139	1	no	—
148	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
149	C₂³	—	139	1	no	—
150	C₂³	—	139	1	no	—
151	Q₈	—	278	2	no	—
152	D₄	—	278	2	no	—
153	C₂×C₄	—	278	2	no	—
154	C₂×C₄	—	278	2	no	—
155	C₂×C₄	—	278	2	no	—
156	C₂×C₄	—	278	2	no	—
157	C₂×C₄	—	139	1	no	—
158	C₂×C₄	—	139	1	no	—
159	C₂×C₄	—	139	1	no	—
160	C₂×C₄	—	139	1	no	—
161	C₂×C₄	—	139	1	no	—
162	C₂×C₄	—	139	1	no	—
163	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
165	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
168	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
170	C₂²	—	278	1	no	—
171	C₂²	—	278	1	no	—
172	C₂²	—	278	1	no	—
173	C₂²	—	278	1	no	—
174	C₂²	—	278	1	no	—
175	C₂²	—	139	2	no	—
176	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	278	1	no	—
185	C₄	—	278	1	no	—
186	C₄	—	278	1	no	—
187	C₄	—	278	1	no	—
188	C₄	—	278	1	no	—
189	C₄	—	139	2	no	—
190	C₂	—	278	2	no	—
191	C₂	—	556	1	no	—
192	C₂	#41×D₁₃₉	1	1	yes	characteristic
193	C₂	#41×D₁₃₉	1	1	yes	characteristic
194	C₂	#17984890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984837

C₂

yes

maximal, characteristic, complemented

#17984837

C₂

yes

maximal, characteristic, complemented

#17984832

C₂

yes

maximal, characteristic, complemented

#17984832

C₂

yes

maximal, characteristic, complemented

#17984827

C₂

yes

maximal, characteristic

#17984827

C₂

yes

maximal, characteristic

#17984820

C₂

yes

maximal, characteristic, complemented

#17984820

C₂

yes

maximal, characteristic, complemented

#17984807

C₂

yes

maximal, characteristic, complemented

#17984804

C₂

yes

maximal, characteristic

#17984800

C₂

yes

maximal, characteristic, complemented

#17984799

C₂

yes

maximal, characteristic, complemented

#17984799

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₃₉

C₂

yes

maximal, characteristic, complemented

#125×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

D₅₅₆

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#41

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

#41

yes

—

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

C₂×C₁₃₉

2₋¹⁺⁴

yes

characteristic

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#552

yes

characteristic, complemented, Sylow

#552

—

139

maximal, Sylow

#126

—

139

—

100

#126

—

139

—

101

#125

—

139

—

102

#125

—

139

—

103

#125

D₁₃₉

yes

characteristic, complemented

104

#123

—

139

—

105

#123

—

139

—

106

C₂²⋊Q₈

—

139

—

107

C₂²⋊Q₈

—

139

—

108

C₄×Q₈

—

139

—

109

C₄×D₄

—

139

—

110

#117

—

139

—

111

#117

—

139

—

112

#117

—

139

—

113

C₂×C₄⋊C₄

—

139

—

114

C₂×Q₈

—

139

—

115

C₂×D₄

—

139

—

116

C₂²×C₄

—

139

—

117

C₂²×C₄

—

139

—

118

C₂²×C₄

—

139

—

119

C₂²×C₄

—

139

—

120

C₄⋊C₄

—

139

—

121

C₄⋊C₄

—

139

—

122

C₄⋊C₄

—

139

—

123

C₄⋊C₄

—

139

—

124

C₄⋊C₄

—

139

—

125

C₄⋊C₄

—

139

—

126

C₄⋊C₄

—

139

—

127

C₄⋊C₄

—

139

—

128

C₄⋊C₄

—

139

—

129

C₄⋊C₄

—

139

—

130

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

131

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

132

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

133

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

134

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

135

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

136

C₂²⋊C₄

—

139

—

137

C₂²⋊C₄

—

139

—

138

C₂²⋊C₄

—

139

—

139

C₂²⋊C₄

—

139

—

140

C₂²⋊C₄

—

139

—

141

C₂²⋊C₄

—

139

—

142

C₂²⋊C₄

—

139

—

143

C₂²⋊C₄

—

139

—

144

C₄²

—

139

—

145

C₄²

—

139

—

146

C₄²

—

139

—

147

C₄²

—

139

—

148

C₄²

C₂×D₁₃₉

yes

characteristic

149

C₂³

—

139

—

150

C₂³

—

139

—

151

Q₈

—

278

—

152

D₄

—

278

—

153

C₂×C₄

—

278

—

154

C₂×C₄

—

278

—

155

C₂×C₄

—

278

—

156

C₂×C₄

—

278

—

157

C₂×C₄

—

139

—

158

C₂×C₄

—

139

—

159

C₂×C₄

—

139

—

160

C₂×C₄

—

139

—

161

C₂×C₄

—

139

—

162

C₂×C₄

—

139

—

163

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

164

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

165

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

166

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

167

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

168

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

169

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

170

C₂²

—

278

—

171

C₂²

—

278

—

172

C₂²

—

278

—

173

C₂²

—

278

—

174

C₂²

—

278

—

175

C₂²

—

139

—

176

C₂²

C₂³×D₁₃₉

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

278

—

185

C₄

—

278

—

186

C₄

—

278

—

187

C₄

—

278

—

188

C₄

—

278

—

189

C₄

—

139

—

190

C₂

—

278

—

191

C₂

—

556

—

192

C₂

#41×D₁₃₉

yes

characteristic

193

C₂

#41×D₁₃₉

yes

characteristic

194

C₂

#17984890

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	#125×C₁₃₉	C₂
hypercenter	#125	D₁₃₉
hypocenter	C₁₃₉	#552
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

#125×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #17984837	C₂	semidirect
2. #17984837	C₂	semidirect
3. #17984832	C₂	semidirect
4. #17984832	C₂	semidirect
7. #17984820	C₂	semidirect
8. #17984820	C₂	semidirect
9. #17984807	C₂	semidirect
11. #17984800	C₂	semidirect
12. #17984799	C₂	semidirect
13. #17984799	C₂	semidirect
14. C₄⋊C₄×D₁₃₉	C₂	semidirect
15. #125×C₁₃₉	C₂	semidirect
97. C₁₃₉	#552	semidirect
103. #125	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984837

C₂

semidirect

2. #17984837

C₂

semidirect

3. #17984832

C₂

semidirect

4. #17984832

C₂

semidirect

7. #17984820

C₂

semidirect

8. #17984820

C₂

semidirect

9. #17984807

C₂

semidirect

11. #17984800

C₂

semidirect

12. #17984799

C₂

semidirect

13. #17984799

C₂

semidirect

14. C₄⋊C₄×D₁₃₉

C₂

semidirect

15. #125×C₁₃₉

C₂

semidirect

97. C₁₃₉

#552

semidirect

103. #125

D₁₃₉

semidirect

Subgroups — #22581635

Conjugacy classes of subgroups

Complements