Group #20274694 — subgroups

Subgroups — #20274694

Order 6784.

400 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	2	yes	maximal, complemented
3	C₂×#13133129	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	2	yes	—
10	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄²×D₅₃	—	2	4	no	—
14	#13133146	—	2	4	no	—
15	#13133129	—	2	4	no	—
16	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	2	no	—
26	C₂×C₄×D₅₃	—	2	2	no	—
27	C₂×C₄×D₅₃	—	2	2	no	—
28	C₂×C₄×D₅₃	—	2	2	no	—
29	C₂×C₄×D₅₃	—	2	2	no	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	2	no	—
32	C₂×C₄×D₅₃	C₈	1	4	yes	complemented
33	C₂×C₄×D₅₃	D₄	1	2	yes	—
34	C₂×C₄×D₅₃	Q₈	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	—	2	2	no	—
39	C₄×Dic₅₃	—	2	2	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₄×Dic₅₃	D₄	1	4	yes	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₅₃	—	2	1	no	—
47	C₂²×D₅₃	—	2	1	no	—
48	C₂²×D₅₃	—	2	1	no	—
49	C₂²×D₅₃	—	2	1	no	—
50	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×D₅₃	C₂×C₈	1	2	yes	—
52	C₂²×D₅₃	M₁₆	1	2	yes	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#41	1	2	yes	—
63	C₂×Dic₅₃	—	2	1	no	—
64	C₂×Dic₅₃	—	2	1	no	—
65	C₂×Dic₅₃	—	2	1	no	—
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
71	C₂×Dic₅₃	M₁₆	1	2	yes	—
72	C₄×D₅₃	—	2	8	no	—
73	C₄×D₅₃	—	2	4	no	—
74	C₄×D₅₃	—	2	4	no	—
75	C₄×D₅₃	—	2	8	no	—
76	C₈×C₅₃	—	4	2	no	—
77	#88322	—	4	2	no	—
78	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₅₃	#117	1	1	yes	characteristic
81	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×D₅₃	—	2	2	no	—
86	C₂×D₅₃	—	2	2	no	—
87	C₂×D₅₃	—	2	2	no	—
88	C₂×D₅₃	—	2	2	no	—
89	C₂×D₅₃	—	2	2	no	—
90	C₂×D₅₃	—	2	2	no	—
91	C₂×D₅₃	C₄⋊C₈	1	4	yes	complemented
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	#121	1	2	yes	—
95	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	2	2	no	—
98	Dic₅₃	C₂²⋊C₈	1	4	yes	complemented
99	#1797	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	#380	1	1	yes	characteristic
101	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#407	1	1	yes	characteristic
103	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₅₃	#423	1	1	yes	characteristic
105	C₂×C₅₃	#431	1	1	yes	characteristic
106	C₂×C₅₃	#432	1	1	yes	characteristic
107	D₅₃	—	2	4	no	—
108	C₂²×C₄²	—	53	1	no	—
109	C₂×C₄⋊C₈	—	53	2	no	—
110	C₂×C₂²⋊C₈	—	53	2	no	—
111	#336	—	53	1	no	—
112	#336	D₅₃	1	1	yes	characteristic, complemented
113	C₅₃	#1797	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	—	53	1	no	—
115	C₂³×C₄	—	53	1	no	—
116	C₂³×C₄	—	53	1	no	—
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	C₂×D₅₃	1	2	yes	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	106	4	no	—
123	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
124	C₄⋊C₈	—	106	4	no	—
125	C₂²⋊C₈	—	106	4	no	—
126	C₂⁴	—	53	1	no	—
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	53	1	no	—
130	C₂²×C₄	—	53	2	no	—
131	C₂²×C₄	—	53	2	no	—
132	C₂²×C₄	—	53	4	no	—
133	C₂²×C₄	—	106	2	no	—
134	C₂²×C₄	—	106	2	no	—
135	C₂²×C₄	—	106	2	no	—
136	C₂²×C₄	—	106	2	no	—
137	C₂²×C₄	—	106	2	no	—
138	C₂²×C₄	—	106	2	no	—
139	C₂²×C₄	—	106	2	no	—
140	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	—	106	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	53	4	no	—
151	C₄²	—	106	2	no	—
152	C₄²	—	106	2	no	—
153	C₂³	—	53	2	no	—
154	C₂³	—	53	2	no	—
155	C₂³	—	53	2	no	—
156	C₂³	—	106	1	no	—
157	C₂³	—	106	1	no	—
158	C₂³	—	106	1	no	—
159	C₂³	—	106	1	no	—
160	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	53	2	no	—
169	C₂×C₄	—	53	2	no	—
170	C₂×C₄	—	106	8	no	—
171	C₂×C₄	—	106	8	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	106	1	no	—
174	C₂×C₄	—	106	1	no	—
175	C₂×C₄	—	106	1	no	—
176	C₂×C₄	—	106	4	no	—
177	C₂×C₄	—	106	4	no	—
178	C₂×C₄	C₈×D₅₃	1	2	yes	—
179	C₂×C₄	#11719912	1	2	yes	—
180	C₂×C₄	#11719945	1	2	yes	—
181	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄	D₄×D₅₃	1	2	yes	—
183	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	212	2	no	—
186	C₂²	—	53	4	no	—
187	C₂²	—	106	2	no	—
188	C₂²	—	106	2	no	—
189	C₂²	—	106	2	no	—
190	C₂²	—	106	2	no	—
191	C₂²	—	106	2	no	—
192	C₂²	—	106	2	no	—
193	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
194	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
195	C₂²	#13133174	1	1	yes	characteristic
196	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
197	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
198	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
199	C₂²	C₂×#11719912	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	53	4	no	—
203	C₄	—	106	2	no	—
204	C₄	—	106	2	no	—
205	C₄	#13133190	1	2	yes	—
206	C₄	#13133195	1	2	yes	—
207	C₂	—	106	4	no	—
208	C₂	#16810482	1	1	yes	characteristic
209	C₂	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
210	C₂	#16810585	1	1	yes	characteristic
211	C₂	C₄⋊C₈×D₅₃	1	1	yes	characteristic
212	C₂	#16810645	1	1	yes	characteristic
213	C₂	#16810674	1	1	yes	characteristic
214	C₂	#16810678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, complemented

C₂×#13133129

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic, complemented

C₅₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

characteristic

C₂²×C₄×D₅₃

C₄

yes

characteristic

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₄²×D₅₃

—

#13133146

—

#13133129

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₈

yes

complemented

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₂×C₈

yes

—

C₂²×D₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

M₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

complemented

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊C₈

yes

complemented

#1797

—

maximal, Sylow

100

C₂×C₅₃

#380

yes

characteristic

101

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

102

C₂×C₅₃

#407

yes

characteristic

103

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

104

C₂×C₅₃

#423

yes

characteristic

105

C₂×C₅₃

#431

yes

characteristic

106

C₂×C₅₃

#432

yes

characteristic

107

D₅₃

—

108

C₂²×C₄²

—

109

C₂×C₄⋊C₈

—

110

C₂×C₂²⋊C₈

—

111

#336

—

112

#336

D₅₃

yes

characteristic, complemented

113

C₅₃

#1797

yes

characteristic, complemented, Sylow

114

C₂³×C₄

—

115

C₂³×C₄

—

116

C₂³×C₄

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×D₅₃

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

—

106

—

123

C₂×C₄²

C₂×D₅₃

yes

characteristic

124

C₄⋊C₈

—

106

—

125

C₂²⋊C₈

—

106

—

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

106

—

134

C₂²×C₄

—

106

—

135

C₂²×C₄

—

106

—

136

C₂²×C₄

—

106

—

137

C₂²×C₄

—

106

—

138

C₂²×C₄

—

106

—

139

C₂²×C₄

—

106

—

140

C₂²×C₄

C₄×D₅₃

yes

characteristic

141

C₂²×C₄

C₄×D₅₃

yes

characteristic

142

C₂²×C₄

C₂²×D₅₃

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

106

—

148

C₂×C₈

—

106

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

106

—

152

C₄²

—

106

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

106

—

157

C₂³

—

106

—

158

C₂³

—

106

—

159

C₂³

—

106

—

160

C₂³

C₂×C₄×D₅₃

yes

characteristic

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

C₈×D₅₃

yes

—

179

C₂×C₄

#11719912

yes

—

180

C₂×C₄

#11719945

yes

—

181

C₂×C₄

D₄×D₅₃

yes

characteristic

182

C₂×C₄

D₄×D₅₃

yes

—

183

C₂×C₄

Q₈×D₅₃

yes

characteristic

184

C₈

—

185

C₈

—

212

—

186

C₂²

—

187

C₂²

—

106

—

188

C₂²

—

106

—

189

C₂²

—

106

—

190

C₂²

—

106

—

191

C₂²

—

106

—

192

C₂²

—

106

—

193

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

194

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

195

C₂²

#13133174

yes

characteristic

196

C₂²

C₂×C₈×D₅₃

yes

characteristic

197

C₂²

M₁₆×D₅₃

yes

characteristic

198

C₂²

M₁₆×D₅₃

yes

characteristic

199

C₂²

C₂×#11719912

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

106

—

204

C₄

—

106

—

205

C₄

#13133190

yes

—

206

C₄

#13133195

yes

—

207

C₂

—

106

—

208

C₂

#16810482

yes

characteristic

209

C₂

C₂²⋊C₈×D₅₃

yes

characteristic

210

C₂

#16810585

yes

characteristic

211

C₂

C₄⋊C₈×D₅₃

yes

characteristic

212

C₂

#16810645

yes

characteristic

213

C₂

#16810674

yes

characteristic

214

C₂

#16810678

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1797
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²×C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×#13133146	C₂	semidirect
4. #16810378	C₂	semidirect
5. C₅₃×#336	C₂	semidirect
32. C₂×C₄×D₅₃	C₈	semidirect
91. C₂×D₅₃	C₄⋊C₈	semidirect
98. Dic₅₃	C₂²⋊C₈	semidirect
112. #336	D₅₃	semidirect
113. C₅₃	#1797	semidirect

H ≅ G/N

type

2. C₂×#13133146

C₂

semidirect

4. #16810378

C₂

semidirect

5. C₅₃×#336

C₂

semidirect

32. C₂×C₄×D₅₃

C₈

semidirect

91. C₂×D₅₃

C₄⋊C₈

semidirect

98. Dic₅₃

C₂²⋊C₈

semidirect

112. #336

D₅₃

semidirect

113. C₅₃

#1797

semidirect

Subgroups — #20274694

Conjugacy classes of subgroups

Complements