Group #23773790 — subgroups

Subgroups — #23773790

Order 10112.

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

normal subgroups only

This group has 424 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	#19099111	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098921	C₂	1	4	yes	maximal, complemented
4	#19098915	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098914	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098909	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098561	C₂	1	4	yes	maximal, complemented
8	C₄×#11929607	C₂	1	1	yes	maximal, characteristic, complemented
9	#486×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929623	C₂²	1	2	yes	—
11	C₂×Q₈×D₇₉	—	2	1	no	—
12	C₂×#11929607	C₂²	1	2	yes	—
13	#14579161	C₂²	1	1	yes	characteristic
14	#14579161	C₂²	1	1	yes	characteristic
15	#14579160	—	2	1	no	—
16	#14579160	C₂²	1	2	yes	—
17	#14579159	—	2	1	no	—
18	#14579148	C₂²	1	2	yes	—
19	#14579132	—	2	1	no	—
20	#14579132	—	2	1	no	—
21	D₄×Dic₇₉	—	2	1	no	—
22	D₄×Dic₇₉	—	2	1	no	—
23	C₄⋊C₄×D₇₉	—	2	1	no	—
24	C₄⋊C₄×D₇₉	C₂²	1	2	yes	—
25	C₄²×D₇₉	C₂²	1	1	yes	characteristic
26	#14579086	C₂²	1	4	yes	—
27	#14579084	C₂²	1	4	yes	—
28	#14579082	C₂²	1	4	yes	—
29	#14579071	C₂²	1	2	yes	—
30	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
32	#127×C₇₉	C₂²	1	1	yes	characteristic
33	#102×C₇₉	C₂²	1	4	yes	—
34	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	#11929623	—	2	8	no	—
36	C₂×Dic₁₅₈	—	2	2	no	—
37	C₂×Dic₁₅₈	—	2	1	no	—
38	C₂×Dic₁₅₈	C₂³	1	2	yes	—
39	Q₈×D₇₉	—	4	4	no	—
40	C₂×C₂²⋊D₇₉	—	2	2	no	—
41	C₂²×Dic₇₉	—	2	2	no	—
42	C₂×C₄×D₇₉	—	2	2	no	—
43	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
46	#11929607	—	2	4	no	—
47	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₇₉	—	2	1	no	—
49	C₄⋊Dic₇₉	—	2	1	no	—
50	C₄⋊Dic₇₉	—	2	1	no	—
51	C₄⋊Dic₇₉	C₂³	1	2	yes	—
52	Dic₇₉⋊C₄	—	2	2	no	—
53	Dic₇₉⋊C₄	—	2	2	no	—
54	C₂²⋊Dic₇₉	—	2	2	no	—
55	C₂²⋊Dic₇₉	—	2	2	no	—
56	C₄×Dic₇₉	D₄	1	1	yes	characteristic
57	C₄×Dic₇₉	D₄	1	1	yes	characteristic
58	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂×Q₈×C₇₉	—	2	1	no	—
60	C₂×D₄×C₇₉	—	2	1	no	—
61	C₂×D₄×C₇₉	—	2	1	no	—
62	C₂×D₄×C₇₉	C₂³	1	2	yes	—
63	C₂×C₈×C₇₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
66	C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	C₂³×C₇₉	—	2	2	no	—
68	C₂²×D₇₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₇₉	—	4	2	no	—
70	D₄×C₇₉	—	2	4	no	—
71	D₄×C₇₉	—	4	2	no	—
72	D₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₇₉	—	4	4	no	—
78	C₂×Dic₇₉	—	2	2	no	—
79	C₂×Dic₇₉	—	2	2	no	—
80	C₂×Dic₇₉	—	4	4	no	—
81	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
84	C₄×D₇₉	—	2	2	no	—
85	C₄×D₇₉	—	2	2	no	—
86	C₄×D₇₉	—	4	4	no	—
87	C₄×D₇₉	#41	1	4	yes	—
88	Dic₁₅₈	—	2	4	no	—
89	Dic₁₅₈	—	4	4	no	—
90	Dic₁₅₈	—	4	2	no	—
91	C₈×C₇₉	—	2	2	no	—
92	C₇₉⋊C₈	—	2	2	no	—
93	C₂²×C₇₉	—	4	2	no	—
94	C₂²×C₇₉	—	4	2	no	—
95	C₂²×C₇₉	—	4	2	no	—
96	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₇₉	—	2	1	no	—
98	C₂×D₇₉	—	2	1	no	—
99	C₂×D₇₉	#124	1	2	yes	—
100	C₄×C₇₉	—	4	2	no	—
101	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₇₉	#137	1	2	yes	—
103	C₄×C₇₉	C₂×#41	1	2	yes	—
104	Dic₇₉	—	2	1	no	—
105	Dic₇₉	—	2	1	no	—
106	Dic₇₉	—	4	2	no	—
107	Dic₇₉	—	4	2	no	—
108	Dic₇₉	#124	1	2	yes	—
109	C₂×C₇₉	—	8	2	no	—
110	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
111	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
113	D₇₉	—	2	2	no	—
114	#3141	—	79	1	no	maximal, Sylow
115	C₇₉	#3141	1	1	yes	characteristic, complemented, Sylow
116	#532	—	79	1	no	—
117	C₂×C₄⋊Q₈	—	79	1	no	—
118	#489	—	79	4	no	—
119	#487	—	79	1	no	—
120	#487	—	79	1	no	—
121	#486	—	79	1	no	—
122	#486	D₇₉	1	1	yes	characteristic, complemented
123	#417	—	79	4	no	—
124	C₄×M₁₆	—	79	1	no	—
125	C₂×#41	—	79	2	no	—
126	C₂²×Q₈	—	158	1	no	—
127	C₂×M₁₆	—	79	2	no	—
128	C₄⋊Q₈	—	79	2	no	—
129	C₄⋊Q₈	—	79	1	no	—
130	C₄⋊Q₈	—	79	1	no	—
131	C₄⋊Q₈	—	158	1	no	—
132	C₄⋊Q₈	—	158	1	no	—
133	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
134	#127	C₂×D₇₉	1	1	yes	characteristic
135	#124	—	79	2	no	—
136	#121	—	158	1	no	—
137	#121	—	158	1	no	—
138	C₄×D₄	—	158	1	no	—
139	C₄×D₄	—	158	1	no	—
140	C₂×C₄⋊C₄	—	79	2	no	—
141	C₂×C₄⋊C₄	—	158	1	no	—
142	C₂×C₄²	—	79	1	no	—
143	#103	—	79	4	no	—
144	#103	—	79	4	no	—
145	#102	—	79	4	no	—
146	#102	C₂×D₇₉	1	4	yes	—
147	#97	—	79	2	no	—
148	C₄×C₈	—	79	1	no	—
149	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
150	#41	—	158	8	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	316	4	no	—
153	C₂×Q₈	—	79	2	no	—
154	C₂×Q₈	—	158	2	no	—
155	C₂×Q₈	—	158	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	158	2	no	—
159	C₂×D₄	C₂²×D₇₉	1	2	yes	—
160	C₂²×C₄	—	79	1	no	—
161	C₂²×C₄	—	79	1	no	—
162	C₂²×C₄	—	79	1	no	—
163	C₂²×C₄	—	158	2	no	—
164	C₂²×C₄	—	158	2	no	—
165	M₁₆	—	158	4	no	—
166	C₂×C₈	—	79	2	no	—
167	C₂×C₈	C₂²×D₇₉	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	79	2	no	—
170	C₄⋊C₄	—	158	2	no	—
171	C₄⋊C₄	—	158	2	no	—
172	C₄⋊C₄	—	158	1	no	—
173	C₄⋊C₄	—	158	1	no	—
174	C₄⋊C₄	—	158	1	no	—
175	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
176	C₂²⋊C₄	—	158	2	no	—
177	C₂²⋊C₄	—	158	2	no	—
178	C₄²	—	79	1	no	—
179	C₄²	—	79	1	no	—
180	C₄²	—	79	1	no	—
181	C₄²	C₂²×D₇₉	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	79	1	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	316	4	no	—
186	Q₈	—	316	2	no	—
187	Q₈	—	158	4	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	D₄	—	4	2	no	—
191	D₄	—	316	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	316	4	no	—
194	C₂×C₄	—	316	4	no	—
195	C₂×C₄	—	79	1	no	—
196	C₂×C₄	—	79	4	no	—
197	C₂×C₄	—	79	1	no	—
198	C₂×C₄	—	79	1	no	—
199	C₂×C₄	—	158	2	no	—
200	C₂×C₄	—	158	2	no	—
201	C₂×C₄	—	158	2	no	—
202	C₂×C₄	—	158	2	no	—
203	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
204	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	158	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	79	2	no	—
212	C₂²	—	158	1	no	—
213	C₂²	—	158	1	no	—
214	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	316	2	no	—
217	C₄	—	316	2	no	—
218	C₄	—	79	2	no	—
219	C₄	—	158	1	no	—
220	C₄	—	158	1	no	—
221	C₄	#14579185	1	2	yes	—
222	C₄	#14579190	1	2	yes	—
223	C₄	#41×D₇₉	1	2	yes	—
224	C₂	—	8	2	no	—
225	C₂	—	158	2	no	—
226	C₂	#124×D₇₉	1	1	yes	characteristic
227	C₂	C₂×#14579185	1	1	yes	characteristic
228	C₂	C₂×#14579190	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099111

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₇₉

C₂

yes

maximal, characteristic, complemented

#19098921

C₂

yes

maximal, complemented

#19098915

C₂

yes

maximal, characteristic, complemented

#19098914

C₂

yes

maximal, characteristic, complemented

#19098909

C₂

yes

maximal, characteristic, complemented

#19098561

C₂

yes

maximal, complemented

C₄×#11929607

C₂

yes

maximal, characteristic, complemented

#486×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

C₂²

yes

—

C₂×Q₈×D₇₉

—

C₂×#11929607

C₂²

yes

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

characteristic

#14579160

—

#14579160

C₂²

yes

—

#14579159

—

#14579148

C₂²

yes

—

#14579132

—

#14579132

—

D₄×Dic₇₉

—

D₄×Dic₇₉

—

C₄⋊C₄×D₇₉

—

C₄⋊C₄×D₇₉

C₂²

yes

—

C₄²×D₇₉

C₂²

yes

characteristic

#14579086

C₂²

yes

—

#14579084

C₂²

yes

—

#14579082

C₂²

yes

—

#14579071

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

#102×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929623

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

Q₈×D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

C₂³

yes

characteristic

#11929607

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

D₄

yes

characteristic

C₄×Dic₇₉

D₄

yes

characteristic

C₄×Dic₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

#41

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

#124

yes

—

100

C₄×C₇₉

—

101

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

102

C₄×C₇₉

#137

yes

—

103

C₄×C₇₉

C₂×#41

yes

—

104

Dic₇₉

—

105

Dic₇₉

—

106

Dic₇₉

—

107

Dic₇₉

—

108

Dic₇₉

#124

yes

—

109

C₂×C₇₉

—

110

C₂×C₇₉

C₂×#124

yes

characteristic

111

C₂×C₇₉

C₂×AGL(1,ℤ/8)

yes

characteristic

112

C₂×C₇₉

C₂×#137

yes

characteristic

113

D₇₉

—

114

#3141

—

maximal, Sylow

115

C₇₉

#3141

yes

characteristic, complemented, Sylow

116

#532

—

117

C₂×C₄⋊Q₈

—

118

#489

—

119

#487

—

120

#487

—

121

#486

—

122

#486

D₇₉

yes

characteristic, complemented

123

#417

—

124

C₄×M₁₆

—

125

C₂×#41

—

126

C₂²×Q₈

—

158

—

127

C₂×M₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

158

—

132

C₄⋊Q₈

—

158

—

133

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

134

#127

C₂×D₇₉

yes

characteristic

135

#124

—

136

#121

—

158

—

137

#121

—

158

—

138

C₄×D₄

—

158

—

139

C₄×D₄

—

158

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

158

—

142

C₂×C₄²

—

143

#103

—

144

#103

—

145

#102

—

146

#102

C₂×D₇₉

yes

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

C₂×D₇₉

yes

characteristic

150

#41

—

158

—

151

C₂×Q₈

—

152

C₂×Q₈

—

316

—

153

C₂×Q₈

—

154

C₂×Q₈

—

158

—

155

C₂×Q₈

—

158

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

158

—

159

C₂×D₄

C₂²×D₇₉

yes

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

158

—

164

C₂²×C₄

—

158

—

165

M₁₆

—

158

—

166

C₂×C₈

—

167

C₂×C₈

C₂²×D₇₉

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

158

—

171

C₄⋊C₄

—

158

—

172

C₄⋊C₄

—

158

—

173

C₄⋊C₄

—

158

—

174

C₄⋊C₄

—

158

—

175

C₄⋊C₄

C₂²×D₇₉

yes

—

176

C₂²⋊C₄

—

158

—

177

C₂²⋊C₄

—

158

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

C₂²×D₇₉

yes

characteristic

182

C₂³

—

183

C₂³

—

184

Q₈

—

185

Q₈

—

316

—

186

Q₈

—

316

—

187

Q₈

—

158

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

316

—

192

C₂×C₄

—

193

C₂×C₄

—

316

—

194

C₂×C₄

—

316

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

158

—

200

C₂×C₄

—

158

—

201

C₂×C₄

—

158

—

202

C₂×C₄

—

158

—

203

C₂×C₄

D₄×D₇₉

yes

characteristic

204

C₂×C₄

D₄×D₇₉

yes

characteristic

205

C₂×C₄

C₂³×D₇₉

yes

characteristic

206

C₈

—

207

C₈

—

158

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

158

—

213

C₂²

—

158

—

214

C₂²

C₂×D₄×D₇₉

yes

characteristic

215

C₄

—

216

C₄

—

316

—

217

C₄

—

316

—

218

C₄

—

219

C₄

—

158

—

220

C₄

—

158

—

221

C₄

#14579185

yes

—

222

C₄

#14579190

yes

—

223

C₄

#41×D₇₉

yes

—

224

C₂

—

225

C₂

—

158

—

226

C₂

#124×D₇₉

yes

characteristic

227

C₂

C₂×#14579185

yes

characteristic

228

C₂

C₂×#14579190

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#486×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099111	C₂
hypercenter	#486	D₇₉
hypocenter	C₇₉	#3141
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#486×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. C₄⋊Q₈×D₇₉	C₂	semidirect
3. #19098921	C₂	semidirect
4. #19098915	C₂	semidirect
5. #19098914	C₂	semidirect
6. #19098909	C₂	semidirect
7. #19098561	C₂	semidirect
8. C₄×#11929607	C₂	semidirect
9. #486×C₇₉	C₂	semidirect
115. C₇₉	#3141	semidirect
122. #486	D₇₉	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₇₉

C₂

semidirect

3. #19098921

C₂

semidirect

4. #19098915

C₂

semidirect

5. #19098914

C₂

semidirect

6. #19098909

C₂

semidirect

7. #19098561

C₂

semidirect

8. C₄×#11929607

C₂

semidirect

9. #486×C₇₉

C₂

semidirect

115. C₇₉

#3141

semidirect

122. #486

D₇₉

semidirect

Subgroups — #23773790

Conjugacy classes of subgroups

Complements