Group #18077850 — subgroups

Subgroups — #18077850

Order 4864.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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	#14573957	C₂	1	1	yes	maximal, characteristic
2	#14562227	C₂	1	1	yes	maximal, characteristic
3	#14562055	C₂	1	2	yes	maximal, complemented
4	#14561471	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2979	C₂	1	1	yes	maximal, characteristic
6	#11927101	C₂²	1	1	yes	characteristic
7	#11926898	—	2	2	no	—
8	#11926894	C₄	1	2	yes	—
9	#11926890	C₄	1	2	yes	—
10	#11926540	—	2	2	no	—
11	C₄×#64520	C₂²	1	2	yes	—
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	#11926284	—	2	2	no	—
14	#11926274	—	2	2	no	—
15	#11926274	—	2	2	no	—
16	C₁₉×#532	C₂²	1	1	yes	characteristic
17	C₁₉×#436	—	2	2	no	—
18	C₈×D₄×C₁₉	—	2	2	no	—
19	C₁₉×#431	C₂²	1	2	yes	—
20	C₁₉×#408	—	2	2	no	—
21	C₂×#41×C₁₉	D₄	1	2	yes	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
25	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
26	C₁₉×#124	D₄	1	2	yes	—
27	C₁₉×#121	—	2	2	no	—
28	C₄×D₄×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	#10597372	—	2	1	no	—
31	#10597370	—	2	2	no	—
32	C₂×#64520	—	2	2	no	—
33	C₂²×C₁₉⋊C₈	—	2	1	no	—
34	#10597316	—	2	1	no	—
35	#10597315	C₂×C₄	1	2	yes	—
36	C₄×Dic₃₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
38	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
39	C₁₉×C₂²⋊C₈	—	2	4	no	—
40	C₁₉×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₈×C₁₉	—	2	2	no	—
42	#10597292	—	2	4	no	—
43	#10597289	—	2	4	no	—
44	#10597260	—	2	2	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	—	2	2	no	—
47	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
48	#41×C₁₉	—	4	2	no	—
49	#41×C₁₉	—	4	2	no	—
50	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	—	2	2	no	—
53	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	#41	1	2	yes	—
57	C₂×Dic₃₈	—	2	2	no	—
58	M₁₆×C₁₉	—	8	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	8	2	no	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₉	—	2	4	no	—
64	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂×D₄	1	2	yes	—
67	C₂²⋊Dic₁₉	—	4	1	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	—	2	2	no	—
70	Dic₁₉⋊C₄	—	4	2	no	—
71	C₄×Dic₁₉	—	4	1	no	—
72	#64520	—	2	4	no	—
73	#64520	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	2	2	no	—
75	C₂×C₁₉⋊C₈	—	4	1	no	—
76	C₂×C₁₉⋊C₈	—	4	1	no	—
77	C₂×C₁₉⋊C₈	—	4	2	no	—
78	#9067	—	19	1	no	maximal, Sylow
79	C₂³×C₁₉	—	2	2	no	—
80	C₂³×C₁₉	#123	1	1	yes	characteristic
81	Q₈×C₁₉	—	4	2	no	—
82	D₄×C₁₉	—	4	2	no	—
83	D₄×C₁₉	—	4	2	no	—
84	D₄×C₁₉	—	4	2	no	—
85	D₄×C₁₉	—	8	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#121	1	2	yes	—
97	C₂×Dic₁₉	—	4	2	no	—
98	Dic₃₈	—	4	2	no	—
99	Dic₃₈	—	4	2	no	—
100	C₈×C₁₉	—	4	2	no	—
101	C₈×C₁₉	—	4	2	no	—
102	C₁₉⋊C₈	—	4	2	no	—
103	C₁₉⋊C₈	—	4	4	no	—
104	#3131	—	19	1	no	—
105	#2979	D₁₉	1	1	yes	characteristic
106	#1525	—	19	1	no	—
107	#1485	—	19	2	no	—
108	#1278	—	19	2	no	—
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	2	1	no	—
111	C₂²×C₁₉	—	4	4	no	—
112	C₂²×C₁₉	—	4	2	no	—
113	C₂²×C₁₉	#386	1	1	yes	characteristic
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	4	2	no	—
117	C₄×C₁₉	#443	1	2	yes	—
118	C₄×C₁₉	#444	1	2	yes	—
119	C₄×C₁₉	#455	1	1	yes	characteristic
120	C₄×C₁₉	#455	1	1	yes	characteristic
121	Dic₁₉	—	8	2	no	—
122	#533	—	19	1	no	—
123	#532	C₂×D₁₉	1	1	yes	characteristic
124	#488	—	38	2	no	—
125	#487	—	19	2	no	—
126	#486	—	19	2	no	—
127	#436	—	2	2	no	—
128	C₈×D₄	—	2	2	no	—
129	#431	C₂×D₁₉	1	2	yes	—
130	#416	—	38	2	no	—
131	#408	—	2	2	no	—
132	C₄×M₁₆	—	19	2	no	—
133	C₂×C₄×C₈	—	19	1	no	—
134	#329	—	38	2	no	—
135	Q₈⋊C₈	—	38	2	no	—
136	Q₈⋊C₈	—	38	2	no	—
137	C₂×C₁₉	—	4	1	no	—
138	C₂×C₁₉	—	8	2	no	—
139	C₂×C₁₉	#1876	1	2	yes	—
140	C₂×C₁₉	#1885	1	1	yes	characteristic
141	C₂×#41	C₂²⋊D₁₉	1	2	yes	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	38	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	38	1	no	—
146	C₄⋊Q₈	—	19	2	no	—
147	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
148	#127	C₄×D₁₉	1	1	yes	characteristic
149	#125	—	38	1	no	—
150	#124	D₇₆	1	2	yes	—
151	C₂²⋊Q₈	—	38	2	no	—
152	#121	—	2	2	no	—
153	C₄×Q₈	—	38	2	no	—
154	C₄×D₄	—	2	2	no	—
155	#117	—	38	1	no	—
156	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
157	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
158	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
159	#103	—	38	4	no	—
160	#102	—	38	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	2	2	no	—
163	#97	—	38	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	19	2	no	—
166	C₄×C₈	—	38	1	no	—
167	C₄×C₈	—	38	2	no	—
168	C₁₉	#9067	1	1	yes	characteristic, complemented, Sylow
169	#41	—	4	2	no	—
170	#41	—	4	2	no	—
171	C₂×Q₈	—	38	2	no	—
172	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	#64541	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
179	M₁₆	—	8	2	no	—
180	M₁₆	—	38	4	no	—
181	M₁₆	—	76	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	8	2	no	—
185	C₂×C₈	—	38	2	no	—
186	C₂×C₈	—	76	1	no	—
187	C₂×C₈	—	76	1	no	—
188	C₂×C₈	—	76	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	38	2	no	—
191	C₄⋊C₄	—	38	2	no	—
192	C₄⋊C₄	—	76	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	76	1	no	—
195	C₄²	—	76	1	no	—
196	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
197	C₄²	C₂×D₇₆	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	#10597375	1	1	yes	characteristic
201	Q₈	—	4	2	no	—
202	Q₈	—	76	2	no	—
203	Q₈	—	76	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	76	2	no	—
216	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
218	C₂×C₄	#10597376	1	2	yes	—
219	C₂×C₄	#10597399	1	1	yes	characteristic
220	C₈	—	4	2	no	—
221	C₈	—	4	2	no	—
222	C₈	—	76	2	no	—
223	C₈	—	76	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	#11926423	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	152	2	no	—
233	C₄	#11926659	1	2	yes	—
234	C₄	#11926664	1	2	yes	—
235	C₄	#11926722	1	1	yes	characteristic
236	C₄	#11926722	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	8	2	no	—
239	C₂	#14563962	1	2	yes	—
240	C₂	#14564023	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14573957

C₂

yes

maximal, characteristic

#14562227

C₂

yes

maximal, characteristic

#14562055

C₂

yes

maximal, complemented

#14561471

C₂

yes

maximal, complemented

C₁₉×#2979

C₂

yes

maximal, characteristic

#11927101

C₂²

yes

characteristic

#11926898

—

#11926894

C₄

yes

—

#11926890

C₄

yes

—

#11926540

—

C₄×#64520

C₂²

yes

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#436

—

C₈×D₄×C₁₉

—

C₁₉×#431

C₂²

yes

—

C₁₉×#408

—

C₂×#41×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₁₉×C₄⋊Q₈

C₂×C₄

yes

characteristic

C₁₉×#127

C₂×C₄

yes

characteristic

C₁₉×#124

D₄

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

C₂²×C₁₉⋊C₈

—

#10597316

—

#10597315

C₂×C₄

yes

—

C₄×Dic₃₈

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

#10597292

—

#10597289

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

C₂×C₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

C₂²⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9067

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#123

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

100

C₈×C₁₉

—

101

C₈×C₁₉

—

102

C₁₉⋊C₈

—

103

C₁₉⋊C₈

—

104

#3131

—

105

#2979

D₁₉

yes

characteristic

106

#1525

—

107

#1485

—

108

#1278

—

109

C₂²×C₁₉

—

110

C₂²×C₁₉

—

111

C₂²×C₁₉

—

112

C₂²×C₁₉

—

113

C₂²×C₁₉

#386

yes

characteristic

114

C₄×C₁₉

—

115

C₄×C₁₉

—

116

C₄×C₁₉

—

117

C₄×C₁₉

#443

yes

—

118

C₄×C₁₉

#444

yes

—

119

C₄×C₁₉

#455

yes

characteristic

120

C₄×C₁₉

#455

yes

characteristic

121

Dic₁₉

—

122

#533

—

123

#532

C₂×D₁₉

yes

characteristic

124

#488

—

125

#487

—

126

#486

—

127

#436

—

128

C₈×D₄

—

129

#431

C₂×D₁₉

yes

—

130

#416

—

131

#408

—

132

C₄×M₁₆

—

133

C₂×C₄×C₈

—

134

#329

—

135

Q₈⋊C₈

—

136

Q₈⋊C₈

—

137

C₂×C₁₉

—

138

C₂×C₁₉

—

139

C₂×C₁₉

#1876

yes

—

140

C₂×C₁₉

#1885

yes

characteristic

141

C₂×#41

C₂²⋊D₁₉

yes

—

142

C₂×M₁₆

—

143

C₂×M₁₆

—

144

C₂²×C₈

—

145

C₂²×C₈

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

C₄×D₁₉

yes

characteristic

148

#127

C₄×D₁₉

yes

characteristic

149

#125

—

150

#124

D₇₆

yes

—

151

C₂²⋊Q₈

—

152

#121

—

153

C₄×Q₈

—

154

C₄×D₄

—

155

#117

—

156

C₂×C₄²

C₂²×D₁₉

yes

characteristic

157

C₄⋊C₈

C₂²⋊D₁₉

yes

—

158

C₄⋊C₈

C₂²⋊D₁₉

yes

—

159

#103

—

160

#102

—

161

C₂²⋊C₈

—

162

C₂²⋊C₈

—

163

#97

—

164

C₄×C₈

—

165

C₄×C₈

—

166

C₄×C₈

—

167

C₄×C₈

—

168

C₁₉

#9067

yes

characteristic, complemented, Sylow

169

#41

—

170

#41

—

171

C₂×Q₈

—

172

C₂×Q₈

(C₂×D₁₉)⋊C₄

yes

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

(C₂×D₁₉)⋊C₄

yes

—

176

C₂²×C₄

—

177

C₂²×C₄

#64541

yes

—

178

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

179

M₁₆

—

180

M₁₆

—

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₂²⋊C₄

—

194

C₂²⋊C₄

—

195

C₄²

—

196

C₄²

C₂×C₄×D₁₉

yes

characteristic

197

C₄²

C₂×D₇₆

yes

characteristic

198

C₄²

C₂×C₂²⋊D₁₉

yes

—

199

C₂³

—

200

C₂³

#10597375

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

D₄

—

205

D₄

—

206

D₄

—

207

D₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

217

C₂×C₄

C₄×C₂²⋊D₁₉

yes

—

218

C₂×C₄

#10597376

yes

—

219

C₂×C₄

#10597399

yes

characteristic

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

#11926423

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

152

—

233

C₄

#11926659

yes

—

234

C₄

#11926664

yes

—

235

C₄

#11926722

yes

characteristic

236

C₄

#11926722

yes

characteristic

237

C₂

—

238

C₂

—

239

C₂

#14563962

yes

—

240

C₂

#14564023

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597399
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2979	C₂
socle	C₂²×C₁₉	#386
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2979	D₁₉
hypocenter	C₁₉	#9067
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2979

subgroup

G/H

center

C₂×C₄

#10597399

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2979

C₂

socle

C₂²×C₁₉

#386

prime radical

C₁₉×#532

C₂²

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2979

N	H ≅ G/N	type
3. #14562055	C₂	semidirect
4. #14561471	C₂	semidirect
168. C₁₉	#9067	semidirect

H ≅ G/N

type

3. #14562055

C₂

semidirect

4. #14561471

C₂

semidirect

168. C₁₉

#9067

semidirect

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

subgroup	index
#14573957	2
#14562227	2
#14562055	2
#14561471	2
C₁₉×#2979	2
#9067	19

subgroup

index

C₁₉×#2979

Subgroups — #18077850

Conjugacy classes of subgroups

Complements

Maximal subgroups