Group #23764215 — subgroups

Subgroups — #23764215

Order 10112.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14579148	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14579084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579082	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#14579148	—	2	2	no	—
11	#14579148	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₉⋊C₄	—	2	2	no	—
14	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
15	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
16	#14579086	—	2	2	no	—
17	#14579084	—	2	2	no	—
18	#14579082	—	2	2	no	—
19	#14579071	—	2	2	no	—
20	C₂²×D₄×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
23	#102×C₇₉	—	2	2	no	—
24	C₂×Dic₁₅₈	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	1	no	—
28	C₂×Dic₁₅₈	—	2	1	no	—
29	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
30	C₂²×Dic₇₉	—	2	1	no	—
31	C₂²×Dic₇₉	—	2	1	no	—
32	C₂²×Dic₇₉	D₄	1	2	yes	—
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
36	C₄⋊Dic₇₉	—	2	1	no	—
37	C₄⋊Dic₇₉	—	2	1	no	—
38	Dic₇₉⋊C₄	—	4	4	no	—
39	C₂²⋊Dic₇₉	—	2	8	no	—
40	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
41	C₄×Dic₇₉	D₄	1	2	yes	—
42	C₂⁴×C₇₉	—	2	1	no	—
43	C₂×D₄×C₇₉	—	2	1	no	—
44	C₂×D₄×C₇₉	—	2	1	no	—
45	C₂×D₄×C₇₉	—	2	1	no	—
46	C₂×D₄×C₇₉	—	2	1	no	—
47	C₂×D₄×C₇₉	—	2	1	no	—
48	C₂×D₄×C₇₉	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	—	2	1	no	—
50	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₇₉	—	2	1	no	—
52	C₂×C₈×C₇₉	—	2	1	no	—
53	C₂×C₈×C₇₉	D₄	1	2	yes	—
54	C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂³×C₇₉	—	2	2	no	—
57	C₂³×C₇₉	—	4	1	no	—
58	C₂³×C₇₉	—	4	1	no	—
59	C₂³×C₇₉	—	4	2	no	—
60	C₂³×C₇₉	—	4	2	no	—
61	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
62	D₄×C₇₉	—	2	2	no	—
63	D₄×C₇₉	—	2	2	no	—
64	D₄×C₇₉	—	4	1	no	—
65	D₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	—	4	1	no	—
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	—	2	2	no	—
77	C₂×Dic₇₉	—	4	1	no	—
78	C₂×Dic₇₉	—	4	1	no	—
79	C₂×Dic₇₉	—	4	1	no	—
80	C₂×Dic₇₉	—	4	2	no	—
81	C₂×Dic₇₉	C₂²⋊C₄	1	4	yes	—
82	Dic₁₅₈	—	2	2	no	—
83	Dic₁₅₈	—	2	2	no	—
84	Dic₁₅₈	—	4	1	no	—
85	Dic₁₅₈	—	4	1	no	—
86	C₈×C₇₉	—	2	2	no	—
87	C₇₉⋊C₈	—	2	2	no	—
88	C₂²×C₇₉	—	4	2	no	—
89	C₂²×C₇₉	—	4	4	no	—
90	C₂²×C₇₉	—	8	2	no	—
91	C₂²×C₇₉	—	8	1	no	—
92	C₂²×C₇₉	—	8	1	no	—
93	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₇₉	#124	1	1	yes	characteristic
95	C₂²×C₇₉	#127	1	1	yes	characteristic
96	C₂²×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
97	C₂²×C₇₉	#137	1	2	yes	—
98	C₄×C₇₉	—	8	1	no	—
99	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
100	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
101	C₄×C₇₉	#121	1	1	yes	characteristic
102	C₄×C₇₉	#121	1	1	yes	characteristic
103	Dic₇₉	—	2	4	no	—
104	Dic₇₉	—	4	2	no	—
105	Dic₇₉	—	8	1	no	—
106	C₂×C₇₉	—	8	2	no	—
107	C₂×C₇₉	#390	1	1	yes	characteristic
108	C₂×C₇₉	#417	1	2	yes	—
109	C₂×C₇₉	#489	1	2	yes	—
110	C₂×C₇₉	#496	1	1	yes	characteristic
111	C₂×C₇₉	#497	1	1	yes	characteristic
112	#1959	—	79	1	no	maximal, Sylow
113	C₇₉	#1959	1	1	yes	characteristic, complemented, Sylow
114	C₂×#124	—	79	1	no	—
115	C₂×#103	—	79	1	no	—
116	C₂×#103	—	79	1	no	—
117	C₂×#102	—	79	1	no	—
118	C₂×#102	D₇₉	1	1	yes	characteristic
119	C₂×#97	—	79	1	no	—
120	#384	—	79	1	no	—
121	C₂²×Q₈	—	79	1	no	—
122	C₂²×D₄	C₂×D₇₉	1	1	yes	characteristic
123	C₂²×C₈	—	79	1	no	—
124	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
125	#124	—	79	4	no	—
126	#124	—	158	2	no	—
127	C₂×C₄⋊C₄	—	79	1	no	—
128	C₂×C₄⋊C₄	—	158	2	no	—
129	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	—	158	2	no	—
131	C₂×C₄²	—	79	1	no	—
132	#103	—	158	2	no	—
133	#103	—	158	2	no	—
134	#102	—	2	2	no	—
135	#102	—	158	2	no	—
136	#97	—	158	2	no	—
137	C₂⁴	—	2	1	no	—
138	C₂×Q₈	—	79	2	no	—
139	C₂×Q₈	—	158	1	no	—
140	C₂×Q₈	—	158	1	no	—
141	C₂×Q₈	—	158	1	no	—
142	C₂×Q₈	—	158	1	no	—
143	C₂×Q₈	—	158	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	C₄×D₇₉	1	2	yes	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	79	2	no	—
152	C₂²×C₄	—	158	1	no	—
153	C₂²×C₄	—	158	1	no	—
154	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	79	2	no	—
158	C₂×C₈	—	158	1	no	—
159	C₂×C₈	—	158	1	no	—
160	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	316	4	no	—
164	C₄⋊C₄	—	158	1	no	—
165	C₄⋊C₄	—	158	1	no	—
166	C₂²⋊C₄	—	158	8	no	—
167	C₄²	—	79	2	no	—
168	C₄²	—	79	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	4	1	no	—
171	C₂³	—	4	1	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	2	no	—
174	C₂³	D₄×D₇₉	1	1	yes	characteristic
175	Q₈	—	316	1	no	—
176	Q₈	—	316	1	no	—
177	Q₈	—	158	2	no	—
178	Q₈	—	158	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	4	1	no	—
182	D₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	316	1	no	—
186	C₂×C₄	—	316	1	no	—
187	C₂×C₄	—	316	1	no	—
188	C₂×C₄	—	316	2	no	—
189	C₂×C₄	—	79	4	no	—
190	C₂×C₄	—	158	2	no	—
191	C₂×C₄	—	158	2	no	—
192	C₂×C₄	—	158	2	no	—
193	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
194	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
195	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
197	C₂×C₄	#11929623	1	1	yes	characteristic
198	C₂×C₄	#11929624	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	158	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	1	no	—
205	C₂²	—	8	1	no	—
206	C₂²	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
207	C₂²	#14579147	1	1	yes	characteristic
208	C₂²	#14579157	1	1	yes	characteristic
209	C₂²	#14579185	1	2	yes	—
210	C₂²	#14579190	1	2	yes	—
211	C₄	—	8	1	no	—
212	C₄	—	316	2	no	—
213	C₄	—	632	1	no	—
214	C₄	—	158	4	no	—
215	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
216	C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
217	C₄	#14579133	1	1	yes	characteristic
218	C₄	#14579134	1	1	yes	characteristic
219	C₂	—	8	2	no	—
220	C₂	#19098463	1	1	yes	characteristic
221	C₂	#19098561	1	2	yes	—
222	C₂	#19098921	1	2	yes	—
223	C₂	#19098951	1	1	yes	characteristic
224	C₂	#19098954	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579148

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic, complemented

C₂×#14579084

C₂

yes

maximal, characteristic, complemented

C₂×#14579082

C₂

yes

maximal, characteristic

C₂×#14579071

C₂

yes

maximal, characteristic, complemented

#19098432

C₂

yes

maximal, characteristic, complemented

C₂×#102×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579148

—

#14579148

C₄

yes

complemented

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₂²⋊Dic₇₉

—

C₂×C₄×Dic₇₉

C₂²

yes

—

—

—

—

C₂²×D₄×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

#102×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

D₄

yes

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²⋊C₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

100

C₄×C₇₉

C₄×D₄

yes

characteristic

101

C₄×C₇₉

#121

yes

characteristic

102

C₄×C₇₉

#121

yes

characteristic

103

Dic₇₉

—

104

Dic₇₉

—

105

Dic₇₉

—

106

C₂×C₇₉

—

107

C₂×C₇₉

#390

yes

characteristic

108

C₂×C₇₉

#417

yes

—

109

C₂×C₇₉

#489

yes

—

110

C₂×C₇₉

#496

yes

characteristic

111

C₂×C₇₉

yes

112

—

113

C₇₉

#1959

yes

characteristic, complemented, Sylow

114

C₂×#124

—

115

C₂×#103

—

116

C₂×#103

—

117

C₂×#102

—

118

C₂×#102

D₇₉

yes

characteristic

119

C₂×#97

—

120

#384

—

121

C₂²×Q₈

—

122

C₂²×D₄

C₂×D₇₉

yes

characteristic

123

C₂²×C₈

—

124

C₂²×C₈

C₂×D₇₉

yes

characteristic

125

#124

—

126

#124

—

158

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

158

—

129

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

130

C₂×C₂²⋊C₄

—

158

—

131

C₂×C₄²

—

132

#103

—

158

—

133

#103

—

158

—

134

#102

—

135

#102

—

158

—

136

#97

—

158

—

137

C₂⁴

—

138

C₂×Q₈

—

139

C₂×Q₈

—

158

—

140

C₂×Q₈

—

158

—

141

C₂×Q₈

—

158

—

142

C₂×Q₈

—

158

—

143

C₂×Q₈

—

158

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₄×D₇₉

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

158

—

153

C₂²×C₄

—

158

—

154

C₂²×C₄

C₂²×D₇₉

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

158

—

159

C₂×C₈

—

158

—

160

C₂×C₈

C₂²⋊D₇₉

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

316

—

164

C₄⋊C₄

—

158

—

165

C₄⋊C₄

—

158

—

166

C₂²⋊C₄

—

158

—

167

C₄²

—

168

C₄²

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

D₄×D₇₉

yes

characteristic

175

Q₈

—

316

—

176

Q₈

—

316

—

177

Q₈

—

158

—

178

Q₈

—

158

—

179

D₄

—

180

D₄

—

181

D₄

—

182

D₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

316

—

186

C₂×C₄

—

316

—

187

C₂×C₄

—

316

—

188

C₂×C₄

—

316

—

189

C₂×C₄

—

190

C₂×C₄

—

158

—

191

C₂×C₄

—

158

—

192

C₂×C₄

—

158

—

193

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

194

C₂×C₄

D₄×D₇₉

yes

characteristic

195

C₂×C₄

D₄×D₇₉

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

197

C₂×C₄

#11929623

yes

characteristic

198

C₂×C₄

#11929624

yes

characteristic

199

C₈

—

200

C₈

—

158

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

C₂²⋊C₄×D₇₉

yes

characteristic

207

C₂²

#14579147

yes

characteristic

208

C₂²

#14579157

yes

characteristic

209

C₂²

#14579185

yes

—

210

C₂²

#14579190

yes

—

211

C₄

—

212

C₄

—

316

—

213

C₄

—

632

—

214

C₄

—

158

—

215

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

216

C₄

C₄×C₂²⋊D₇₉

yes

characteristic

217

C₄

#14579133

yes

characteristic

218

C₄

#14579134

yes

characteristic

219

C₂

—

220

C₂

#19098463

yes

characteristic

221

C₂

#19098561

yes

—

222

C₂

#19098921

yes

—

223

C₂

#19098951

yes

characteristic

224

C₂

#19098954

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#102×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂²×D₄×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#102	D₇₉
hypocenter	C₇₉	#1959
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#102×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂²×D₄×C₇₉

C₂²

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

N	H ≅ G/N	type
2. C₂×#14579086	C₂	semidirect
3. C₂×#14579084	C₂	semidirect
5. C₂×#14579071	C₂	semidirect
6. #19098432	C₂	semidirect
11. #14579148	C₄	semidirect
113. C₇₉	#1959	semidirect

H ≅ G/N

type

2. C₂×#14579086

C₂

semidirect

3. C₂×#14579084

C₂

semidirect

5. C₂×#14579071

C₂

semidirect

6. #19098432

C₂

semidirect

11. #14579148

C₄

semidirect

113. C₇₉

#1959

semidirect

Subgroups — #23764215

Conjugacy classes of subgroups

Complements