Group #19140419 — subgroups

Subgroups — #19140419

Order 5504.

488 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063282	C₂	1	1	yes	maximal, characteristic
3	#14582839	C₂	1	2	yes	maximal, complemented
4	#14582836	C₂	1	2	yes	maximal
5	#14582820	C₂	1	2	yes	maximal, complemented
6	#14582812	C₂	1	2	yes	maximal
7	#14582798	C₂	1	2	yes	maximal, complemented
8	#14582756	C₂	1	2	yes	maximal
9	C₄₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₈₆	C₂²	1	2	yes	—
11	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
12	#13063325	—	2	4	no	—
13	#13063282	—	2	4	no	—
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₃	C₂²	1	4	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄₃⋊#95	C₂²	1	4	yes	—
24	C₄₃⋊#95	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	#95×C₄₃	C₂²	1	2	yes	—
28	#95×C₄₃	C₂²	1	2	yes	—
29	C₂³×D₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₃	C₂³	1	4	yes	—
36	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₄₃	—	2	2	no	—
39	C₂×C₄×D₄₃	—	2	2	no	—
40	C₂×C₄×D₄₃	Q₈	1	4	yes	complemented
41	C₂×Dic₈₆	—	2	4	no	—
42	C₂×Dic₈₆	—	4	4	no	—
43	C₄⋊C₄×C₄₃	—	2	4	no	—
44	C₄²×C₄₃	D₄	1	4	yes	—
45	(C₂×D₄₃)⋊C₄	—	2	2	no	—
46	(C₂×D₄₃)⋊C₄	—	2	2	no	—
47	(C₂×D₄₃)⋊C₄	—	4	4	no	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	C₄⋊Dic₄₃	—	2	4	no	—
50	C₄⋊Dic₄₃	—	4	4	no	—
51	Dic₄₃⋊C₄	—	2	2	no	—
52	Dic₄₃⋊C₄	—	2	2	no	—
53	C₄×Dic₄₃	—	2	2	no	—
54	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×D₄₃	—	2	1	no	—
56	C₂²×D₄₃	—	2	1	no	—
57	C₂²×D₄₃	—	2	2	no	—
58	C₂²×D₄₃	—	2	2	no	—
59	C₂²×D₄₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	2	4	no	—
64	C₂×C₄×C₄₃	—	2	4	no	—
65	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₄₃	#41	1	2	yes	—
70	C₂×Dic₄₃	—	2	1	no	—
71	C₂×Dic₄₃	—	2	1	no	—
72	C₂×Dic₄₃	—	2	1	no	—
73	C₂×Dic₄₃	—	2	1	no	—
74	C₂×Dic₄₃	—	2	4	no	—
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	—	2	4	no	—
77	C₂×Dic₄₃	—	2	4	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
80	C₄×D₄₃	—	2	8	no	—
81	Dic₈₆	—	4	8	no	—
82	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
85	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
86	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
87	C₂×D₄₃	—	2	2	no	—
88	C₂×D₄₃	—	2	4	no	—
89	C₂×D₄₃	—	4	1	no	—
90	C₂×D₄₃	—	4	1	no	—
91	C₂×D₄₃	—	4	2	no	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	4	4	no	—
95	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
96	Dic₄₃	—	2	2	no	—
97	Dic₄₃	—	4	4	no	—
98	Dic₄₃	—	4	1	no	—
99	Dic₄₃	—	4	1	no	—
100	#2628	—	43	1	no	maximal, Sylow
101	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄₃	#535	1	1	yes	characteristic
103	C₂×C₄₃	#536	1	1	yes	characteristic
104	C₂×C₄₃	#542	1	2	yes	—
105	C₂×C₄₃	#544	1	2	yes	—
106	D₄₃	—	4	2	no	—
107	C₂×C₄⋊Q₈	—	43	1	no	—
108	C₂×#117	—	43	1	no	—
109	#400	—	43	2	no	—
110	#399	—	43	2	no	—
111	#395	—	43	2	no	—
112	#393	—	43	2	no	—
113	#389	—	43	2	no	—
114	#382	D₄₃	1	1	yes	characteristic, complemented
115	#380	—	43	2	no	—
116	C₄₃	#2628	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	—	43	2	no	—
118	C₂³×C₄	—	43	1	no	—
119	C₄⋊Q₈	—	86	4	no	—
120	#117	—	86	4	no	—
121	C₂×C₄⋊C₄	—	43	1	no	—
122	C₂×C₄⋊C₄	—	43	1	no	—
123	C₂×C₄⋊C₄	—	43	2	no	—
124	C₂×C₄⋊C₄	—	43	2	no	—
125	C₂×C₄⋊C₄	—	43	4	no	—
126	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
127	C₂×C₂²⋊C₄	—	43	1	no	—
128	C₂×C₂²⋊C₄	—	43	1	no	—
129	C₂×C₂²⋊C₄	—	43	4	no	—
130	C₂×C₄²	—	43	1	no	—
131	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
132	#95	—	43	4	no	—
133	#95	—	43	4	no	—
134	#95	C₂×D₄₃	1	2	yes	—
135	#95	C₂×D₄₃	1	2	yes	—
136	C₂⁴	—	43	1	no	—
137	C₂×Q₈	—	86	4	no	—
138	C₂×Q₈	—	172	4	no	—
139	C₂²×C₄	—	43	1	no	—
140	C₂²×C₄	—	43	4	no	—
141	C₂²×C₄	—	43	1	no	—
142	C₂²×C₄	—	43	4	no	—
143	C₂²×C₄	—	43	1	no	—
144	C₂²×C₄	—	86	2	no	—
145	C₂²×C₄	—	86	2	no	—
146	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
147	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	86	2	no	—
152	C₄⋊C₄	—	86	2	no	—
153	C₄⋊C₄	—	86	4	no	—
154	C₄⋊C₄	—	86	4	no	—
155	C₄⋊C₄	—	172	4	no	—
156	C₂²⋊C₄	—	86	2	no	—
157	C₂²⋊C₄	—	86	2	no	—
158	C₂²⋊C₄	—	172	4	no	—
159	C₄²	—	86	2	no	—
160	C₄²	D₁₇₂	1	4	yes	—
161	C₂³	—	43	2	no	—
162	C₂³	—	86	1	no	—
163	C₂³	—	86	1	no	—
164	C₂³	—	86	2	no	—
165	C₂³	—	86	2	no	—
166	C₂³	C₂³×D₄₃	1	1	yes	characteristic
167	Q₈	—	172	8	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	43	2	no	—
174	C₂×C₄	—	86	1	no	—
175	C₂×C₄	—	86	1	no	—
176	C₂×C₄	—	86	8	no	—
177	C₂×C₄	—	86	1	no	—
178	C₂×C₄	—	86	1	no	—
179	C₂×C₄	—	86	4	no	—
180	C₂×C₄	—	86	4	no	—
181	C₂×C₄	—	86	2	no	—
182	C₂×C₄	—	86	4	no	—
183	C₂×C₄	—	86	2	no	—
184	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
185	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
186	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
187	C₂×C₄	#10621900	1	2	yes	—
188	C₂×C₄	Q₈×D₄₃	1	2	yes	—
189	C₂²	—	86	2	no	—
190	C₂²	—	86	4	no	—
191	C₂²	—	172	1	no	—
192	C₂²	—	172	1	no	—
193	C₂²	—	172	2	no	—
194	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
195	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
196	C₂²	C₂×#10621900	1	1	yes	characteristic
197	C₂²	#13063371	1	2	yes	—
198	C₂²	#13063372	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	4	no	—
202	C₄	—	86	2	no	—
203	C₄	—	172	4	no	—
204	C₄	—	172	1	no	—
205	C₄	—	172	1	no	—
206	C₄	#13063305	1	4	yes	—
207	C₂	—	172	2	no	—
208	C₂	C₂×#13063305	1	1	yes	characteristic
209	C₂	#14583468	1	1	yes	characteristic
210	C₂	#14583475	1	1	yes	characteristic
211	C₂	#14583518	1	2	yes	—
212	C₂	#14583529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic, complemented

C₂×#13063282

C₂

yes

maximal, characteristic

#14582839

C₂

yes

maximal, complemented

#14582836

C₂

yes

maximal

#14582820

C₂

yes

maximal, complemented

#14582812

C₂

yes

maximal

#14582798

C₂

yes

maximal, complemented

#14582756

C₂

yes

maximal

C₄₃×#382

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

—

C₂²×C₄×D₄₃

C₂²

yes

characteristic

#13063325

—

#13063282

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

#95×C₄₃

C₂²

yes

—

C₂³×D₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

Q₈

yes

complemented

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₄×D₄₃

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

100

#2628

—

maximal, Sylow

101

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

characteristic

102

C₂×C₄₃

#535

yes

characteristic

103

C₂×C₄₃

#536

yes

characteristic

104

C₂×C₄₃

#542

yes

—

105

C₂×C₄₃

#544

yes

—

106

D₄₃

—

107

C₂×C₄⋊Q₈

—

108

C₂×#117

—

109

#400

—

110

#399

—

111

#395

—

112

#393

—

113

#389

—

114

#382

D₄₃

yes

characteristic, complemented

115

#380

—

116

C₄₃

#2628

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

—

118

C₂³×C₄

—

119

C₄⋊Q₈

—

120

#117

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

127

C₂×C₂²⋊C₄

—

128

C₂×C₂²⋊C₄

—

129

C₂×C₂²⋊C₄

—

130

C₂×C₄²

—

131

C₂×C₄²

C₂×D₄₃

yes

characteristic

132

#95

—

133

#95

—

134

#95

C₂×D₄₃

yes

—

135

#95

C₂×D₄₃

yes

—

136

C₂⁴

—

137

C₂×Q₈

—

138

C₂×Q₈

—

172

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²×D₄₃

yes

—

147

C₂²×C₄

C₂²×D₄₃

yes

characteristic

148

C₂²×C₄

C₂²×D₄₃

yes

characteristic

149

C₂²×C₄

C₂²×D₄₃

yes

characteristic

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

172

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

172

—

159

C₄²

—

160

C₄²

D₁₇₂

yes

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

C₂³×D₄₃

yes

characteristic

167

Q₈

—

172

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂×D₁₇₂

yes

—

185

C₂×C₄

C₂×D₁₇₂

yes

—

186

C₂×C₄

C₂×D₁₇₂

yes

—

187

C₂×C₄

#10621900

yes

—

188

C₂×C₄

Q₈×D₄₃

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

172

—

192

C₂²

—

172

—

193

C₂²

—

172

—

194

C₂²

C₂²×D₁₇₂

yes

characteristic

195

C₂²

C₂×Q₈×D₄₃

yes

characteristic

196

C₂²

C₂×#10621900

yes

characteristic

197

C₂²

#13063371

yes

—

198

C₂²

#13063372

yes

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

172

—

204

C₄

—

172

—

205

C₄

—

172

—

206

C₄

#13063305

yes

—

207

C₂

—

172

—

208

C₂

C₂×#13063305

yes

characteristic

209

C₂

#14583468

yes

characteristic

210

C₂

#14583475

yes

characteristic

211

C₂

#14583518

yes

—

212

C₂

#14583529

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#382	C₂
hypercenter	#382	D₄₃
hypocenter	C₄₃	#2628
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄₃×#382

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
1. C₂×#13063325	C₂	semidirect
3. #14582839	C₂	semidirect
5. #14582820	C₂	semidirect
7. #14582798	C₂	semidirect
9. C₄₃×#382	C₂	semidirect
40. C₂×C₄×D₄₃	Q₈	semidirect
114. #382	D₄₃	semidirect
116. C₄₃	#2628	semidirect

H ≅ G/N

type

1. C₂×#13063325

C₂

semidirect

3. #14582839

C₂

semidirect

5. #14582820

C₂

semidirect

7. #14582798

C₂

semidirect

9. C₄₃×#382

C₂

semidirect

40. C₂×C₄×D₄₃

Q₈

semidirect

114. #382

D₄₃

semidirect

116. C₄₃

#2628

semidirect

Subgroups — #19140419

Conjugacy classes of subgroups

Complements