Group #19147630 — subgroups

Subgroups — #19147630

Order 5504.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14583624	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583619	C₂	1	1	yes	maximal, characteristic
3	#14583290	C₂	1	2	yes	maximal
4	#14583278	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583119	C₂	1	2	yes	maximal
6	#14583110	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583108	C₂	1	2	yes	maximal, complemented
8	#14583026	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583003	C₂	1	2	yes	maximal, complemented
10	C₄×#10621883	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#461	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₁₇₂	C₂²	1	2	yes	—
13	C₂×#10621885	C₂²	1	2	yes	—
14	C₂×#10621885	C₂²	1	2	yes	—
15	C₂×#10621883	C₂²	1	1	yes	characteristic
16	#13063325	C₂²	1	1	yes	characteristic
17	#13063323	C₂²	1	2	yes	—
18	#13063318	—	2	2	no	—
19	#13063316	—	2	1	no	—
20	#13063315	—	2	2	no	—
21	#13063312	C₂²	1	2	yes	—
22	#13063304	—	2	1	no	—
23	Q₈×Dic₄₃	C₂²	1	2	yes	—
24	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
26	#13063290	—	2	1	no	—
27	#13063290	C₂²	1	2	yes	—
28	#13063284	—	2	1	no	—
29	#13063256	C₂²	1	1	yes	characteristic
30	Dic₄₃⋊C₈	C₂²	1	2	yes	—
31	#13063250	C₂²	1	2	yes	—
32	#13063248	C₂²	1	1	yes	characteristic
33	#13063246	C₂²	1	1	yes	characteristic
34	#13063235	C₂²	1	2	yes	—
35	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₄₃	C₂²	1	2	yes	—
37	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
40	#103×C₄₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
42	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	—	2	1	no	—
44	C₂×Dic₈₆	D₄	1	1	yes	characteristic
45	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
46	C₂×Dic₈₆	C₂³	1	2	yes	—
47	SD₁₆×C₄₃	—	4	2	no	—
48	C₂×C₈×C₄₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²⋊C₄×C₄₃	—	2	1	no	—
53	C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₄₃	—	2	2	no	—
55	C₂²⋊Dic₄₃	—	2	2	no	—
56	#10621885	—	2	4	no	—
57	#10621885	—	4	2	no	—
58	#10621883	—	2	2	no	—
59	C₄⋊Dic₄₃	—	2	1	no	—
60	C₄⋊Dic₄₃	—	2	1	no	—
61	C₄⋊Dic₄₃	—	2	2	no	—
62	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
63	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
64	Dic₄₃⋊C₄	—	2	1	no	—
65	Dic₄₃⋊C₄	—	2	2	no	—
66	Dic₄₃⋊C₄	—	2	2	no	—
67	Dic₄₃⋊C₄	—	2	2	no	—
68	Dic₄₃⋊C₄	D₄	1	2	yes	—
69	C₄×Dic₄₃	—	2	1	no	—
70	C₄×Dic₄₃	—	2	2	no	—
71	C₄×Dic₄₃	—	2	2	no	—
72	C₄×Dic₄₃	C₂³	1	2	yes	—
73	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	Dic₁₇₂	—	4	2	no	—
76	C₂³×C₄₃	—	2	1	no	—
77	Q₈×C₄₃	—	2	2	no	—
78	Q₈×C₄₃	—	2	2	no	—
79	D₄×C₄₃	—	2	1	no	—
80	D₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	4	2	no	—
84	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₄₃	—	2	1	no	—
88	C₂×Dic₄₃	—	2	1	no	—
89	C₂×Dic₄₃	—	2	2	no	—
90	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
91	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
92	Dic₈₆	—	2	2	no	—
93	Dic₈₆	—	2	1	no	—
94	Dic₈₆	—	2	2	no	—
95	Dic₈₆	—	2	1	no	—
96	Dic₈₆	—	2	2	no	—
97	C₈×C₄₃	—	2	2	no	—
98	C₄₃⋊C₈	—	2	1	no	—
99	C₄₃⋊C₈	#41	1	2	yes	—
100	C₂²×C₄₃	—	4	1	no	—
101	C₂²×C₄₃	—	4	1	no	—
102	C₂²×C₄₃	—	4	1	no	—
103	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	—	4	2	no	—
106	C₄×C₄₃	—	4	1	no	—
107	C₄×C₄₃	#137	1	2	yes	—
108	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
110	Dic₄₃	—	2	2	no	—
111	Dic₄₃	—	2	2	no	—
112	Dic₄₃	—	4	1	no	—
113	Dic₄₃	—	4	2	no	—
114	Dic₄₃	—	4	1	no	—
115	#3398	—	43	1	no	maximal, Sylow
116	C₂×C₄₃	—	8	1	no	—
117	C₂×C₄₃	#550	1	1	yes	characteristic
118	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
119	C₂×C₄₃	#578	1	1	yes	characteristic
120	#557	—	43	1	no	—
121	#556	—	43	1	no	—
122	#497	—	43	2	no	—
123	#494	—	43	1	no	—
124	#464	—	43	2	no	—
125	#462	—	43	1	no	—
126	#462	—	43	2	no	—
127	#461	D₄₃	1	1	yes	characteristic
128	#446	—	43	1	no	—
129	#441	—	43	2	no	—
130	C₄×SD₁₆	—	43	1	no	—
131	C₄₃	#3398	1	1	yes	characteristic, complemented, Sylow
132	C₂×Q₁₆	—	43	2	no	—
133	C₂×Q₁₆	—	43	2	no	—
134	C₂×Q₁₆	—	43	2	no	—
135	C₂×SD₁₆	—	43	1	no	—
136	C₂×SD₁₆	C₂×D₄₃	1	2	yes	—
137	C₄⋊Q₈	—	43	2	no	—
138	C₄⋊Q₈	—	43	1	no	—
139	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
140	#126	—	86	2	no	—
141	#125	—	86	2	no	—
142	#125	—	86	1	no	—
143	#124	—	43	2	no	—
144	C₂²⋊Q₈	—	86	1	no	—
145	C₄×Q₈	—	43	1	no	—
146	C₄×Q₈	—	43	2	no	—
147	C₄×Q₈	—	43	2	no	—
148	C₄×Q₈	—	43	1	no	—
149	C₄×Q₈	—	86	1	no	—
150	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
151	#117	—	86	1	no	—
152	#106	—	43	1	no	—
153	C₄⋊C₈	—	43	2	no	—
154	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
155	#103	—	43	2	no	—
156	#103	—	43	1	no	—
157	#103	C₂×D₄₃	1	2	yes	—
158	#102	—	43	1	no	—
159	#97	—	43	2	no	—
160	C₄×C₈	—	43	1	no	—
161	C₂×Q₈	—	43	1	no	—
162	C₂×Q₈	—	43	2	no	—
163	C₂×Q₈	—	43	1	no	—
164	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
165	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	Q₁₆	—	86	4	no	—
168	Q₁₆	—	172	2	no	—
169	Q₁₆	—	172	2	no	—
170	SD₁₆	—	4	2	no	—
171	SD₁₆	—	86	2	no	—
172	C₂×C₈	—	43	1	no	—
173	C₂×C₈	—	43	1	no	—
174	C₂×C₈	C₂²×D₄₃	1	2	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	43	1	no	—
178	C₄⋊C₄	—	43	2	no	—
179	C₄⋊C₄	—	43	1	no	—
180	C₄⋊C₄	—	86	2	no	—
181	C₄⋊C₄	—	86	2	no	—
182	C₄⋊C₄	—	86	2	no	—
183	C₄⋊C₄	—	86	2	no	—
184	C₄⋊C₄	—	86	1	no	—
185	C₄⋊C₄	—	86	1	no	—
186	C₄⋊C₄	—	86	1	no	—
187	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	86	2	no	—
190	C₂²⋊C₄	—	86	2	no	—
191	C₄²	—	43	2	no	—
192	C₄²	—	86	2	no	—
193	C₄²	—	86	2	no	—
194	C₄²	—	86	1	no	—
195	C₄²	C₂²×D₄₃	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	86	2	no	—
200	Q₈	—	86	1	no	—
201	Q₈	—	86	2	no	—
202	Q₈	—	86	1	no	—
203	Q₈	—	86	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	43	2	no	—
210	C₂×C₄	—	43	2	no	—
211	C₂×C₄	—	86	1	no	—
212	C₂×C₄	—	86	2	no	—
213	C₂×C₄	—	86	1	no	—
214	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
215	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	43	2	no	—
219	C₈	—	86	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	86	2	no	—
228	C₄	—	86	2	no	—
229	C₄	—	172	2	no	—
230	C₄	—	172	1	no	—
231	C₄	—	172	1	no	—
232	C₄	#13063354	1	2	yes	—
233	C₄	#41×D₄₃	1	1	yes	characteristic
234	C₄	#13063370	1	1	yes	characteristic
235	C₂	—	8	1	no	—
236	C₂	#14583567	1	1	yes	characteristic
237	C₂	C₂×#13063354	1	1	yes	characteristic
238	C₂	#14583715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583624

C₂

yes

maximal, characteristic, complemented

#14583619

C₂

yes

maximal, characteristic

#14583290

C₂

yes

maximal

#14583278

C₂

yes

maximal, characteristic, complemented

#14583119

C₂

yes

maximal

#14583110

C₂

yes

maximal, characteristic, complemented

#14583108

C₂

yes

maximal, complemented

#14583026

C₂

yes

maximal, characteristic, complemented

#14583003

C₂

yes

maximal, complemented

C₄×#10621883

C₂

yes

maximal, characteristic

C₄₃×#461

C₂

yes

maximal, characteristic

C₂×Dic₁₇₂

C₂²

yes

—

C₂×#10621885

C₂²

yes

—

C₂×#10621885

C₂²

yes

—

C₂×#10621883

C₂²

yes

characteristic

#13063325

C₂²

yes

characteristic

#13063323

C₂²

yes

—

—

—

—

C₂²

yes

—

#13063304

—

Q₈×Dic₄₃

C₂²

yes

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

C₂²

yes

characteristic

#13063290

—

#13063290

C₂²

yes

—

#13063284

—

#13063256

C₂²

yes

characteristic

Dic₄₃⋊C₈

C₂²

yes

—

#13063250

C₂²

yes

—

#13063248

C₂²

yes

characteristic

#13063246

C₂²

yes

characteristic

#13063235

C₂²

yes

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄×D₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×Dic₈₆

C₂³

yes

characteristic

C₂×Dic₈₆

C₂³

yes

—

SD₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

#10621885

—

#10621885

—

#10621883

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

characteristic

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

D₄

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

Dic₁₇₂

—

C₂³×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#41

yes

—

100

C₂²×C₄₃

—

101

C₂²×C₄₃

—

102

C₂²×C₄₃

—

103

C₂²×C₄₃

C₂²×D₄

yes

characteristic

104

C₄×C₄₃

—

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

#137

yes

—

108

C₄×C₄₃

C₂×#41

yes

characteristic

109

C₄×C₄₃

H(5,𝔽₂)

yes

characteristic

110

Dic₄₃

—

111

Dic₄₃

—

112

Dic₄₃

—

113

Dic₄₃

—

114

Dic₄₃

—

115

#3398

—

maximal, Sylow

116

C₂×C₄₃

—

117

C₂×C₄₃

#550

yes

characteristic

118

C₂×C₄₃

C₂×#137

yes

characteristic

119

C₂×C₄₃

#578

yes

characteristic

120

#557

—

121

#556

—

122

#497

—

123

#494

—

124

#464

—

125

#462

—

126

#462

—

127

#461

D₄₃

yes

characteristic

128

#446

—

129

#441

—

130

C₄×SD₁₆

—

131

C₄₃

#3398

yes

characteristic, complemented, Sylow

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×SD₁₆

—

136

C₂×SD₁₆

C₂×D₄₃

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

140

#126

—

141

#125

—

142

#125

—

143

#124

—

144

C₂²⋊Q₈

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

C₄×D₄

C₂×D₄₃

yes

characteristic

151

#117

—

152

#106

—

153

C₄⋊C₈

—

154

C₄⋊C₈

C₂×D₄₃

yes

characteristic

155

#103

—

156

#103

—

157

#103

C₂×D₄₃

yes

—

158

#102

—

159

#97

—

160

C₄×C₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

C₂²×D₄₃

yes

—

165

C₂×D₄

C₂²×D₄₃

yes

characteristic

166

C₂²×C₄

—

167

Q₁₆

—

168

Q₁₆

—

172

—

169

Q₁₆

—

172

—

170

SD₁₆

—

171

SD₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

C₂²×D₄₃

yes

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

C₂²×D₄₃

yes

characteristic

188

C₂²⋊C₄

—

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

193

C₄²

—

194

C₄²

—

195

C₄²

C₂²×D₄₃

yes

characteristic

196

C₂³

—

197

Q₈

—

198

Q₈

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

D₄

—

205

D₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

D₄×D₄₃

yes

characteristic

215

C₂×C₄

D₄×D₄₃

yes

characteristic

216

C₂×C₄

C₂³×D₄₃

yes

characteristic

217

C₈

—

218

C₈

—

219

C₈

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

C₂×D₄×D₄₃

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

172

—

230

C₄

—

172

—

231

C₄

—

172

—

232

C₄

#13063354

yes

—

233

C₄

#41×D₄₃

yes

characteristic

234

C₄

#13063370

yes

characteristic

235

C₂

—

236

C₂

#14583567

yes

characteristic

237

C₂

C₂×#13063354

yes

characteristic

238

C₂

#14583715

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#461	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#461	D₄₃
hypocenter	C₄₃	#3398
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#461

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #14583624	C₂	semidirect
4. #14583278	C₂	semidirect
6. #14583110	C₂	semidirect
7. #14583108	C₂	semidirect
8. #14583026	C₂	semidirect
9. #14583003	C₂	semidirect
131. C₄₃	#3398	semidirect

H ≅ G/N

type

1. #14583624

C₂

semidirect

4. #14583278

C₂

semidirect

6. #14583110

C₂

semidirect

7. #14583108

C₂

semidirect

8. #14583026

C₂

semidirect

9. #14583003

C₂

semidirect

131. C₄₃

#3398

semidirect

Subgroups — #19147630

Conjugacy classes of subgroups

Complements