Group #19136136 — subgroups

Subgroups — #19136136

Order 5504.

254 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13063257	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063256	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063249	C₂	1	1	yes	maximal, characteristic
4	#14582806	C₂	1	1	yes	maximal, characteristic
5	#14582768	C₂	1	1	yes	maximal, characteristic
6	#14582653	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	#13063257	—	2	2	no	—
14	#13063256	—	2	2	no	—
15	#13063249	—	2	2	no	—
16	C₄₃⋊#95	—	2	2	no	—
17	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	#103×C₄₃	—	2	2	no	—
21	C₂×Q₈×C₄₃	—	2	1	no	—
22	C₂×Q₈×C₄₃	—	2	1	no	—
23	C₂×Q₈×C₄₃	—	2	1	no	—
24	C₂×Q₈×C₄₃	—	2	1	no	—
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₈×C₄₃	Q₈	1	2	yes	—
27	C₂²×C₄×C₄₃	—	2	1	no	—
28	C₂²×C₄×C₄₃	—	2	1	no	—
29	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₃	—	2	1	no	—
31	C₂²×Dic₄₃	—	2	1	no	—
32	C₂²×Dic₄₃	D₄	1	2	yes	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₂×C₈×C₄₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₄⋊C₄×C₄₃	—	2	1	no	—
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄⋊Dic₄₃	—	2	1	no	—
42	Dic₄₃⋊C₄	—	4	4	no	—
43	C₄×Dic₄₃	—	2	1	no	—
44	C₄×Dic₄₃	—	2	1	no	—
45	C₂×C₄₃⋊C₈	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
48	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₄₃	—	2	2	no	—
50	Q₈×C₄₃	—	2	2	no	—
51	Q₈×C₄₃	—	4	1	no	—
52	Q₈×C₄₃	—	4	1	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	—	4	2	no	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×Dic₄₃	—	2	2	no	—
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	4	1	no	—
67	C₂×Dic₄₃	—	4	1	no	—
68	C₂×Dic₄₃	—	4	2	no	—
69	C₂×Dic₄₃	—	4	2	no	—
70	C₂×Dic₄₃	SD₁₆	1	4	yes	—
71	C₈×C₄₃	—	2	2	no	—
72	C₄₃⋊C₈	—	2	2	no	—
73	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₄₃	#123	1	1	yes	characteristic
75	C₂²×C₄₃	#127	1	1	yes	characteristic
76	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₄₃	#137	1	2	yes	—
78	C₄×C₄₃	—	4	2	no	—
79	C₄×C₄₃	—	8	1	no	—
80	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₄₃	#123	1	1	yes	characteristic
83	C₄×C₄₃	#123	1	1	yes	characteristic
84	Dic₄₃	—	2	4	no	—
85	Dic₄₃	—	8	1	no	—
86	Dic₄₃	—	8	1	no	—
87	#2096	—	43	1	no	maximal, Sylow
88	C₂×C₄₃	#399	1	1	yes	characteristic
89	C₂×C₄₃	#475	1	2	yes	—
90	C₂×C₄₃	#483	1	2	yes	—
91	C₂×C₄₃	#492	1	1	yes	characteristic
92	C₂×C₄₃	#497	1	1	yes	characteristic
93	C₂×#106	—	43	1	no	—
94	C₂×#106	—	43	1	no	—
95	C₂×#103	—	43	1	no	—
96	C₂×#103	D₄₃	1	1	yes	characteristic
97	#391	—	43	1	no	—
98	#384	—	43	1	no	—
99	#336	—	43	1	no	—
100	C₄₃	#2096	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
102	C₂²×C₈	—	43	1	no	—
103	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	43	1	no	—
105	C₂×C₄⋊C₄	—	43	1	no	—
106	C₂×C₄⋊C₄	—	86	2	no	—
107	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄²	—	43	1	no	—
109	#106	—	86	2	no	—
110	#106	—	86	2	no	—
111	#103	—	2	2	no	—
112	#103	—	86	2	no	—
113	#95	—	86	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₈₆	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	43	2	no	—
123	C₂²×C₄	—	86	1	no	—
124	C₂²×C₄	—	86	1	no	—
125	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	43	2	no	—
129	C₂×C₈	—	86	1	no	—
130	C₂×C₈	—	86	1	no	—
131	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	86	1	no	—
135	C₄⋊C₄	—	86	1	no	—
136	C₄⋊C₄	—	86	1	no	—
137	C₄⋊C₄	—	86	1	no	—
138	C₄⋊C₄	—	172	4	no	—
139	C₄²	—	86	1	no	—
140	C₄²	—	86	1	no	—
141	C₂³	D₄×D₄₃	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	43	4	no	—
152	C₂×C₄	—	86	2	no	—
153	C₂×C₄	—	86	2	no	—
154	C₂×C₄	—	172	1	no	—
155	C₂×C₄	—	172	1	no	—
156	C₂×C₄	—	172	2	no	—
157	C₂×C₄	—	172	2	no	—
158	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
159	C₂×C₄	#10621898	1	1	yes	characteristic
160	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
161	C₂×C₄	#10621900	1	1	yes	characteristic
162	C₂×C₄	#10621900	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	86	2	no	—
166	C₂²	#13063302	1	1	yes	characteristic
167	C₂²	#13063307	1	1	yes	characteristic
168	C₂²	#13063321	1	1	yes	characteristic
169	C₂²	SD₁₆×D₄₃	1	2	yes	—
170	C₂²	#13063357	1	2	yes	—
171	C₄	—	4	2	no	—
172	C₄	—	8	1	no	—
173	C₄	—	86	4	no	—
174	C₄	—	344	1	no	—
175	C₄	—	344	1	no	—
176	C₄	#13063302	1	1	yes	characteristic
177	C₄	#13063304	1	1	yes	characteristic
178	C₄	#13063306	1	1	yes	characteristic
179	C₄	#13063307	1	1	yes	characteristic
180	C₂	#14582835	1	1	yes	characteristic
181	C₂	#14583182	1	2	yes	—
182	C₂	#14583228	1	2	yes	—
183	C₂	#14583270	1	1	yes	characteristic
184	C₂	#14583292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063257

C₂

yes

maximal, characteristic

C₂×#13063256

C₂

yes

maximal, characteristic

C₂×#13063249

C₂

yes

maximal, characteristic

#14582806

C₂

yes

maximal, characteristic

#14582768

C₂

yes

maximal, characteristic

#14582653

C₂

yes

maximal, characteristic

C₂×#103×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

C₂²

yes

—

—

—

C₄₃⋊#95

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

#103×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×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

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

SD₁₆

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²⋊Q₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×SD₁₆

yes

—

C₂²×C₄₃

#137

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#483

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

D₄₃

yes

—

—

—

100

C₄₃

#2096

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

C₂×D₄₃

yes

characteristic

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₄₃

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

108

C₂×C₄²

—

109

#106

—

110

#106

—

111

#103

—

112

#103

—

113

#95

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₈₆

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₄₃

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₂²⋊D₄₃

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

172

—

139

C₄²

—

140

C₄²

—

141

C₂³

D₄×D₄₃

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

172

—

155

C₂×C₄

—

172

—

156

C₂×C₄

—

172

—

157

C₂×C₄

—

172

—

158

C₂×C₄

C₂×Dic₈₆

yes

characteristic

159

C₂×C₄

#10621898

yes

characteristic

160

C₂×C₄

D₄×D₄₃

yes

characteristic

161

C₂×C₄

#10621900

yes

characteristic

162

C₂×C₄

#10621900

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

164

C₈

—

165

C₈

—

166

C₂²

#13063302

yes

characteristic

167

C₂²

#13063307

yes

characteristic

168

C₂²

#13063321

yes

characteristic

169

C₂²

SD₁₆×D₄₃

yes

—

170

C₂²

#13063357

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

344

—

175

C₄

—

344

—

176

C₄

#13063302

yes

characteristic

177

C₄

#13063304

yes

characteristic

178

C₄

#13063306

yes

characteristic

179

C₄

#13063307

yes

characteristic

180

C₂

#14582835

yes

characteristic

181

C₂

#14583182

yes

—

182

C₂

#14583228

yes

—

183

C₂

#14583270

yes

characteristic

184

C₂

#14583292

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#103×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#103	D₄₃
hypocenter	C₄₃	#2096
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#103×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
100. C₄₃	#2096	semidirect

H ≅ G/N

type

100. C₄₃

#2096

semidirect

Subgroups — #19136136

Conjugacy classes of subgroups

Complements