Group #19135018 — subgroups

Subgroups — #19135018

Order 5504.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13063304	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	2	yes	maximal, complemented
3	#14582759	C₂	1	1	yes	maximal, characteristic
4	#14582662	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	#13063304	—	2	2	no	—
8	#13063304	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₄	—	2	1	no	—
12	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
13	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
14	#13063250	—	2	4	no	—
15	C₄₃⋊#95	—	2	1	no	—
16	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₄₃	—	2	2	no	—
18	C₂²×C₈×C₄₃	—	2	4	no	—
19	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
20	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	—	2	2	no	—
23	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₃	D₄	1	2	yes	—
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₃	—	2	1	no	—
28	C₂²×Dic₄₃	—	2	1	no	—
29	C₂²×Dic₄₃	—	2	2	no	—
30	C₂×Dic₈₆	—	2	2	no	—
31	C₂×Dic₈₆	—	2	2	no	—
32	C₂×Dic₈₆	—	2	1	no	—
33	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₄₃	—	2	8	no	—
35	C₂×C₈×C₄₃	—	2	2	no	—
36	C₂×C₈×C₄₃	—	2	2	no	—
37	C₂×C₈×C₄₃	D₄	1	4	yes	—
38	C₂²⋊Dic₄₃	—	2	4	no	—
39	C₂²⋊Dic₄₃	—	4	2	no	—
40	C₄⋊Dic₄₃	—	2	1	no	—
41	C₄⋊Dic₄₃	—	2	2	no	—
42	C₄⋊Dic₄₃	—	2	2	no	—
43	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
44	Dic₄₃⋊C₄	—	2	4	no	—
45	C₂³×C₄₃	—	2	2	no	—
46	C₂³×C₄₃	—	2	2	no	—
47	C₂³×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₄₃	SD₁₆	1	2	yes	—
49	C₂³×C₄₃	Q₁₆	1	2	yes	—
50	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	4	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#41	1	2	yes	—
60	C₂×Dic₄₃	—	2	2	no	—
61	C₂×Dic₄₃	—	2	2	no	—
62	C₂×Dic₄₃	—	4	2	no	—
63	C₂×Dic₄₃	—	4	2	no	—
64	C₂×Dic₄₃	—	4	2	no	—
65	C₂×Dic₄₃	—	4	2	no	—
66	C₂×Dic₄₃	—	4	2	no	—
67	Dic₈₆	—	2	4	no	—
68	Dic₈₆	—	4	1	no	—
69	Dic₈₆	—	4	1	no	—
70	C₈×C₄₃	—	2	4	no	—
71	C₂²×C₄₃	—	2	2	no	—
72	C₂²×C₄₃	—	2	2	no	—
73	C₂²×C₄₃	—	2	4	no	—
74	C₂²×C₄₃	—	2	4	no	—
75	C₂²×C₄₃	#103	1	4	yes	complemented
76	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄₃	#121	1	2	yes	—
78	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂²×C₄₃	#135	1	2	yes	—
81	C₄×C₄₃	—	2	2	no	—
82	C₄×C₄₃	C₄×D₄	1	2	yes	—
83	C₄×C₄₃	#120	1	1	yes	characteristic
84	C₄×C₄₃	#123	1	1	yes	characteristic
85	Dic₄₃	—	4	2	no	—
86	Dic₄₃	—	4	2	no	—
87	Dic₄₃	—	8	2	no	—
88	#1892	—	43	1	no	maximal, Sylow
89	C₂×C₄₃	—	2	4	no	—
90	C₂×C₄₃	#386	1	1	yes	characteristic
91	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
92	C₂×C₄₃	#416	1	1	yes	characteristic
93	C₂×C₄₃	#465	1	2	yes	—
94	C₂×C₄₃	#467	1	2	yes	—
95	C₂³×C₈	D₄₃	1	1	yes	characteristic
96	C₂×C₂²⋊Q₈	—	43	1	no	—
97	C₂×#103	—	43	2	no	—
98	#380	—	43	1	no	—
99	#340	—	43	2	no	—
100	C₄₃	#1892	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	43	1	no	—
102	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	C₂×D₄₃	1	2	yes	—
106	C₂²⋊Q₈	—	43	4	no	—
107	C₂²⋊Q₈	—	86	2	no	—
108	C₂×C₄⋊C₄	—	43	1	no	—
109	C₂×C₄⋊C₄	—	43	2	no	—
110	C₂×C₄⋊C₄	—	86	1	no	—
111	C₂×C₂²⋊C₄	—	86	1	no	—
112	C₂×C₂²⋊C₄	—	86	1	no	—
113	#103	—	86	4	no	—
114	#95	—	86	1	no	—
115	C₂⁴	D₁₇₂	1	1	yes	characteristic
116	C₂×Q₈	—	43	2	no	—
117	C₂×Q₈	—	86	1	no	—
118	C₂×Q₈	—	86	2	no	—
119	C₂×Q₈	—	86	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	86	1	no	—
123	C₂²×C₄	—	86	1	no	—
124	C₂²×C₄	—	86	2	no	—
125	C₂²×C₄	C₄×D₄₃	1	2	yes	—
126	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	C₂²⋊D₄₃	1	4	yes	—
133	C₄⋊C₄	—	43	2	no	—
134	C₄⋊C₄	—	86	4	no	—
135	C₄⋊C₄	—	86	1	no	—
136	C₄⋊C₄	—	86	2	no	—
137	C₄⋊C₄	—	86	2	no	—
138	C₂²⋊C₄	—	86	4	no	—
139	C₂²⋊C₄	—	172	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	#10621873	1	2	yes	—
143	C₂³	Dic₁₇₂	1	2	yes	—
144	C₂³	(C₂×D₄₃)⋊C₄	1	2	yes	—
145	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
146	Q₈	—	86	4	no	—
147	Q₈	—	172	1	no	—
148	Q₈	—	172	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	86	2	no	—
154	C₂×C₄	—	86	2	no	—
155	C₂×C₄	—	172	2	no	—
156	C₂×C₄	—	172	2	no	—
157	C₂×C₄	—	172	2	no	—
158	C₂×C₄	—	172	2	no	—
159	C₂×C₄	—	172	2	no	—
160	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
162	C₂×C₄	#10621898	1	2	yes	—
163	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
164	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	#13063250	1	4	yes	complemented
171	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
172	C₂²	#13063299	1	2	yes	—
173	C₂²	C₂×#10621873	1	1	yes	characteristic
174	C₂²	C₂×Dic₁₇₂	1	1	yes	characteristic
175	C₂²	#13063347	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	344	2	no	—
178	C₄	—	172	2	no	—
179	C₄	—	172	2	no	—
180	C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
181	C₄	#13063294	1	1	yes	characteristic
182	C₄	#13063308	1	1	yes	characteristic
183	C₂	—	2	4	no	—
184	C₂	#14582778	1	1	yes	characteristic
185	C₂	C₂×#13063250	1	1	yes	characteristic
186	C₂	#14582896	1	1	yes	characteristic
187	C₂	#14583128	1	2	yes	—
188	C₂	#14583140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063304

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, complemented

#14582759

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal, complemented

C₂³×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

#13063304

—

#13063304

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₂²⋊Dic₄₃

—

C₂×C₂²⋊Dic₄₃

—

#13063250

—

C₄₃⋊#95

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

SD₁₆

yes

—

C₂³×C₄₃

Q₁₆

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#103

yes

complemented

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₂²×C₄₃

#135

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#1892

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#467

yes

—

C₂³×C₈

D₄₃

yes

characteristic

C₂×C₂²⋊Q₈

—

C₂×#103

—

#380

—

#340

—

100

C₄₃

#1892

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂³×C₄

C₂×D₄₃

yes

characteristic

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂×D₄₃

yes

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

#103

—

114

#95

—

115

C₂⁴

D₁₇₂

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₄×D₄₃

yes

—

126

C₂²×C₄

D₁₇₂

yes

characteristic

127

C₂²×C₄

C₂²⋊D₄₃

yes

—

128

C₂²×C₄

C₂²×D₄₃

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₄₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

172

—

140

C₂³

—

141

C₂³

—

142

C₂³

#10621873

yes

—

143

C₂³

Dic₁₇₂

yes

—

144

C₂³

(C₂×D₄₃)⋊C₄

yes

—

145

C₂³

C₂×D₁₇₂

yes

characteristic

146

Q₈

—

147

Q₈

—

172

—

148

Q₈

—

172

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

172

—

156

C₂×C₄

—

172

—

157

C₂×C₄

—

172

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

161

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

162

C₂×C₄

#10621898

yes

—

163

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

164

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#13063250

yes

complemented

171

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

172

C₂²

#13063299

yes

—

173

C₂²

C₂×#10621873

yes

characteristic

174

C₂²

C₂×Dic₁₇₂

yes

characteristic

175

C₂²

#13063347

yes

—

176

C₄

—

177

C₄

—

344

—

178

C₄

—

172

—

179

C₄

—

172

—

180

C₄

C₄×C₂²⋊D₄₃

yes

—

181

C₄

#13063294

yes

characteristic

182

C₄

#13063308

yes

characteristic

183

C₂

—

184

C₂

#14582778

yes

characteristic

185

C₂

C₂×#13063250

yes

characteristic

186

C₂

#14582896

yes

characteristic

187

C₂

#14583128

yes

—

188

C₂

#14583140

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂³×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂³×C₈	D₄₃
hypocenter	C₄₃	#1892
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂³×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂³×C₈

D₄₃

hypocenter

C₄₃

#1892

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. C₂×#13063250	C₂	semidirect
4. #14582662	C₂	semidirect
8. #13063304	C₄	semidirect
75. C₂²×C₄₃	#103	semidirect
100. C₄₃	#1892	semidirect
170. C₂²	#13063250	semidirect

H ≅ G/N

type

2. C₂×#13063250

C₂

semidirect

4. #14582662

C₂

semidirect

8. #13063304

C₄

semidirect

75. C₂²×C₄₃

#103

semidirect

100. C₄₃

#1892

semidirect

170. C₂²

#13063250

semidirect

Subgroups — #19135018

Conjugacy classes of subgroups

Complements