Group #19133299 — subgroups

Subgroups — #19133299

Order 5504.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14583450	C₂	1	1	yes	maximal, characteristic
2	#14582955	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582951	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582645	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621902	—	2	1	no	—
9	#13063322	C₂²	1	1	yes	characteristic
10	#13063314	—	2	1	no	—
11	#13063300	—	2	1	no	—
12	#13063300	C₄	1	2	yes	—
13	#13063287	—	2	1	no	—
14	#13063287	C₄	1	2	yes	—
15	C₄²×D₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	(C₂×D₄₃)⋊C₈	—	2	1	no	—
20	#13063235	—	2	1	no	—
21	C₈×Dic₄₃	—	2	1	no	—
22	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	#10621902	—	4	2	no	—
27	#10621902	—	4	2	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	2	no	—
31	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₄₃	—	2	2	no	—
33	C₂×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
40	C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	(C₂×D₄₃)⋊C₄	—	2	2	no	—
42	(C₂×D₄₃)⋊C₄	—	2	2	no	—
43	C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₄₃	D₄	1	1	yes	characteristic
45	C₄×Dic₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂²×D₄₃	—	2	2	no	—
49	C₂²×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	Q₈×C₄₃	—	4	1	no	—
51	Q₈×C₄₃	—	4	1	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	D₁₇₂	—	4	2	no	—
60	D₁₇₂	—	4	4	no	—
61	D₁₇₂	—	4	1	no	—
62	D₁₇₂	—	4	1	no	—
63	D₁₇₂	—	4	1	no	—
64	D₁₇₂	—	4	1	no	—
65	C₄×D₄₃	—	2	1	no	—
66	C₄×D₄₃	—	2	1	no	—
67	C₄×D₄₃	—	2	1	no	—
68	C₄×D₄₃	—	2	1	no	—
69	C₄×D₄₃	—	4	1	no	—
70	C₄×D₄₃	—	4	4	no	—
71	C₈×C₄₃	—	4	1	no	—
72	C₈×C₄₃	—	4	1	no	—
73	C₄₃⋊C₈	—	4	1	no	—
74	C₄₃⋊C₈	—	4	1	no	—
75	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×D₄₃	—	2	1	no	—
77	C₂×D₄₃	—	2	1	no	—
78	C₂×D₄₃	—	2	1	no	—
79	C₂×D₄₃	—	4	2	no	—
80	C₂×D₄₃	—	4	2	no	—
81	C₂×D₄₃	—	4	2	no	—
82	C₄×C₄₃	—	2	1	no	—
83	C₄×C₄₃	—	4	2	no	—
84	C₄×C₄₃	#135	1	1	yes	characteristic
85	C₄×C₄₃	#135	1	1	yes	characteristic
86	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	Dic₄₃	—	2	1	no	—
89	Dic₄₃	—	2	1	no	—
90	Dic₄₃	—	2	1	no	—
91	#1553	—	43	1	no	maximal, Sylow
92	C₂×C₄₃	#413	1	1	yes	characteristic
93	C₂×C₄₃	#416	1	1	yes	characteristic
94	C₂×C₄₃	#418	1	1	yes	characteristic
95	D₄₃	—	4	1	no	—
96	D₄₃	—	8	2	no	—
97	#532	—	43	1	no	—
98	#432	—	43	1	no	—
99	#431	—	43	1	no	—
100	#332	—	43	1	no	—
101	#332	D₄₃	1	1	yes	characteristic, complemented
102	#331	—	43	1	no	—
103	#331	—	43	1	no	—
104	C₄₃	#1553	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	86	1	no	—
106	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
107	#127	—	43	1	no	—
108	#124	—	86	1	no	—
109	#121	—	43	2	no	—
110	#121	—	86	1	no	—
111	C₄×D₄	—	43	2	no	—
112	C₄×D₄	—	86	1	no	—
113	C₂×C₄²	—	43	1	no	—
114	C₄⋊C₈	—	43	1	no	—
115	C₄⋊C₈	—	43	1	no	—
116	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
118	C₂²⋊C₈	—	86	1	no	—
119	C₂²⋊C₈	—	86	1	no	—
120	#97	—	86	1	no	—
121	C₄×C₈	—	86	1	no	—
122	#41	—	172	2	no	—
123	#41	—	172	2	no	—
124	C₂×Q₈	—	2	1	no	—
125	C₂×D₄	—	43	2	no	—
126	C₂×D₄	—	86	2	no	—
127	C₂×D₄	—	86	1	no	—
128	C₂×D₄	—	86	1	no	—
129	C₂²×C₄	—	43	1	no	—
130	C₂²×C₄	—	43	1	no	—
131	C₂²×C₄	—	43	1	no	—
132	C₂²×C₄	—	86	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	86	1	no	—
136	C₂×C₈	—	86	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	C₄×D₄₃	1	2	yes	—
139	C₂²⋊C₄	—	86	2	no	—
140	C₂²⋊C₄	—	86	2	no	—
141	C₄²	—	43	1	no	—
142	C₄²	—	43	1	no	—
143	C₄²	—	43	1	no	—
144	C₄²	C₂²×D₄₃	1	1	yes	characteristic
145	C₂³	—	43	1	no	—
146	C₂³	—	86	2	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	D₄	—	172	2	no	—
150	D₄	—	172	4	no	—
151	D₄	—	172	1	no	—
152	D₄	—	172	1	no	—
153	D₄	—	172	1	no	—
154	D₄	—	172	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	43	1	no	—
157	C₂×C₄	—	43	1	no	—
158	C₂×C₄	—	43	1	no	—
159	C₂×C₄	—	86	1	no	—
160	C₂×C₄	—	86	1	no	—
161	C₂×C₄	—	86	1	no	—
162	C₂×C₄	—	86	1	no	—
163	C₂×C₄	—	172	1	no	—
164	C₂×C₄	—	172	4	no	—
165	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
166	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
167	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	172	1	no	—
171	C₈	—	172	1	no	—
172	C₂²	—	86	1	no	—
173	C₂²	—	86	1	no	—
174	C₂²	—	86	1	no	—
175	C₂²	—	172	2	no	—
176	C₂²	—	172	2	no	—
177	C₂²	—	172	2	no	—
178	C₂²	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	4	2	no	—
181	C₄	—	86	1	no	—
182	C₄	—	86	1	no	—
183	C₄	—	86	1	no	—
184	C₄	#13063346	1	1	yes	characteristic
185	C₄	#13063348	1	1	yes	characteristic
186	C₄	#13063349	1	1	yes	characteristic
187	C₄	#13063352	1	1	yes	characteristic
188	C₂	—	344	2	no	—
189	C₂	—	172	1	no	—
190	C₂	#14582882	1	1	yes	characteristic
191	C₂	#14582894	1	1	yes	characteristic
192	C₂	#14582902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583450

C₂

yes

maximal, characteristic

#14582955

C₂

yes

maximal, characteristic, complemented

#14582951

C₂

yes

maximal, characteristic, complemented

#14582645

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

C₄₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#10621902

—

#13063322

C₂²

yes

—

—

C₄

yes

—

#13063287

—

#13063287

C₄

yes

—

C₄²×D₄₃

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

(C₂×D₄₃)⋊C₈

—

(C₂×D₄₃)⋊C₈

—

#13063235

—

C₈×Dic₄₃

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

#10621902

—

#10621902

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

C₂×C₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄×Dic₄₃

C₂×C₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

#532

—

#432

—

#431

—

100

#332

—

101

#332

D₄₃

yes

characteristic, complemented

102

#331

—

103

#331

—

104

C₄₃

#1553

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

107

#127

—

108

#124

—

109

#121

—

110

#121

—

111

C₄×D₄

—

112

C₄×D₄

—

113

C₂×C₄²

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

C₂×D₄₃

yes

characteristic

117

C₄⋊C₈

C₂×D₄₃

yes

characteristic

118

C₂²⋊C₈

—

119

C₂²⋊C₈

—

120

#97

—

121

C₄×C₈

—

122

#41

—

172

—

123

#41

—

172

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₄⋊C₄

—

138

C₄⋊C₄

C₄×D₄₃

yes

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²×D₄₃

yes

characteristic

145

C₂³

—

146

C₂³

—

147

Q₈

—

148

Q₈

—

149

D₄

—

172

—

150

D₄

—

172

—

151

D₄

—

172

—

152

D₄

—

172

—

153

D₄

—

172

—

154

D₄

—

172

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

172

—

164

C₂×C₄

—

172

—

165

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

166

C₂×C₄

D₄×D₄₃

yes

characteristic

167

C₂×C₄

D₄×D₄₃

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

172

—

171

C₈

—

172

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

172

—

176

C₂²

—

172

—

177

C₂²

—

172

—

178

C₂²

C₂²⋊C₄×D₄₃

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#13063346

yes

characteristic

185

C₄

#13063348

yes

characteristic

186

C₄

#13063349

yes

characteristic

187

C₄

#13063352

yes

characteristic

188

C₂

—

344

—

189

C₂

—

172

—

190

C₂

#14582882

yes

characteristic

191

C₂

#14582894

yes

characteristic

192

C₂

#14582902

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#332	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#332	D₄₃
hypocenter	C₄₃	#1553
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#332

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #14582955	C₂	semidirect
3. #14582951	C₂	semidirect
4. #14582645	C₂	semidirect
5. #14582642	C₂	semidirect
6. #14582642	C₂	semidirect
7. C₄₃×#332	C₂	semidirect
101. #332	D₄₃	semidirect
104. C₄₃	#1553	semidirect

H ≅ G/N

type

2. #14582955

C₂

semidirect

3. #14582951

C₂

semidirect

4. #14582645

C₂

semidirect

5. #14582642

C₂

semidirect

6. #14582642

C₂

semidirect

7. C₄₃×#332

C₂

semidirect

101. #332

D₄₃

semidirect

104. C₄₃

#1553

semidirect

Subgroups — #19133299

Conjugacy classes of subgroups

Complements