Group #19134072 — subgroups

Subgroups — #19134072

Order 5504.

198 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14583275	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583094	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582977	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582649	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582643	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621882	—	2	1	no	—
9	C₂×#10621882	—	2	1	no	—
10	C₂×#10621882	—	2	1	no	—
11	C₂×#10621877	—	2	1	no	—
12	#13063322	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#13063245	—	2	1	no	—
16	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
17	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₄₃	—	2	1	no	—
19	#127×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	#102×C₄₃	—	2	1	no	—
23	C₂×D₄×C₄₃	—	2	1	no	—
24	C₂×D₄×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	—	2	1	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	D₈×C₄₃	—	4	2	no	—
32	C₂×C₈×C₄₃	—	2	1	no	—
33	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₄₃	—	2	1	no	—
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	#10621882	—	4	2	no	—
37	#10621882	—	4	2	no	—
38	#10621882	—	4	2	no	—
39	#10621877	—	4	2	no	—
40	C₂×C₄₃⋊C₈	—	2	1	no	—
41	C₂×C₄₃⋊C₈	—	2	1	no	—
42	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
44	C₂³×C₄₃	—	2	1	no	—
45	C₂³×C₄₃	—	4	1	no	—
46	C₂²×D₄₃	—	4	1	no	—
47	D₄×C₄₃	—	2	1	no	—
48	D₄×C₄₃	—	2	1	no	—
49	D₄×C₄₃	—	4	1	no	—
50	D₄×C₄₃	—	4	1	no	—
51	D₄×C₄₃	—	4	1	no	—
52	D₄×C₄₃	—	4	1	no	—
53	D₄×C₄₃	—	8	1	no	—
54	C₂×C₄×C₄₃	—	2	1	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
60	D₁₇₂	—	4	1	no	—
61	D₁₇₂	—	4	1	no	—
62	D₁₇₂	—	4	1	no	—
63	D₁₇₂	—	4	1	no	—
64	D₁₇₂	—	8	1	no	—
65	C₈×C₄₃	—	4	1	no	—
66	C₄₃⋊C₈	—	2	1	no	—
67	C₄₃⋊C₈	—	4	1	no	—
68	C₄₃⋊C₈	—	4	1	no	—
69	C₄₃⋊C₈	D₈	1	2	yes	complemented
70	C₂²×C₄₃	—	4	1	no	—
71	C₂²×C₄₃	—	4	1	no	—
72	C₂²×C₄₃	—	4	1	no	—
73	C₂²×C₄₃	—	8	1	no	—
74	C₂²×C₄₃	—	8	1	no	—
75	C₂²×C₄₃	—	8	1	no	—
76	C₂²×C₄₃	#121	1	1	yes	characteristic
77	C₂×D₄₃	—	8	1	no	—
78	C₂×D₄₃	—	8	1	no	—
79	C₂×D₄₃	—	8	1	no	—
80	C₄×C₄₃	—	2	1	no	—
81	C₄×C₄₃	—	4	1	no	—
82	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
83	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	#1687	—	43	1	no	maximal, Sylow
87	C₂×C₄₃	—	8	1	no	—
88	C₂×C₄₃	—	16	1	no	—
89	C₂×C₄₃	#459	1	1	yes	characteristic
90	C₂×C₄₃	#469	1	1	yes	characteristic
91	C₂×C₄₃	#472	1	1	yes	characteristic
92	D₄₃	—	16	1	no	—
93	#493	—	43	1	no	—
94	#459	—	43	1	no	—
95	#459	D₄₃	1	1	yes	characteristic, complemented
96	#436	—	43	1	no	—
97	#334	—	43	1	no	—
98	#331	—	43	1	no	—
99	#331	—	43	1	no	—
100	C₄₃	#1687	1	1	yes	characteristic, complemented, Sylow
101	C₂×D₈	—	2	1	no	—
102	C₂×D₈	—	86	1	no	—
103	C₂×D₈	—	86	1	no	—
104	C₂×D₈	—	86	1	no	—
105	C₂×M₁₆	—	86	1	no	—
106	#127	—	43	1	no	—
107	#127	C₂×D₄₃	1	1	yes	characteristic
108	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
109	C₄⋊C₈	—	43	1	no	—
110	C₄⋊C₈	—	43	1	no	—
111	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
112	#102	—	2	1	no	—
113	#102	—	86	1	no	—
114	C₂²⋊C₈	—	86	1	no	—
115	C₄×C₈	—	43	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	—	86	1	no	—
120	C₂×D₄	—	86	1	no	—
121	C₂×D₄	—	86	1	no	—
122	C₂×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₂²×C₄	—	2	1	no	—
124	D₈	—	4	2	no	—
125	D₈	—	172	2	no	—
126	D₈	—	172	2	no	—
127	D₈	—	172	2	no	—
128	M₁₆	—	172	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	43	1	no	—
131	C₂×C₈	—	43	1	no	—
132	C₂×C₈	—	86	1	no	—
133	C₂×C₈	—	86	1	no	—
134	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
135	C₂²⋊C₄	—	2	1	no	—
136	C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	—	4	1	no	—
139	C₂³	—	172	1	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	8	1	no	—
147	D₄	—	344	1	no	—
148	D₄	—	172	1	no	—
149	D₄	—	172	1	no	—
150	D₄	—	172	1	no	—
151	D₄	—	172	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
156	C₂×C₄	#10621900	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	43	2	no	—
160	C₈	—	86	1	no	—
161	C₈	—	172	1	no	—
162	C₈	—	172	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	344	1	no	—
170	C₂²	—	344	1	no	—
171	C₂²	—	344	1	no	—
172	C₂²	#13063297	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	D₈×D₄₃	1	1	yes	characteristic
176	C₄	#13063349	1	1	yes	characteristic
177	C₄	#13063350	1	1	yes	characteristic
178	C₄	#13063351	1	1	yes	characteristic
179	C₂	—	8	1	no	—
180	C₂	—	16	1	no	—
181	C₂	—	688	1	no	—
182	C₂	#14583092	1	1	yes	characteristic
183	C₂	#14583150	1	1	yes	characteristic
184	C₂	#14583167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583275

C₂

yes

maximal, characteristic, complemented

#14583094

C₂

yes

maximal, characteristic, complemented

#14582977

C₂

yes

maximal, characteristic, complemented

#14582649

C₂

yes

maximal, characteristic, complemented

#14582643

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

C₄₃×#459

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063245

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₄₃

—

#127×C₄₃

C₂²

yes

characteristic

C₄×D₄×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

#102×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

D₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

—

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂²×D₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×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

#41

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

D₈

yes

complemented

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#493

—

#459

—

#459

D₄₃

yes

characteristic, complemented

—

—

—

—

100

C₄₃

#1687

yes

characteristic, complemented, Sylow

101

C₂×D₈

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₂×D₈

—

105

C₂×M₁₆

—

106

#127

—

107

#127

C₂×D₄₃

yes

characteristic

108

C₄×D₄

C₂×D₄₃

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₄₃

yes

characteristic

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

C₂²⋊D₄₃

yes

characteristic

123

C₂²×C₄

—

124

D₈

—

125

D₈

—

172

—

126

D₈

—

172

—

127

D₈

—

172

—

128

M₁₆

—

172

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

135

C₂²⋊C₄

—

136

C₄²

C₂²×D₄₃

yes

characteristic

137

C₂³

—

138

C₂³

—

139

C₂³

—

172

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

344

—

148

D₄

—

172

—

149

D₄

—

172

—

150

D₄

—

172

—

151

D₄

—

172

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

D₄×D₄₃

yes

characteristic

156

C₂×C₄

#10621900

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

172

—

162

C₈

—

172

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

344

—

170

C₂²

—

344

—

171

C₂²

—

344

—

172

C₂²

#13063297

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

D₈×D₄₃

yes

characteristic

176

C₄

#13063349

yes

characteristic

177

C₄

#13063350

yes

characteristic

178

C₄

#13063351

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

—

688

—

182

C₂

#14583092

yes

characteristic

183

C₂

#14583150

yes

characteristic

184

C₂

#14583167

yes

characteristic

	subgroup	G/H
center	C₂²	#13063297
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#459	C₂
socle	C₂²×C₄₃	#121
hypercenter	#459	D₄₃
hypocenter	C₄₃	#1687
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#13063297

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#459

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. #14583275	C₂	semidirect
2. #14583094	C₂	semidirect
3. #14582977	C₂	semidirect
4. #14582649	C₂	semidirect
5. #14582643	C₂	semidirect
6. #14582642	C₂	semidirect
7. C₄₃×#459	C₂	semidirect
69. C₄₃⋊C₈	D₈	semidirect
95. #459	D₄₃	semidirect
100. C₄₃	#1687	semidirect

H ≅ G/N

type

1. #14583275

C₂

semidirect

2. #14583094

C₂

semidirect

3. #14582977

C₂

semidirect

4. #14582649

C₂

semidirect

5. #14582643

C₂

semidirect

6. #14582642

C₂

semidirect

7. C₄₃×#459

C₂

semidirect

69. C₄₃⋊C₈

D₈

semidirect

95. #459

D₄₃

semidirect

100. C₄₃

#1687

semidirect

Subgroups — #19134072

Conjugacy classes of subgroups

Complements