Group #19143670 — subgroups

Subgroups — #19143670

Order 5504.

402 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 161 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	#14582978	C₂	1	2	yes	maximal, complemented
3	#14582969	C₂	1	4	yes	maximal, complemented
4	C₈×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	#14582956	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582864	C₂	1	4	yes	maximal, complemented
7	C₄₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621898	C₂²	1	2	yes	—
9	C₂×#10621872	C₂²	1	4	yes	—
10	C₂×C₈×D₄₃	C₂²	1	4	yes	—
11	#13063325	C₄	1	1	yes	characteristic
12	#13063322	C₄	1	1	yes	characteristic
13	#13063313	C₄	1	2	yes	—
14	#13063299	C₄	1	2	yes	—
15	#13063299	C₄	1	2	yes	—
16	C₄×D₁₇₂	C₂²	1	2	yes	—
17	C₄×D₁₇₂	C₂²	1	2	yes	—
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
19	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
20	(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	—
21	(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	—
22	#13063236	C₂²	1	2	yes	—
23	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
26	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₄₃	C₂×C₄	1	4	yes	—
32	#10621898	—	4	4	no	—
33	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
34	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
35	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₄₃	C₂³	1	4	yes	—
37	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₄₃	C₂³	1	2	yes	—
39	C₂×C₈×C₄₃	C₂³	1	2	yes	—
40	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	(C₂×D₄₃)⋊C₄	C₂×C₄	1	4	yes	—
45	(C₂×D₄₃)⋊C₄	C₂×C₄	1	4	yes	—
46	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
49	#10621872	—	4	4	no	—
50	C₈×D₄₃	—	4	4	no	—
51	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₄₃	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₂²⋊D₄₃	—	4	4	no	—
63	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
64	D₁₇₂	—	2	4	no	—
65	D₁₇₂	—	4	2	no	—
66	D₁₇₂	—	4	2	no	—
67	C₄×D₄₃	—	2	4	no	—
68	C₄×D₄₃	—	2	4	no	—
69	Dic₈₆	—	4	2	no	—
70	C₈×C₄₃	—	2	2	no	—
71	C₈×C₄₃	—	2	2	no	—
72	C₄₃⋊C₈	—	2	4	no	—
73	C₂²×C₄₃	—	2	1	no	—
74	C₂²×C₄₃	—	2	1	no	—
75	C₂²×C₄₃	—	2	1	no	—
76	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₄₃	—	2	4	no	—
78	C₂×D₄₃	—	2	4	no	—
79	C₂×D₄₃	—	2	4	no	—
80	C₄×C₄₃	—	2	1	no	—
81	C₄×C₄₃	—	2	1	no	—
82	C₄×C₄₃	—	2	1	no	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	#131	1	2	yes	—
85	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
86	Dic₄₃	—	2	4	no	—
87	#2981	—	43	1	no	maximal, Sylow
88	C₂×C₄₃	—	4	1	no	—
89	C₂×C₄₃	#518	1	1	yes	characteristic
90	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
91	C₂×C₄₃	#568	1	1	yes	characteristic
92	D₄₃	—	4	4	no	—
93	#532	—	43	1	no	—
94	#436	—	43	2	no	—
95	#435	—	43	4	no	—
96	C₈×D₄	—	43	2	no	—
97	#432	—	43	1	no	—
98	#431	D₄₃	1	1	yes	characteristic, complemented
99	#408	—	43	4	no	—
100	C₄₃	#2981	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	43	2	no	—
102	C₂×M₁₆	—	43	4	no	—
103	C₂²×C₈	—	43	4	no	—
104	C₄⋊Q₈	—	43	1	no	—
105	#127	—	43	1	no	—
106	#124	—	43	2	no	—
107	#121	—	43	2	no	—
108	#121	—	43	2	no	—
109	C₄×D₄	—	43	2	no	—
110	C₄×D₄	—	43	2	no	—
111	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
112	C₄⋊C₈	—	43	2	no	—
113	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
114	C₂²⋊C₈	—	43	2	no	—
115	C₂²⋊C₈	—	43	4	no	—
116	C₂²⋊C₈	—	43	4	no	—
117	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
118	#97	—	43	2	no	—
119	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
120	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
121	#41	—	172	4	no	—
122	C₂×Q₈	—	43	2	no	—
123	C₂×D₄	—	43	4	no	—
124	C₂×D₄	—	43	2	no	—
125	C₂×D₄	—	43	2	no	—
126	C₂×D₄	—	43	2	no	—
127	C₂²×C₄	—	43	4	no	—
128	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
131	M₁₆	—	172	4	no	—
132	C₂×C₈	—	43	4	no	—
133	C₂×C₈	—	172	4	no	—
134	C₂×C₈	C₂²×D₄₃	1	2	yes	—
135	C₂×C₈	C₂²×D₄₃	1	2	yes	—
136	C₄⋊C₄	—	43	2	no	—
137	C₄⋊C₄	—	43	2	no	—
138	C₂²⋊C₄	—	43	4	no	—
139	C₂²⋊C₄	—	43	4	no	—
140	C₄²	C₄×D₄₃	1	1	yes	characteristic
141	C₄²	C₄×D₄₃	1	1	yes	characteristic
142	C₄²	C₂²×D₄₃	1	1	yes	characteristic
143	C₄²	C₂²×D₄₃	1	1	yes	characteristic
144	C₂³	—	43	4	no	—
145	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
146	Q₈	—	172	2	no	—
147	D₄	—	86	4	no	—
148	D₄	—	172	2	no	—
149	D₄	—	172	2	no	—
150	D₄	—	172	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	43	4	no	—
155	C₂×C₄	—	86	4	no	—
156	C₂×C₄	—	86	4	no	—
157	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	86	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	86	4	no	—
170	C₂²	—	86	4	no	—
171	C₂²	—	86	4	no	—
172	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	86	4	no	—
178	C₄	#13063331	1	2	yes	—
179	C₄	#13063371	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	172	4	no	—
182	C₂	#14583373	1	1	yes	characteristic
183	C₂	C₂×#13063331	1	1	yes	characteristic
184	C₂	#14583665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14582978

C₂

yes

maximal, complemented

#14582969

C₂

yes

maximal, complemented

C₈×D₁₇₂

C₂

yes

maximal, complemented

#14582956

C₂

yes

maximal, characteristic, complemented

#14582864

C₂

yes

maximal, complemented

C₄₃×#431

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

—

C₂×#10621872

C₂²

yes

—

C₂×C₈×D₄₃

C₂²

yes

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

characteristic

#13063313

C₄

yes

—

#13063299

C₄

yes

—

#13063299

C₄

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

(C₂×D₄₃)⋊C₈

C₂²

yes

—

(C₂×D₄₃)⋊C₈

C₂²

yes

—

#13063236

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

C₂×C₄

yes

—

#10621898

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

C₂³

yes

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

—

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂²×D₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#131

yes

—

C₄×C₄₃

H(5,𝔽₂)

yes

—

Dic₄₃

—

#2981

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#436

—

#435

—

C₈×D₄

—

#432

—

#431

D₄₃

yes

characteristic, complemented

#408

—

100

C₄₃

#2981

yes

characteristic, complemented, Sylow

101

C₂×#41

—

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₄⋊Q₈

—

105

#127

—

106

#124

—

107

#121

—

108

#121

—

109

C₄×D₄

—

110

C₄×D₄

—

111

C₂×C₄²

C₂×D₄₃

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

C₂×D₄₃

yes

—

114

C₂²⋊C₈

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

117

C₂²⋊C₈

C₂×D₄₃

yes

—

118

#97

—

119

C₄×C₈

C₂×D₄₃

yes

characteristic

120

C₄×C₈

C₂×D₄₃

yes

characteristic

121

#41

—

172

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₄×D₄₃

yes

characteristic

129

C₂²×C₄

C₄×D₄₃

yes

characteristic

130

C₂²×C₄

C₂²×D₄₃

yes

characteristic

131

M₁₆

—

172

—

132

C₂×C₈

—

133

C₂×C₈

—

172

—

134

C₂×C₈

C₂²×D₄₃

yes

—

135

C₂×C₈

C₂²×D₄₃

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

C₄×D₄₃

yes

characteristic

141

C₄²

C₄×D₄₃

yes

characteristic

142

C₄²

C₂²×D₄₃

yes

characteristic

143

C₄²

C₂²×D₄₃

yes

characteristic

144

C₂³

—

145

C₂³

C₂×C₄×D₄₃

yes

characteristic

146

Q₈

—

172

—

147

D₄

—

148

D₄

—

172

—

149

D₄

—

172

—

150

D₄

—

172

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

158

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

159

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

160

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

161

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

162

C₂×C₄

C₂³×D₄₃

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂²×C₄×D₄₃

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#13063331

yes

—

179

C₄

#13063371

yes

—

180

C₂

—

181

C₂

—

172

—

182

C₂

#14583373

yes

characteristic

183

C₂

C₂×#13063331

yes

characteristic

184

C₂

#14583665

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#431	C₂
hypercenter	#431	D₄₃
hypocenter	C₄₃	#2981
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₃

commutator, socle

C₂²×C₄₃

C₂³×C₄

Fitting

C₄₃×#431

C₂

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #14582978	C₂	semidirect
3. #14582969	C₂	semidirect
4. C₈×D₁₇₂	C₂	semidirect
5. #14582956	C₂	semidirect
6. #14582864	C₂	semidirect
7. C₄₃×#431	C₂	semidirect
98. #431	D₄₃	semidirect
100. C₄₃	#2981	semidirect

H ≅ G/N

type

2. #14582978

C₂

semidirect

3. #14582969

C₂

semidirect

4. C₈×D₁₇₂

C₂

semidirect

5. #14582956

C₂

semidirect

6. #14582864

C₂

semidirect

7. C₄₃×#431

C₂

semidirect

98. #431

D₄₃

semidirect

100. C₄₃

#2981

semidirect

Subgroups — #19143670

Conjugacy classes of subgroups

Complements