Group #19139597 — subgroups

Subgroups — #19139597

Order 5504.

676 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063313	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582817	C₂	1	4	yes	maximal, complemented
5	#14582801	C₂	1	2	yes	maximal, complemented
6	#14582800	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂²×D₁₇₂	C₂²	1	2	yes	—
10	C₂²×D₁₇₂	C₂²	1	2	yes	—
11	C₂²×D₁₇₂	C₂²	1	2	yes	—
12	#13063322	—	2	4	no	—
13	#13063313	—	2	8	no	—
14	C₄×Dic₈₆	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	8	yes	—
18	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	8	yes	—
19	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	#95×C₄₃	C₂²	1	2	yes	—
24	C₂³×D₄₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₃	C₂³	1	2	yes	—
30	C₂²×Dic₄₃	C₂³	1	2	yes	—
31	C₂×D₁₇₂	—	2	4	no	—
32	C₂×D₁₇₂	—	2	4	no	—
33	C₂×D₁₇₂	—	2	4	no	—
34	C₂×D₁₇₂	—	2	4	no	—
35	C₂×D₁₇₂	—	2	8	no	—
36	C₂×D₁₇₂	—	2	4	no	—
37	C₂×D₁₇₂	—	4	4	no	—
38	C₂×Dic₈₆	—	2	2	no	—
39	C₂×Dic₈₆	—	2	2	no	—
40	C₂×Dic₈₆	D₄	1	4	yes	complemented
41	C₄⋊C₄×C₄₃	—	2	4	no	—
42	C₄²×C₄₃	—	2	4	no	—
43	C₄²×C₄₃	D₄	1	4	yes	—
44	(C₂×D₄₃)⋊C₄	—	2	16	no	—
45	(C₂×D₄₃)⋊C₄	—	4	8	no	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	Dic₄₃⋊C₄	—	2	4	no	—
48	C₄×Dic₄₃	—	2	4	no	—
49	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₂²×D₄₃	—	2	4	no	—
51	C₂²×D₄₃	—	2	4	no	—
52	C₂²×D₄₃	—	2	8	no	—
53	C₂²×D₄₃	—	2	4	no	—
54	C₂²×D₄₃	—	2	8	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	4	no	—
57	C₂×C₄×C₄₃	—	2	4	no	—
58	C₂×C₄×C₄₃	—	2	4	no	—
59	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₄₃	#41	1	4	yes	—
64	C₂×C₄×C₄₃	#41	1	2	yes	—
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	2	2	no	—
67	C₂×Dic₄₃	—	2	2	no	—
68	C₂×Dic₄₃	—	2	4	no	—
69	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
70	D₁₇₂	—	4	8	no	—
71	D₁₇₂	—	4	8	no	—
72	D₁₇₂	—	4	8	no	—
73	Dic₈₆	—	2	8	no	—
74	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₄₃	C₂×#41	1	2	yes	—
77	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
78	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
79	C₂×D₄₃	—	4	4	no	—
80	C₂×D₄₃	—	4	4	no	—
81	C₂×D₄₃	—	4	4	no	—
82	C₂×D₄₃	—	4	8	no	—
83	C₂×D₄₃	—	4	8	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	4	4	no	—
86	C₄×C₄₃	#121	1	4	yes	—
87	C₄×C₄₃	#124	1	4	yes	complemented
88	Dic₄₃	—	2	4	no	—
89	Dic₄₃	—	4	2	no	—
90	#2548	—	43	1	no	maximal, Sylow
91	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
92	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
93	C₂×C₄₃	#535	1	1	yes	characteristic
94	C₂×C₄₃	#550	1	2	yes	—
95	C₂×C₄₃	#559	1	2	yes	—
96	D₄₃	—	8	4	no	—
97	C₂×#127	—	43	1	no	—
98	C₂×#124	—	43	2	no	—
99	C₂×C₄×Q₈	—	43	1	no	—
100	#394	—	43	4	no	—
101	#390	—	43	4	no	—
102	#390	—	43	2	no	—
103	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
104	C₄₃	#2548	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	43	1	no	—
106	C₂²×D₄	—	43	2	no	—
107	C₂²×D₄	—	43	2	no	—
108	C₂²×D₄	—	43	2	no	—
109	#127	—	86	4	no	—
110	#124	—	86	8	no	—
111	C₄×Q₈	—	86	4	no	—
112	C₂×C₄⋊C₄	—	43	2	no	—
113	C₂×C₄⋊C₄	—	43	1	no	—
114	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
115	C₂×C₂²⋊C₄	—	43	8	no	—
116	C₂×C₂²⋊C₄	—	43	8	no	—
117	C₂×C₄²	—	43	2	no	—
118	C₂×C₄²	C₂×D₄₃	1	2	yes	—
119	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
120	#95	C₂×D₄₃	1	2	yes	—
121	C₂⁴	—	43	4	no	—
122	C₂×Q₈	—	43	4	no	—
123	C₂×Q₈	—	86	2	no	—
124	C₂×Q₈	—	86	2	no	—
125	C₂×D₄	—	86	4	no	—
126	C₂×D₄	—	86	4	no	—
127	C₂×D₄	—	86	4	no	—
128	C₂×D₄	—	86	8	no	—
129	C₂×D₄	—	86	4	no	—
130	C₂×D₄	—	86	4	no	—
131	C₂×D₄	—	172	4	no	—
132	C₂²×C₄	—	43	2	no	—
133	C₂²×C₄	—	43	2	no	—
134	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
135	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	86	4	no	—
140	C₄⋊C₄	—	86	2	no	—
141	C₂²⋊C₄	—	86	16	no	—
142	C₂²⋊C₄	—	172	8	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	86	4	no	—
145	C₄²	D₁₇₂	1	4	yes	—
146	C₂³	—	86	4	no	—
147	C₂³	—	86	4	no	—
148	C₂³	—	86	8	no	—
149	C₂³	—	86	4	no	—
150	C₂³	—	86	8	no	—
151	C₂³	C₂³×D₄₃	1	1	yes	characteristic
152	Q₈	—	86	8	no	—
153	D₄	—	172	8	no	—
154	D₄	—	172	8	no	—
155	D₄	—	172	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	43	4	no	—
161	C₂×C₄	—	86	2	no	—
162	C₂×C₄	—	86	2	no	—
163	C₂×C₄	—	86	2	no	—
164	C₂×C₄	—	86	4	no	—
165	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
166	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
167	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
168	C₂×C₄	#10621898	1	4	yes	—
169	C₂×C₄	D₄×D₄₃	1	2	yes	—
170	C₂×C₄	#10621902	1	2	yes	—
171	C₂²	—	172	8	no	—
172	C₂²	—	172	8	no	—
173	C₂²	—	172	4	no	—
174	C₂²	—	172	4	no	—
175	C₂²	—	172	4	no	—
176	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
177	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
178	C₂²	C₂×#10621898	1	2	yes	—
179	C₂²	C₂×#10621902	1	1	yes	characteristic
180	C₂²	#13063371	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	4	4	no	—
183	C₄	—	86	4	no	—
184	C₄	—	172	2	no	—
185	C₄	#13063300	1	4	yes	—
186	C₄	#13063313	1	4	yes	complemented
187	C₂	—	344	4	no	—
188	C₂	C₂×#13063300	1	1	yes	characteristic
189	C₂	C₂×#13063313	1	1	yes	characteristic
190	C₂	#14583468	1	1	yes	characteristic
191	C₂	#14583570	1	2	yes	—
192	C₂	#14583634	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063322

C₂

yes

maximal, characteristic

C₂×#13063313

C₂

yes

maximal, complemented

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#14582817

C₂

yes

maximal, complemented

#14582801

C₂

yes

maximal, complemented

#14582800

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

#13063322

—

#13063313

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

C₂³×D₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

complemented

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

C₄×C₄₃

#124

yes

complemented

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#559

yes

—

D₄₃

—

C₂×#127

—

C₂×#124

—

C₂×C₄×Q₈

—

100

#394

—

101

#390

—

102

#390

—

103

C₄×C₄⋊C₄

D₄₃

yes

characteristic, complemented

104

C₄₃

#2548

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂²×D₄

—

107

C₂²×D₄

—

108

C₂²×D₄

—

109

#127

—

110

#124

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂×D₄₃

yes

—

119

C₂×C₄²

C₂×D₄₃

yes

characteristic

120

#95

C₂×D₄₃

yes

—

121

C₂⁴

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

172

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂²×D₄₃

yes

—

135

C₂²×C₄

C₂²×D₄₃

yes

characteristic

136

C₂²×C₄

C₂²×D₄₃

yes

characteristic

137

C₂²×C₄

C₂²×D₄₃

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

172

—

143

C₄²

—

144

C₄²

—

145

C₄²

D₁₇₂

yes

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

C₂³×D₄₃

yes

characteristic

152

Q₈

—

153

D₄

—

172

—

154

D₄

—

172

—

155

D₄

—

172

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×D₁₇₂

yes

—

166

C₂×C₄

C₂×D₁₇₂

yes

—

167

C₂×C₄

C₂×D₁₇₂

yes

—

168

C₂×C₄

#10621898

yes

—

169

C₂×C₄

D₄×D₄₃

yes

—

170

C₂×C₄

#10621902

yes

—

171

C₂²

—

172

—

172

C₂²

—

172

—

173

C₂²

—

172

—

174

C₂²

—

172

—

175

C₂²

—

172

—

176

C₂²

C₂×D₄×D₄₃

yes

characteristic

177

C₂²

C₂²×D₁₇₂

yes

characteristic

178

C₂²

C₂×#10621898

yes

—

179

C₂²

C₂×#10621902

yes

characteristic

180

C₂²

#13063371

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

172

—

185

C₄

#13063300

yes

—

186

C₄

#13063313

yes

complemented

187

C₂

—

344

—

188

C₂

C₂×#13063300

yes

characteristic

189

C₂

C₂×#13063313

yes

characteristic

190

C₂

#14583468

yes

characteristic

191

C₂

#14583570

yes

—

192

C₂

#14583634

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2548
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

hypercenter

C₄×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×#13063313	C₂	semidirect
3. C₂×C₄×Dic₈₆	C₂	semidirect
4. #14582817	C₂	semidirect
5. #14582801	C₂	semidirect
6. #14582800	C₂	semidirect
7. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
40. C₂×Dic₈₆	D₄	semidirect
87. C₄×C₄₃	#124	semidirect
103. C₄×C₄⋊C₄	D₄₃	semidirect
104. C₄₃	#2548	semidirect
186. C₄	#13063313	semidirect

H ≅ G/N

type

2. C₂×#13063313

C₂

semidirect

3. C₂×C₄×Dic₈₆

C₂

semidirect

4. #14582817

C₂

semidirect

5. #14582801

C₂

semidirect

6. #14582800

C₂

semidirect

7. C₄×C₄⋊C₄×C₄₃

C₂

semidirect

40. C₂×Dic₈₆

D₄

semidirect

87. C₄×C₄₃

#124

semidirect

103. C₄×C₄⋊C₄

D₄₃

semidirect

104. C₄₃

#2548

semidirect

186. C₄

#13063313

semidirect

Subgroups — #19139597

Conjugacy classes of subgroups

Complements