Group #19135606 — subgroups

Subgroups — #19135606

Order 5504.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063305	C₂	1	2	yes	maximal
3	C₂×#13063248	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₄₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₈₆	C₂²	1	2	yes	—
7	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₇₂	—	2	4	no	—
9	C₂×Dic₁₇₂	—	2	2	no	—
10	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	#13063305	—	2	4	no	—
12	#13063305	—	2	4	no	—
13	C₂×C₄⋊Dic₄₃	—	2	2	no	—
14	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	#13063248	—	2	4	no	—
16	(C₂×D₄₃)⋊C₈	—	2	2	no	—
17	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₂³×D₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₃	—	2	2	no	—
23	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
24	C₂×C₄×D₄₃	—	2	1	no	—
25	C₂×C₄×D₄₃	—	2	2	no	—
26	C₂×C₄×D₄₃	—	2	2	no	—
27	C₂×C₄×D₄₃	—	2	1	no	—
28	C₂×Dic₈₆	—	2	2	no	—
29	C₂×Dic₈₆	—	2	2	no	—
30	C₂×Dic₈₆	—	2	2	no	—
31	C₂×Dic₈₆	—	2	2	no	—
32	C₂×Dic₈₆	—	2	2	no	—
33	C₂×Dic₈₆	D₄	1	4	yes	—
34	C₂×C₈×C₄₃	—	2	2	no	—
35	C₂×C₈×C₄₃	D₄	1	2	yes	—
36	C₄⋊C₄×C₄₃	—	2	2	no	—
37	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
38	(C₂×D₄₃)⋊C₄	—	2	8	no	—
39	C₄⋊Dic₄₃	—	2	8	no	—
40	C₂×C₄₃⋊C₈	—	2	1	no	—
41	C₂×C₄₃⋊C₈	—	2	2	no	—
42	Dic₁₇₂	—	2	8	no	—
43	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×D₄₃	—	2	2	no	—
45	C₂²×D₄₃	—	2	1	no	—
46	C₂²×D₄₃	—	2	1	no	—
47	C₂²×D₄₃	—	2	2	no	—
48	C₂²×D₄₃	Q₁₆	1	2	yes	complemented
49	C₂×C₄×C₄₃	—	2	4	no	—
50	C₂×C₄×C₄₃	—	4	2	no	—
51	C₂×C₄×C₄₃	—	4	2	no	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	2	4	no	—
58	C₂×Dic₄₃	—	2	1	no	—
59	C₂×Dic₄₃	—	4	2	no	—
60	C₂×Dic₄₃	—	4	2	no	—
61	C₄×D₄₃	—	4	1	no	—
62	C₄×D₄₃	—	4	2	no	—
63	C₄×D₄₃	—	4	1	no	—
64	Dic₈₆	—	2	4	no	—
65	Dic₈₆	—	2	4	no	—
66	Dic₈₆	—	4	2	no	—
67	Dic₈₆	—	4	2	no	—
68	C₈×C₄₃	—	2	2	no	—
69	C₄₃⋊C₈	—	4	1	no	—
70	C₂²×C₄₃	#120	1	2	yes	—
71	C₂²×C₄₃	#121	1	1	yes	characteristic
72	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×C₄₃	#135	1	1	yes	characteristic
74	C₂²×C₄₃	#137	1	2	yes	—
75	C₂×D₄₃	—	2	4	no	—
76	C₂×D₄₃	—	2	2	no	—
77	C₂×D₄₃	—	4	1	no	—
78	C₂×D₄₃	—	4	2	no	—
79	C₂×D₄₃	—	4	1	no	—
80	C₄×C₄₃	—	4	4	no	—
81	C₄×C₄₃	#120	1	1	yes	characteristic
82	C₄×C₄₃	#121	1	2	yes	—
83	C₄×C₄₃	#127	1	1	yes	characteristic
84	Dic₄₃	—	4	4	no	—
85	Dic₄₃	—	4	1	no	—
86	#2001	—	43	1	no	maximal, Sylow
87	C₂×C₄₃	#392	1	1	yes	characteristic
88	C₂×C₄₃	#451	1	2	yes	—
89	C₂×C₄₃	#452	1	2	yes	—
90	C₂×C₄₃	#467	1	1	yes	characteristic
91	C₂×C₄₃	#470	1	1	yes	characteristic
92	D₄₃	—	4	2	no	—
93	C₂²×Q₁₆	—	43	1	no	—
94	C₂×C₂²⋊Q₈	—	43	2	no	—
95	C₂×#103	—	43	2	no	—
96	C₂×C₂²⋊C₈	—	43	1	no	—
97	#340	D₄₃	1	1	yes	characteristic, complemented
98	C₄₃	#2001	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	43	2	no	—
100	C₂³×C₄	—	43	1	no	—
101	C₂×Q₁₆	—	86	4	no	—
102	C₂×Q₁₆	—	86	2	no	—
103	C₂²×C₈	—	43	1	no	—
104	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
105	C₂²⋊Q₈	—	86	4	no	—
106	C₂²⋊Q₈	—	86	4	no	—
107	C₂×C₄⋊C₄	—	86	2	no	—
108	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
109	C₂×C₂²⋊C₄	—	86	2	no	—
110	#103	—	86	4	no	—
111	C₂²⋊C₈	—	86	2	no	—
112	C₂⁴	—	43	1	no	—
113	C₂×Q₈	—	43	4	no	—
114	C₂×Q₈	—	86	2	no	—
115	C₂×Q₈	—	86	2	no	—
116	C₂×Q₈	—	86	2	no	—
117	C₂×Q₈	—	86	2	no	—
118	C₂×Q₈	—	86	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	43	1	no	—
121	C₂²×C₄	—	86	2	no	—
122	C₂²×C₄	—	86	1	no	—
123	C₂²×C₄	—	86	2	no	—
124	C₂²×C₄	—	86	2	no	—
125	C₂²×C₄	—	86	1	no	—
126	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
127	Q₁₆	—	86	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	86	1	no	—
130	C₂×C₈	—	86	2	no	—
131	C₂×C₈	D₁₇₂	1	2	yes	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	86	8	no	—
134	C₄⋊C₄	D₁₇₂	1	4	yes	—
135	C₂²⋊C₄	—	86	8	no	—
136	C₂³	—	43	2	no	—
137	C₂³	—	86	2	no	—
138	C₂³	—	86	1	no	—
139	C₂³	—	86	1	no	—
140	C₂³	—	86	2	no	—
141	C₂³	D₄×D₄₃	1	1	yes	characteristic
142	Q₈	—	86	4	no	—
143	Q₈	—	86	4	no	—
144	Q₈	—	172	2	no	—
145	Q₈	—	172	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	86	4	no	—
150	C₂×C₄	—	86	2	no	—
151	C₂×C₄	—	86	1	no	—
152	C₂×C₄	—	172	2	no	—
153	C₂×C₄	—	172	2	no	—
154	C₂×C₄	—	172	1	no	—
155	C₂×C₄	—	172	2	no	—
156	C₂×C₄	—	172	1	no	—
157	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
158	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
159	C₂×C₄	D₄×D₄₃	1	2	yes	—
160	C₂×C₄	#10621902	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	172	1	no	—
163	C₂²	—	86	4	no	—
164	C₂²	—	86	2	no	—
165	C₂²	—	172	2	no	—
166	C₂²	—	172	1	no	—
167	C₂²	—	172	1	no	—
168	C₂²	#13063295	1	2	yes	—
169	C₂²	#13063300	1	1	yes	characteristic
170	C₂²	Q₁₆×D₄₃	1	1	yes	characteristic
171	C₂²	#13063344	1	1	yes	characteristic
172	C₂²	#13063354	1	2	yes	—
173	C₄	—	4	4	no	—
174	C₄	—	172	1	no	—
175	C₄	—	172	4	no	—
176	C₄	#13063295	1	1	yes	characteristic
177	C₄	#13063300	1	2	yes	—
178	C₄	#13063322	1	1	yes	characteristic
179	C₂	—	172	2	no	—
180	C₂	#14582809	1	1	yes	characteristic
181	C₂	#14583056	1	2	yes	—
182	C₂	#14583062	1	2	yes	—
183	C₂	#14583141	1	1	yes	characteristic
184	C₂	#14583158	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

C₂×#13063305

C₂

yes

maximal

C₂×#13063248

C₂

yes

maximal, complemented

C₂×(C₂×D₄₃)⋊C₈

C₂

yes

maximal, characteristic

C₄₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

—

C₂²×C₄×D₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

#13063305

—

#13063305

—

C₂×C₄⋊Dic₄₃

—

C₂×(C₂×D₄₃)⋊C₄

—

#13063248

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂³×D₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

—

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#121

yes

—

C₄×C₄₃

#127

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#452

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂²×Q₁₆

—

C₂×C₂²⋊Q₈

—

C₂×#103

—

C₂×C₂²⋊C₈

—

#340

D₄₃

yes

characteristic, complemented

C₄₃

#2001

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂³×C₄

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂×D₄₃

yes

characteristic

105

C₂²⋊Q₈

—

106

C₂²⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

109

C₂×C₂²⋊C₄

—

110

#103

—

111

C₂²⋊C₈

—

112

C₂⁴

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₄₃

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

D₁₇₂

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

D₁₇₂

yes

—

135

C₂²⋊C₄

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

D₄×D₄₃

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

172

—

145

Q₈

—

172

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

172

—

153

C₂×C₄

—

172

—

154

C₂×C₄

—

172

—

155

C₂×C₄

—

172

—

156

C₂×C₄

—

172

—

157

C₂×C₄

C₂×D₁₇₂

yes

characteristic

158

C₂×C₄

C₂×D₁₇₂

yes

—

159

C₂×C₄

D₄×D₄₃

yes

—

160

C₂×C₄

#10621902

yes

characteristic

161

C₈

—

162

C₈

—

172

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

172

—

166

C₂²

—

172

—

167

C₂²

—

172

—

168

C₂²

#13063295

yes

—

169

C₂²

#13063300

yes

characteristic

170

C₂²

Q₁₆×D₄₃

yes

characteristic

171

C₂²

#13063344

yes

characteristic

172

C₂²

#13063354

yes

—

173

C₄

—

174

C₄

—

172

—

175

C₄

—

172

—

176

C₄

#13063295

yes

characteristic

177

C₄

#13063300

yes

—

178

C₄

#13063322

yes

characteristic

179

C₂

—

172

—

180

C₂

#14582809

yes

characteristic

181

C₂

#14583056

yes

—

182

C₂

#14583062

yes

—

183

C₂

#14583141

yes

characteristic

184

C₂

#14583158

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#340	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#340	D₄₃
hypocenter	C₄₃	#2001
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#340

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
1. C₂²×Dic₁₇₂	C₂	semidirect
3. C₂×#13063248	C₂	semidirect
5. C₄₃×#340	C₂	semidirect
48. C₂²×D₄₃	Q₁₆	semidirect
97. #340	D₄₃	semidirect
98. C₄₃	#2001	semidirect

H ≅ G/N

type

1. C₂²×Dic₁₇₂

C₂

semidirect

3. C₂×#13063248

C₂

semidirect

5. C₄₃×#340

C₂

semidirect

48. C₂²×D₄₃

Q₁₆

semidirect

97. #340

D₄₃

semidirect

98. C₄₃

#2001

semidirect

Subgroups — #19135606

Conjugacy classes of subgroups

Complements