Group #19133700 — subgroups

Subgroups — #19133700

Order 5504.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583246	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583099	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583095	C₂	1	1	yes	maximal, characteristic
5	#14582644	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582628	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₃	—	2	1	no	—
9	C₂×#10621884	—	2	1	no	—
10	C₂×D₃₄₄	—	2	1	no	—
11	#13063325	—	2	1	no	—
12	#13063325	C₂²	1	1	yes	characteristic
13	#13063322	C₂²	1	1	yes	characteristic
14	#13063305	—	2	1	no	—
15	#13063305	—	2	2	no	—
16	C₄×D₁₇₂	—	2	1	no	—
17	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	#13063247	—	2	1	no	—
20	#13063247	—	2	2	no	—
21	#13063245	—	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	D₄	1	1	yes	characteristic
26	Q₈×D₄₃	—	4	2	no	—
27	Q₈×D₄₃	—	4	2	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	C₂×D₁₇₂	D₄	1	1	yes	characteristic
32	C₂×C₄×D₄₃	—	2	2	no	—
33	C₂×C₄×D₄₃	—	2	1	no	—
34	C₂×Dic₈₆	—	2	2	no	—
35	C₂×Dic₈₆	—	2	1	no	—
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	D₄	1	1	yes	characteristic
40	C₄²×C₄₃	—	2	1	no	—
41	C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	#10621884	—	4	2	no	—
43	(C₂×D₄₃)⋊C₄	—	2	2	no	—
44	(C₂×D₄₃)⋊C₄	—	2	1	no	—
45	C₄⋊Dic₄₃	—	2	2	no	—
46	C₄⋊Dic₄₃	—	2	1	no	—
47	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
48	C₄⋊Dic₄₃	D₄	1	2	yes	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	D₃₄₄	—	4	2	no	—
51	C₂²×D₄₃	—	2	1	no	—
52	C₂²×D₄₃	—	4	1	no	—
53	Q₈×C₄₃	—	2	1	no	—
54	Q₈×C₄₃	D₈	1	2	yes	complemented
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	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	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₄₃	—	2	1	no	—
61	C₂×Dic₄₃	—	2	2	no	—
62	D₁₇₂	—	2	1	no	—
63	D₁₇₂	—	4	2	no	—
64	D₁₇₂	—	4	1	no	—
65	D₁₇₂	—	4	1	no	—
66	D₁₇₂	—	4	1	no	—
67	D₁₇₂	—	4	1	no	—
68	D₁₇₂	SD₁₆	1	2	yes	complemented
69	C₄×D₄₃	—	4	4	no	—
70	C₄×D₄₃	—	4	1	no	—
71	C₄×D₄₃	—	4	1	no	—
72	Dic₈₆	—	4	4	no	—
73	Dic₈₆	—	4	1	no	—
74	Dic₈₆	—	4	1	no	—
75	C₈×C₄₃	—	4	1	no	—
76	C₈×C₄₃	—	4	1	no	—
77	C₄₃⋊C₈	—	4	1	no	—
78	C₂²×C₄₃	#120	1	1	yes	characteristic
79	C₂×D₄₃	—	2	2	no	—
80	C₂×D₄₃	—	4	1	no	—
81	C₂×D₄₃	—	4	1	no	—
82	C₂×D₄₃	—	8	1	no	—
83	C₂×D₄₃	—	8	1	no	—
84	C₂×D₄₃	—	8	1	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	4	1	no	—
88	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
89	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄×C₄₃	#137	1	1	yes	characteristic
92	Dic₄₃	—	4	1	no	—
93	Dic₄₃	—	4	2	no	—
94	#1627	—	43	1	no	maximal, Sylow
95	C₂×C₄₃	#447	1	1	yes	characteristic
96	C₂×C₄₃	#448	1	1	yes	characteristic
97	C₂×C₄₃	#454	1	1	yes	characteristic
98	D₄₃	—	4	2	no	—
99	D₄₃	—	16	1	no	—
100	D₄×Q₈	—	43	1	no	—
101	#486	—	43	1	no	—
102	#460	—	43	1	no	—
103	#459	—	43	1	no	—
104	#332	—	43	1	no	—
105	Q₈⋊C₈	D₄₃	1	1	yes	characteristic, complemented
106	D₄⋊C₈	—	43	1	no	—
107	C₄₃	#1627	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	86	1	no	—
109	C₂×SD₁₆	—	86	1	no	—
110	C₂×D₈	—	86	1	no	—
111	C₄⋊Q₈	—	43	1	no	—
112	C₄⋊Q₈	—	86	1	no	—
113	#127	—	43	1	no	—
114	C₂²⋊Q₈	—	86	2	no	—
115	C₂²⋊Q₈	—	86	1	no	—
116	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
117	C₄×D₄	—	43	1	no	—
118	C₄×D₄	—	86	1	no	—
119	C₄⋊C₈	—	43	1	no	—
120	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
121	#102	—	86	1	no	—
122	#102	—	86	2	no	—
123	#102	—	86	1	no	—
124	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
125	C₂×Q₈	—	86	2	no	—
126	C₂×Q₈	—	86	1	no	—
127	C₂×Q₈	—	172	2	no	—
128	C₂×Q₈	—	172	2	no	—
129	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
130	C₂×D₄	—	43	1	no	—
131	C₂×D₄	—	86	1	no	—
132	C₂×D₄	—	86	1	no	—
133	C₂×D₄	—	86	1	no	—
134	C₂²×C₄	—	86	2	no	—
135	C₂²×C₄	—	86	1	no	—
136	SD₁₆	—	172	2	no	—
137	D₈	—	172	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	86	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	43	1	no	—
143	C₄⋊C₄	—	43	2	no	—
144	C₄⋊C₄	—	86	2	no	—
145	C₄⋊C₄	—	86	1	no	—
146	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
147	C₂²⋊C₄	—	86	2	no	—
148	C₂²⋊C₄	—	86	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₂²×D₄₃	1	1	yes	characteristic
151	C₂³	—	86	1	no	—
152	C₂³	—	172	1	no	—
153	Q₈	—	2	1	no	—
154	Q₈	—	172	4	no	—
155	Q₈	—	172	1	no	—
156	Q₈	—	172	1	no	—
157	Q₈	D₃₄₄	1	2	yes	complemented
158	D₄	—	43	2	no	—
159	D₄	—	86	1	no	—
160	D₄	—	172	2	no	—
161	D₄	—	172	1	no	—
162	D₄	—	172	1	no	—
163	D₄	—	172	1	no	—
164	D₄	—	172	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	86	2	no	—
168	C₂×C₄	—	86	1	no	—
169	C₂×C₄	—	172	4	no	—
170	C₂×C₄	—	172	1	no	—
171	C₂×C₄	—	172	1	no	—
172	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
173	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
174	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
175	C₈	—	4	1	no	—
176	C₈	—	4	1	no	—
177	C₈	—	172	1	no	—
178	C₂²	—	86	2	no	—
179	C₂²	—	344	1	no	—
180	C₂²	—	344	1	no	—
181	C₂²	—	344	1	no	—
182	C₂²	—	172	1	no	—
183	C₂²	—	172	1	no	—
184	C₂²	#13063295	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	1	no	—
188	C₄	—	172	2	no	—
189	C₄	—	172	1	no	—
190	C₄	C₂×D₃₄₄	1	1	yes	characteristic
191	C₄	SD₁₆×D₄₃	1	1	yes	characteristic
192	C₄	#13063353	1	1	yes	characteristic
193	C₄	#13063357	1	1	yes	characteristic
194	C₂	—	172	2	no	—
195	C₂	—	688	1	no	—
196	C₂	#14583035	1	1	yes	characteristic
197	C₂	#14583039	1	1	yes	characteristic
198	C₂	#14583073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₇₂

C₂

yes

maximal, characteristic, complemented

#14583246

C₂

yes

maximal, characteristic, complemented

#14583099

C₂

yes

maximal, characteristic, complemented

#14583095

C₂

yes

maximal, characteristic

#14582644

C₂

yes

maximal, characteristic, complemented

#14582628

C₂

yes

maximal, characteristic, complemented

C₄₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₃

—

C₂×#10621884

—

C₂×D₃₄₄

—

#13063325

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

characteristic

#13063305

—

#13063305

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

D₄

yes

characteristic

Q₈×D₄₃

—

Q₈×D₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621884

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

D₃₄₄

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

D₈

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

SD₁₆

yes

complemented

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

100

D₄×Q₈

—

101

#486

—

102

#460

—

103

#459

—

104

#332

—

105

Q₈⋊C₈

D₄₃

yes

characteristic, complemented

106

D₄⋊C₈

—

107

C₄₃

#1627

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

—

109

C₂×SD₁₆

—

110

C₂×D₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

#127

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

C₂×D₄₃

yes

characteristic

117

C₄×D₄

—

118

C₄×D₄

—

119

C₄⋊C₈

—

120

C₄⋊C₈

C₂×D₄₃

yes

characteristic

121

#102

—

122

#102

—

123

#102

—

124

C₄×C₈

C₂×D₄₃

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

172

—

128

C₂×Q₈

—

172

—

129

C₂×Q₈

D₁₇₂

yes

characteristic

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

SD₁₆

—

172

—

137

D₈

—

172

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

D₁₇₂

yes

characteristic

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

C₂²×D₄₃

yes

characteristic

151

C₂³

—

152

C₂³

—

172

—

153

Q₈

—

154

Q₈

—

172

—

155

Q₈

—

172

—

156

Q₈

—

172

—

157

Q₈

D₃₄₄

yes

complemented

158

D₄

—

159

D₄

—

160

D₄

—

172

—

161

D₄

—

172

—

162

D₄

—

172

—

163

D₄

—

172

—

164

D₄

—

172

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

172

—

170

C₂×C₄

—

172

—

171

C₂×C₄

—

172

—

172

C₂×C₄

C₂×D₁₇₂

yes

characteristic

173

C₂×C₄

D₄×D₄₃

yes

characteristic

174

C₂×C₄

D₄×D₄₃

yes

characteristic

175

C₈

—

176

C₈

—

177

C₈

—

172

—

178

C₂²

—

179

C₂²

—

344

—

180

C₂²

—

344

—

181

C₂²

—

344

—

182

C₂²

—

172

—

183

C₂²

—

172

—

184

C₂²

#13063295

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

172

—

189

C₄

—

172

—

190

C₄

C₂×D₃₄₄

yes

characteristic

191

C₄

SD₁₆×D₄₃

yes

characteristic

192

C₄

#13063353

yes

characteristic

193

C₄

#13063357

yes

characteristic

194

C₂

—

172

—

195

C₂

—

688

—

196

C₂

#14583035

yes

characteristic

197

C₂

#14583039

yes

characteristic

198

C₂

#14583073

yes

characteristic

	subgroup	G/H
center	C₂²	#13063295
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×Q₈⋊C₈	C₂
socle	C₂²×C₄₃	#120
hypercenter	Q₈⋊C₈	D₄₃
hypocenter	C₄₃	#1627
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13063295

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×Q₈⋊C₈

C₂

socle

C₂²×C₄₃

#120

hypercenter

Q₈⋊C₈

D₄₃

hypocenter

C₄₃

#1627

LCS

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
1. Q₈×D₁₇₂	C₂	semidirect
2. #14583246	C₂	semidirect
3. #14583099	C₂	semidirect
5. #14582644	C₂	semidirect
6. #14582628	C₂	semidirect
7. C₄₃×Q₈⋊C₈	C₂	semidirect
54. Q₈×C₄₃	D₈	semidirect
68. D₁₇₂	SD₁₆	semidirect
105. Q₈⋊C₈	D₄₃	semidirect
107. C₄₃	#1627	semidirect
157. Q₈	D₃₄₄	semidirect

H ≅ G/N

type

1. Q₈×D₁₇₂

C₂

semidirect

2. #14583246

C₂

semidirect

3. #14583099

C₂

semidirect

5. #14582644

C₂

semidirect

6. #14582628

C₂

semidirect

7. C₄₃×Q₈⋊C₈

C₂

semidirect

54. Q₈×C₄₃

D₈

semidirect

68. D₁₇₂

SD₁₆

semidirect

105. Q₈⋊C₈

D₄₃

semidirect

107. C₄₃

#1627

semidirect

157. Q₈

D₃₄₄

semidirect

Subgroups — #19133700

Conjugacy classes of subgroups

Complements