Group #20242672 — subgroups

Subgroups — #20242672

Order 6016.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689519	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689403	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689328	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#547	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625335	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13093957	—	2	1	no	—
11	#13093955	—	2	2	no	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#13093947	—	2	1	no	—
15	#13093945	—	2	1	no	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#41×C₄₇	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	#123×C₄₇	—	2	2	no	—
20	C₂²⋊Q₈×C₄₇	—	2	1	no	—
21	#121×C₄₇	—	2	1	no	—
22	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
25	#41×C₄₇	—	4	2	no	—
26	#41×C₄₇	—	4	2	no	—
27	C₂×Q₈×C₄₇	—	2	1	no	—
28	C₂×D₄×C₄₇	—	2	2	no	—
29	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	1	no	—
33	C₂²×C₄×C₄₇	—	2	1	no	—
34	C₂×Dic₉₄	—	2	1	no	—
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
40	C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂²⋊C₄×C₄₇	—	2	1	no	—
42	C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#10625335	—	4	2	no	—
45	C₄⋊Dic₄₇	—	2	1	no	—
46	C₄⋊Dic₄₇	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂×C₄₇⋊C₈	—	2	1	no	—
49	C₂×C₄₇⋊C₈	—	2	1	no	—
50	C₂³×C₄₇	—	2	1	no	—
51	C₂³×C₄₇	—	2	1	no	—
52	Q₈×C₄₇	—	4	1	no	—
53	Q₈×C₄₇	—	4	1	no	—
54	D₄×C₄₇	—	2	1	no	—
55	D₄×C₄₇	—	2	1	no	—
56	D₄×C₄₇	—	2	1	no	—
57	D₄×C₄₇	—	4	4	no	—
58	D₄×C₄₇	SD₁₆	1	2	yes	complemented
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	4	4	no	—
63	C₂×C₄×C₄₇	—	4	1	no	—
64	C₂×C₄×C₄₇	—	4	1	no	—
65	C₂×C₄×C₄₇	—	4	1	no	—
66	C₂×C₄×C₄₇	—	4	1	no	—
67	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₄₇	—	4	1	no	—
71	Dic₉₄	—	4	1	no	—
72	Dic₉₄	—	4	1	no	—
73	C₄₇⋊C₈	—	2	2	no	—
74	C₄₇⋊C₈	—	4	1	no	—
75	C₄₇⋊C₈	—	4	1	no	—
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	4	1	no	—
78	C₂²×C₄₇	—	4	1	no	—
79	C₂²×C₄₇	—	4	1	no	—
80	C₂²×C₄₇	—	4	1	no	—
81	C₂²×C₄₇	—	4	1	no	—
82	C₂²×C₄₇	#120	1	1	yes	characteristic
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	4	1	no	—
85	C₄×C₄₇	—	4	1	no	—
86	C₄×C₄₇	—	4	2	no	—
87	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×C₄₇	#135	1	1	yes	characteristic
89	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
91	Dic₄₇	—	8	1	no	—
92	#1654	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	—	4	2	no	—
94	C₂×C₄₇	—	8	1	no	—
95	C₂×C₄₇	#449	1	1	yes	characteristic
96	C₂×C₄₇	#450	1	1	yes	characteristic
97	C₂×C₄₇	#456	1	1	yes	characteristic
98	#547	D₄₇	1	1	yes	characteristic
99	#498	—	47	1	no	—
100	#474	—	47	1	no	—
101	#461	—	47	1	no	—
102	#332	—	47	1	no	—
103	D₄⋊C₈	—	47	1	no	—
104	D₄⋊C₈	—	47	1	no	—
105	C₄₇	#1654	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	2	1	no	—
107	C₂×SD₁₆	—	94	1	no	—
108	C₄⋊Q₈	—	47	1	no	—
109	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
110	#123	—	2	2	no	—
111	C₂²⋊Q₈	—	2	1	no	—
112	#121	—	2	1	no	—
113	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
114	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	#107	—	94	1	no	—
117	#106	—	94	2	no	—
118	C₄⋊C₈	—	47	1	no	—
119	C₄⋊C₈	—	47	1	no	—
120	#103	—	94	1	no	—
121	#102	—	94	1	no	—
122	C₄×C₈	—	47	1	no	—
123	#41	—	4	2	no	—
124	#41	—	4	2	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	94	1	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
129	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	SD₁₆	—	188	2	no	—
134	C₂×C₈	—	94	1	no	—
135	C₂×C₈	—	94	1	no	—
136	C₂×C₈	—	94	1	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	94	1	no	—
140	C₄⋊C₄	—	94	1	no	—
141	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
143	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	1	no	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	C₂²×D₄₇	1	1	yes	characteristic
148	C₂³	—	2	1	no	—
149	C₂³	—	2	1	no	—
150	Q₈	—	4	1	no	—
151	Q₈	—	4	1	no	—
152	Q₈	—	188	1	no	—
153	Q₈	—	188	1	no	—
154	D₄	—	2	1	no	—
155	D₄	—	2	1	no	—
156	D₄	—	2	1	no	—
157	D₄	—	4	4	no	—
158	D₄	#10625335	1	2	yes	complemented
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	188	1	no	—
168	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
171	C₈	—	94	2	no	—
172	C₈	—	188	1	no	—
173	C₈	—	188	1	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	#13093993	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	376	1	no	—
186	C₄	C₂×#10625335	1	1	yes	characteristic
187	C₄	#13094044	1	1	yes	characteristic
188	C₄	#13094049	1	1	yes	characteristic
189	C₄	#13094050	1	1	yes	characteristic
190	C₂	—	4	2	no	—
191	C₂	—	8	1	no	—
192	C₂	#15689268	1	1	yes	characteristic
193	C₂	#15689274	1	1	yes	characteristic
194	C₂	#15689306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689519

C₂

yes

maximal, characteristic, complemented

#15689403

C₂

yes

maximal, characteristic, complemented

#15689328

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₄₇×#547

C₂

yes

maximal, characteristic

C₂×#10625335

—

#13094024

C₂²

yes

characteristic

#13093957

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093947

—

#13093945

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×#41×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#123×C₄₇

—

C₂²⋊Q₈×C₄₇

—

#121×C₄₇

—

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

SD₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1654

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#498

—

100

#474

—

101

#461

—

102

#332

—

103

D₄⋊C₈

—

104

D₄⋊C₈

—

105

C₄₇

#1654

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂×SD₁₆

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

110

#123

—

111

C₂²⋊Q₈

—

112

#121

—

113

C₄×D₄

C₂×D₄₇

yes

characteristic

114

C₄×D₄

C₂×D₄₇

yes

characteristic

115

C₂×C₄⋊C₄

—

116

#107

—

117

#106

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

#103

—

121

#102

—

122

C₄×C₈

—

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×D₄

—

128

C₂×D₄

C₂²⋊D₄₇

yes

characteristic

129

C₂×D₄

C₂²⋊D₄₇

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

SD₁₆

—

188

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

142

C₄⋊C₄

C₂²⋊D₄₇

yes

—

143

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

C₂²×D₄₇

yes

characteristic

148

C₂³

—

149

C₂³

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

188

—

153

Q₈

—

188

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

#10625335

yes

complemented

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

188

—

168

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

169

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

170

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

171

C₈

—

172

C₈

—

188

—

173

C₈

—

188

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

#13093993

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

376

—

186

C₄

C₂×#10625335

yes

characteristic

187

C₄

#13094044

yes

characteristic

188

C₄

#13094049

yes

characteristic

189

C₄

#13094050

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

#15689268

yes

characteristic

193

C₂

#15689274

yes

characteristic

194

C₂

#15689306

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#547	C₂
socle	C₂²×C₄₇	#120
hypercenter	#547	D₄₇
hypocenter	C₄₇	#1654
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #547

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#547

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #547

N	H ≅ G/N	type
1. #15689519	C₂	semidirect
2. #15689403	C₂	semidirect
3. #15689328	C₂	semidirect
4. #15688869	C₂	semidirect
5. D₄⋊(C₄₇⋊C₈)	C₂	semidirect
6. D₄⋊(C₄₇⋊C₈)	C₂	semidirect
58. D₄×C₄₇	SD₁₆	semidirect
105. C₄₇	#1654	semidirect
158. D₄	#10625335	semidirect

H ≅ G/N

type

1. #15689519

C₂

semidirect

2. #15689403

C₂

semidirect

3. #15689328

C₂

semidirect

4. #15688869

C₂

semidirect

5. D₄⋊(C₄₇⋊C₈)

C₂

semidirect

6. D₄⋊(C₄₇⋊C₈)

C₂

semidirect

58. D₄×C₄₇

SD₁₆

semidirect

105. C₄₇

#1654

semidirect

158. D₄

#10625335

semidirect

Subgroups — #20242672

Conjugacy classes of subgroups

Complements