Group #20242866 — subgroups

Subgroups — #20242866

Order 6016.

198 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 51 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	#15689499	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689319	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689315	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689200	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688866	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃₇₆	—	2	1	no	—
9	C₂×#10625334	—	2	1	no	—
10	C₂×#10625334	—	2	1	no	—
11	C₂×#10625334	—	2	1	no	—
12	C₂×#10625324	—	2	1	no	—
13	#13094021	C₂²	1	1	yes	characteristic
14	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#13093946	—	2	1	no	—
17	#13093945	—	2	1	no	—
18	(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	C₂×D₁₈₈	D₄	1	1	yes	characteristic
30	C₂×C₄×D₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#10625334	—	4	2	no	—
35	#10625334	—	4	2	no	—
36	#10625334	—	4	2	no	—
37	(C₂×D₄₇)⋊C₄	—	2	1	no	—
38	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	D₃₇₆	—	4	2	no	—
43	#10625324	—	4	2	no	—
44	C₂³×C₄₇	—	4	1	no	—
45	C₂²×D₄₇	—	2	1	no	—
46	C₂²×D₄₇	—	4	1	no	—
47	D₄×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	4	1	no	—
49	D₄×C₄₇	—	4	1	no	—
50	D₄×C₄₇	—	4	1	no	—
51	D₄×C₄₇	—	8	1	no	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	1	no	—
56	D₁₈₈	—	2	1	no	—
57	D₁₈₈	—	2	1	no	—
58	D₁₈₈	—	4	1	no	—
59	D₁₈₈	—	4	1	no	—
60	D₁₈₈	—	4	1	no	—
61	D₁₈₈	—	4	1	no	—
62	D₁₈₈	—	8	1	no	—
63	C₄×D₄₇	—	4	1	no	—
64	C₄×D₄₇	—	4	1	no	—
65	C₈×C₄₇	—	4	1	no	—
66	C₈×C₄₇	—	4	1	no	—
67	C₄₇⋊C₈	—	2	1	no	—
68	C₄₇⋊C₈	—	4	1	no	—
69	C₄₇⋊C₈	D₈	1	2	yes	complemented
70	C₂²×C₄₇	—	8	1	no	—
71	C₂²×C₄₇	—	8	1	no	—
72	C₂²×C₄₇	—	8	1	no	—
73	C₂²×C₄₇	#121	1	1	yes	characteristic
74	C₂×D₄₇	—	4	1	no	—
75	C₂×D₄₇	—	4	1	no	—
76	C₂×D₄₇	—	4	1	no	—
77	C₂×D₄₇	—	8	1	no	—
78	C₂×D₄₇	—	8	1	no	—
79	C₂×D₄₇	—	8	1	no	—
80	C₄×C₄₇	—	2	1	no	—
81	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	Dic₄₇	—	4	1	no	—
86	#1687	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	16	1	no	—
88	C₂×C₄₇	#459	1	1	yes	characteristic
89	C₂×C₄₇	#469	1	1	yes	characteristic
90	C₂×C₄₇	#472	1	1	yes	characteristic
91	D₄₇	—	8	1	no	—
92	D₄₇	—	16	1	no	—
93	#493	—	47	1	no	—
94	#459	—	47	1	no	—
95	#459	—	47	1	no	—
96	#436	—	47	1	no	—
97	#334	—	47	1	no	—
98	#331	—	47	1	no	—
99	#331	D₄₇	1	1	yes	characteristic, complemented
100	C₄₇	#1687	1	1	yes	characteristic, complemented, Sylow
101	C₂×D₈	—	94	1	no	—
102	C₂×D₈	—	94	1	no	—
103	C₂×D₈	—	94	1	no	—
104	C₂×D₈	—	94	1	no	—
105	C₂×M₁₆	—	94	1	no	—
106	#127	—	47	1	no	—
107	#127	C₂×D₄₇	1	1	yes	characteristic
108	C₄×D₄	—	47	1	no	—
109	C₄⋊C₈	—	47	1	no	—
110	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
112	#102	—	94	1	no	—
113	#102	—	94	1	no	—
114	C₂²⋊C₈	—	94	1	no	—
115	C₄×C₈	—	47	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	—	47	1	no	—
120	C₂×D₄	—	94	1	no	—
121	C₂×D₄	—	94	1	no	—
122	C₂×D₄	—	94	1	no	—
123	C₂²×C₄	—	94	1	no	—
124	D₈	—	188	2	no	—
125	D₈	—	188	2	no	—
126	D₈	—	188	2	no	—
127	D₈	—	188	2	no	—
128	M₁₆	—	188	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	47	1	no	—
132	C₂×C₈	—	47	1	no	—
133	C₂×C₈	—	94	1	no	—
134	C₄⋊C₄	—	47	1	no	—
135	C₂²⋊C₄	—	94	1	no	—
136	C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	C₂³	—	4	1	no	—
138	C₂³	—	94	1	no	—
139	C₂³	—	188	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	94	1	no	—
146	D₄	—	94	1	no	—
147	D₄	—	376	1	no	—
148	D₄	—	188	1	no	—
149	D₄	—	188	1	no	—
150	D₄	—	188	1	no	—
151	D₄	—	188	1	no	—
152	C₂×C₄	—	94	1	no	—
153	C₂×C₄	—	188	1	no	—
154	C₂×C₄	—	188	1	no	—
155	C₂×C₄	#10625350	1	1	yes	characteristic
156	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	4	1	no	—
160	C₈	—	47	2	no	—
161	C₈	—	94	1	no	—
162	C₈	—	188	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	376	1	no	—
167	C₂²	—	376	1	no	—
168	C₂²	—	376	1	no	—
169	C₂²	—	188	1	no	—
170	C₂²	—	188	1	no	—
171	C₂²	—	188	1	no	—
172	C₂²	#13093997	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	188	1	no	—
175	C₄	D₈×D₄₇	1	1	yes	characteristic
176	C₄	#13094048	1	1	yes	characteristic
177	C₄	#13094051	1	1	yes	characteristic
178	C₄	#13094051	1	1	yes	characteristic
179	C₂	—	16	1	no	—
180	C₂	—	376	1	no	—
181	C₂	—	752	1	no	—
182	C₂	#15689317	1	1	yes	characteristic
183	C₂	#15689375	1	1	yes	characteristic
184	C₂	#15689392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689499

C₂

yes

maximal, characteristic, complemented

#15689319

C₂

yes

maximal, characteristic, complemented

#15689315

C₂

yes

maximal, characteristic, complemented

#15689200

C₂

yes

maximal, characteristic, complemented

#15688873

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

—

—

—

—

—

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093946

—

#13093945

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

—

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

D₃₇₆

—

#10625324

—

C₂³×C₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

D₈

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1687

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

D₄₇

—

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

100

C₄₇

#1687

yes

characteristic, complemented, Sylow

101

C₂×D₈

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₂×D₈

—

105

C₂×M₁₆

—

106

#127

—

107

#127

C₂×D₄₇

yes

characteristic

108

C₄×D₄

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₄₇

yes

characteristic

111

C₄⋊C₈

C₂×D₄₇

yes

characteristic

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

D₈

—

188

—

125

D₈

—

188

—

126

D₈

—

188

—

127

D₈

—

188

—

128

M₁₆

—

188

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

C₂²×D₄₇

yes

characteristic

137

C₂³

—

138

C₂³

—

139

C₂³

—

188

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

376

—

148

D₄

—

188

—

149

D₄

—

188

—

150

D₄

—

188

—

151

D₄

—

188

—

152

C₂×C₄

—

153

C₂×C₄

—

188

—

154

C₂×C₄

—

188

—

155

C₂×C₄

#10625350

yes

characteristic

156

C₂×C₄

D₄×D₄₇

yes

characteristic

157

C₂×C₄

D₄×D₄₇

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

188

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

376

—

167

C₂²

—

376

—

168

C₂²

—

376

—

169

C₂²

—

188

—

170

C₂²

—

188

—

171

C₂²

—

188

—

172

C₂²

#13093997

yes

characteristic

173

C₄

—

174

C₄

—

188

—

175

C₄

D₈×D₄₇

yes

characteristic

176

C₄

#13094048

yes

characteristic

177

C₄

#13094051

yes

characteristic

178

C₄

#13094051

yes

characteristic

179

C₂

—

180

C₂

—

376

—

181

C₂

—

752

—

182

C₂

#15689317

yes

characteristic

183

C₂

#15689375

yes

characteristic

184

C₂

#15689392

yes

characteristic

	subgroup	G/H
center	C₂²	#13093997
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#331	C₂
socle	C₂²×C₄₇	#121
hypercenter	#331	D₄₇
hypocenter	C₄₇	#1687
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#13093997

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#331

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #15689499	C₂	semidirect
2. #15689319	C₂	semidirect
3. #15689315	C₂	semidirect
4. #15689200	C₂	semidirect
5. #15688873	C₂	semidirect
6. #15688866	C₂	semidirect
7. C₄₇×#331	C₂	semidirect
69. C₄₇⋊C₈	D₈	semidirect
99. #331	D₄₇	semidirect
100. C₄₇	#1687	semidirect

H ≅ G/N

type

1. #15689499

C₂

semidirect

2. #15689319

C₂

semidirect

3. #15689315

C₂

semidirect

4. #15689200

C₂

semidirect

5. #15688873

C₂

semidirect

6. #15688866

C₂

semidirect

7. C₄₇×#331

C₂

semidirect

69. C₄₇⋊C₈

D₈

semidirect

99. #331

D₄₇

semidirect

100. C₄₇

#1687

semidirect

Subgroups — #20242866

Conjugacy classes of subgroups

Complements