Group #20244696 — subgroups

Subgroups — #20244696

Order 6016.

360 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 89 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×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal, complemented
3	#15689026	C₂	1	2	yes	maximal, complemented
4	#15688886	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
9	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
10	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	Dic₄₇⋊C₈	—	2	2	no	—
12	#13093945	—	2	4	no	—
13	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
14	C₂×D₈×C₄₇	—	2	4	no	—
15	C₂×D₈×C₄₇	—	2	2	no	—
16	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄₇	—	2	2	no	—
18	C₂×D₄×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	2	no	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	D₄	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
27	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
28	D₈×C₄₇	—	2	8	no	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	D₄	1	2	yes	—
31	C₂²⋊Dic₄₇	—	4	4	no	—
32	C₂²⋊Dic₄₇	—	4	4	no	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	—	2	2	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₄×Dic₄₇	—	2	1	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂³×C₄₇	—	2	4	no	—
40	C₂³×C₄₇	—	4	2	no	—
41	C₂³×C₄₇	—	4	2	no	—
42	C₂³×C₄₇	—	4	2	no	—
43	C₂³×C₄₇	—	4	2	no	—
44	C₂³×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	4	2	no	—
46	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
47	D₄×C₄₇	—	2	4	no	—
48	D₄×C₄₇	—	2	4	no	—
49	D₄×C₄₇	—	4	2	no	—
50	D₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	2	yes	—
55	C₂×Dic₄₇	—	2	1	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	1	no	—
58	C₂×Dic₄₇	—	2	2	no	—
59	C₂×Dic₄₇	—	4	2	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	—	4	2	no	—
62	C₂×Dic₄₇	D₈	1	2	yes	complemented
63	C₈×C₄₇	—	2	2	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₂²×C₄₇	—	4	4	no	—
66	C₂²×C₄₇	—	4	4	no	—
67	C₂²×C₄₇	—	4	4	no	—
68	C₂²×C₄₇	—	8	2	no	—
69	C₂²×C₄₇	—	8	2	no	—
70	C₂²×C₄₇	—	8	2	no	—
71	C₂²×C₄₇	—	8	2	no	—
72	C₂²×C₄₇	#121	1	1	yes	characteristic
73	C₂²×C₄₇	#121	1	2	yes	—
74	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
75	C₂²×C₄₇	#135	1	1	yes	characteristic
76	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₄×C₄₇	#120	1	1	yes	characteristic
78	C₄×C₄₇	#123	1	2	yes	—
79	C₄×C₄₇	#124	1	1	yes	characteristic
80	Dic₄₇	—	2	2	no	—
81	Dic₄₇	—	4	1	no	—
82	Dic₄₇	—	8	2	no	—
83	#2054	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	—	8	4	no	—
85	C₂×C₄₇	#394	1	1	yes	characteristic
86	C₂×C₄₇	#459	1	2	yes	—
87	C₂×C₄₇	#463	1	2	yes	—
88	C₂×C₄₇	#466	1	1	yes	characteristic
89	C₂×C₄₇	#469	1	1	yes	characteristic
90	C₂²×D₈	D₄₇	1	1	yes	characteristic
91	C₂×C₄⋊C₈	—	47	1	no	—
92	C₂×#102	—	47	2	no	—
93	#390	—	47	2	no	—
94	#340	—	47	1	no	—
95	C₄₇	#2054	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	C₂×D₄₇	1	2	yes	—
97	C₂×D₈	—	2	4	no	—
98	C₂×D₈	—	2	2	no	—
99	C₂²×C₈	—	47	1	no	—
100	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	47	2	no	—
102	C₂×C₂²⋊C₄	—	94	2	no	—
103	C₂×C₂²⋊C₄	—	94	2	no	—
104	C₂×C₄²	—	47	1	no	—
105	C₄⋊C₈	—	94	2	no	—
106	#102	—	94	4	no	—
107	C₂⁴	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	C₂²⋊D₄₇	1	4	yes	—
114	C₂²×C₄	—	47	1	no	—
115	C₂²×C₄	—	47	1	no	—
116	C₂²×C₄	—	94	2	no	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	D₈	—	2	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	94	2	no	—
121	C₂×C₈	—	94	1	no	—
122	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
123	C₄⋊C₄	—	94	2	no	—
124	C₄⋊C₄	—	94	2	no	—
125	C₂²⋊C₄	—	188	4	no	—
126	C₂²⋊C₄	—	188	4	no	—
127	C₄²	—	94	1	no	—
128	C₄²	—	94	1	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	2	no	—
136	C₂³	D₄×D₄₇	1	1	yes	characteristic
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	2	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	47	2	no	—
142	C₂×C₄	—	94	1	no	—
143	C₂×C₄	—	94	1	no	—
144	C₂×C₄	—	94	1	no	—
145	C₂×C₄	—	94	2	no	—
146	C₂×C₄	—	188	2	no	—
147	C₂×C₄	—	188	2	no	—
148	C₂×C₄	—	188	2	no	—
149	C₂×C₄	#10625352	1	1	yes	characteristic
150	C₂×C₄	#10625352	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	188	1	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	—	8	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	#13093995	1	1	yes	characteristic
163	C₂²	#13093996	1	2	yes	—
164	C₂²	D₈×D₄₇	1	1	yes	characteristic
165	C₂²	#13094043	1	1	yes	characteristic
166	C₂²	#13094048	1	2	yes	—
167	C₄	—	94	2	no	—
168	C₄	—	376	2	no	—
169	C₄	—	188	1	no	—
170	C₄	#13093993	1	1	yes	characteristic
171	C₄	#13094005	1	2	yes	—
172	C₄	#13094011	1	1	yes	characteristic
173	C₂	—	8	4	no	—
174	C₂	#15689037	1	1	yes	characteristic
175	C₂	#15689316	1	2	yes	—
176	C₂	#15689336	1	2	yes	—
177	C₂	#15689356	1	1	yes	characteristic
178	C₂	#15689373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#13093945

C₂

yes

maximal, complemented

#15689026

C₂

yes

maximal, complemented

#15688886

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093945

—

C₂²×D₄×C₄₇

C₂²

yes

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

D₈

yes

complemented

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂×D₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

#120

yes

characteristic

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#124

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2054

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#394

yes

characteristic

C₂×C₄₇

#459

yes

—

C₂×C₄₇

#463

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×D₈

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#102

—

#390

—

#340

—

C₄₇

#2054

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₄₇

yes

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

100

C₂²×C₈

C₂×D₄₇

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

—

105

C₄⋊C₈

—

106

#102

—

107

C₂⁴

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²⋊D₄₇

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

D₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₄₇

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

188

—

126

C₂²⋊C₄

—

188

—

127

C₄²

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

D₄×D₄₇

yes

characteristic

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

188

—

147

C₂×C₄

—

188

—

148

C₂×C₄

—

188

—

149

C₂×C₄

#10625352

yes

characteristic

150

C₂×C₄

#10625352

yes

—

151

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

152

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

153

C₈

—

154

C₈

—

188

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#13093995

yes

characteristic

163

C₂²

#13093996

yes

—

164

C₂²

D₈×D₄₇

yes

characteristic

165

C₂²

#13094043

yes

characteristic

166

C₂²

#13094048

yes

—

167

C₄

—

168

C₄

—

376

—

169

C₄

—

188

—

170

C₄

#13093993

yes

characteristic

171

C₄

#13094005

yes

—

172

C₄

#13094011

yes

characteristic

173

C₂

—

174

C₂

#15689037

yes

characteristic

175

C₂

#15689316

yes

—

176

C₂

#15689336

yes

—

177

C₂

#15689356

yes

characteristic

178

C₂

#15689373

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂²×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂²×D₈	D₄₇
hypocenter	C₄₇	#2054
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂²×D₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂²×D₈

D₄₇

hypocenter

C₄₇

#2054

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×Dic₄₇⋊C₈	C₂	semidirect
2. C₂×#13093945	C₂	semidirect
3. #15689026	C₂	semidirect
4. #15688886	C₂	semidirect
62. C₂×Dic₄₇	D₈	semidirect
95. C₄₇	#2054	semidirect

H ≅ G/N

type

1. C₂×Dic₄₇⋊C₈

C₂

semidirect

2. C₂×#13093945

C₂

semidirect

3. #15689026

C₂

semidirect

4. #15688886

C₂

semidirect

62. C₂×Dic₄₇

D₈

semidirect

95. C₄₇

#2054

semidirect

Subgroups — #20244696

Conjugacy classes of subgroups

Complements