Group #20244318 — subgroups

Subgroups — #20244318

Order 6016.

344 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13094012	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093949	C₂	1	2	yes	maximal, complemented
3	C₂×#13093946	C₂	1	2	yes	maximal
4	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
8	#13094012	—	2	2	no	—
9	#13094012	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
12	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₄₇	—	2	1	no	—
14	#13093949	—	2	4	no	—
15	#13093946	—	2	4	no	—
16	C₄₇⋊#95	—	2	1	no	—
17	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂³×D₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	—	2	2	no	—
25	C₂²×Dic₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	2	no	—
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×Dic₉₄	—	2	2	no	—
32	C₂×Dic₉₄	—	2	1	no	—
33	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₄²×C₄₇	D₄	1	2	yes	—
39	(C₂×D₄₇)⋊C₄	—	2	4	no	—
40	(C₂×D₄₇)⋊C₄	—	2	4	no	—
41	C₄⋊Dic₄₇	—	2	2	no	—
42	C₄⋊Dic₄₇	D₄	1	4	yes	—
43	C₄×Dic₄₇	—	4	2	no	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	2	no	—
46	C₂²×D₄₇	—	4	2	no	—
47	C₂²×D₄₇	—	4	2	no	—
48	C₂²×D₄₇	—	4	1	no	—
49	C₂²×D₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#41	1	2	yes	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	4	2	no	—
62	C₂×Dic₄₇	—	4	2	no	—
63	C₂×Dic₄₇	—	4	2	no	—
64	D₁₈₈	—	2	4	no	—
65	D₁₈₈	—	4	1	no	—
66	D₁₈₈	—	4	1	no	—
67	Dic₉₄	—	2	4	no	—
68	Dic₉₄	—	4	1	no	—
69	Dic₉₄	—	4	1	no	—
70	C₈×C₄₇	—	4	2	no	—
71	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄₇	#121	1	2	yes	—
73	C₂²×C₄₇	#135	1	2	yes	—
74	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂²×C₄₇	#137	1	1	yes	characteristic
76	C₂×D₄₇	—	4	2	no	—
77	C₂×D₄₇	—	4	2	no	—
78	C₂×D₄₇	—	4	2	no	—
79	C₂×D₄₇	—	8	2	no	—
80	C₂×D₄₇	—	8	2	no	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	C₄×D₄	1	2	yes	—
84	C₄×C₄₇	#124	1	1	yes	characteristic
85	C₄×C₄₇	#127	1	1	yes	characteristic
86	Dic₄₇	—	4	2	no	—
87	Dic₄₇	—	4	4	no	—
88	#1983	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	#390	1	1	yes	characteristic
90	C₂×C₄₇	#416	1	1	yes	characteristic
91	C₂×C₄₇	#417	1	1	yes	characteristic
92	C₂×C₄₇	#463	1	2	yes	—
93	C₂×C₄₇	#464	1	2	yes	—
94	D₄₇	—	8	2	no	—
95	C₂×#124	—	47	1	no	—
96	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
97	C₂×#103	—	47	2	no	—
98	C₂×#102	—	47	2	no	—
99	C₄×C₄⋊C₄	—	47	1	no	—
100	C₄₇	#1983	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	47	1	no	—
102	C₂²×D₄	—	47	1	no	—
103	C₂²×C₈	C₂×D₄₇	1	2	yes	—
104	#124	—	47	4	no	—
105	#124	—	94	2	no	—
106	C₂×C₄⋊C₄	—	47	2	no	—
107	C₂×C₂²⋊C₄	—	94	1	no	—
108	C₂×C₂²⋊C₄	—	94	1	no	—
109	C₂×C₄²	—	94	1	no	—
110	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	C₄⋊C₈	—	2	2	no	—
112	#103	—	94	4	no	—
113	#102	—	94	4	no	—
114	#95	—	94	1	no	—
115	C₂⁴	—	94	1	no	—
116	C₂×Q₈	—	47	2	no	—
117	C₂×Q₈	—	94	1	no	—
118	C₂×Q₈	—	94	2	no	—
119	C₂×Q₈	—	94	2	no	—
120	C₂×D₄	—	47	2	no	—
121	C₂×D₄	—	94	2	no	—
122	C₂×D₄	—	94	2	no	—
123	C₂×D₄	—	94	1	no	—
124	C₂²×C₄	—	94	2	no	—
125	C₂²×C₄	—	94	1	no	—
126	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
127	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₄⋊C₄	—	47	4	no	—
133	C₄⋊C₄	—	94	2	no	—
134	C₂²⋊C₄	—	94	4	no	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₄²	—	188	2	no	—
137	C₄²	C₄×D₄₇	1	2	yes	—
138	C₄²	C₂²⋊D₄₇	1	2	yes	—
139	C₂³	—	94	2	no	—
140	C₂³	—	188	2	no	—
141	C₂³	—	188	2	no	—
142	C₂³	—	188	1	no	—
143	C₂³	—	188	1	no	—
144	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
145	Q₈	—	94	4	no	—
146	Q₈	—	188	1	no	—
147	Q₈	—	188	1	no	—
148	D₄	—	94	4	no	—
149	D₄	—	188	1	no	—
150	D₄	—	188	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	188	2	no	—
157	C₂×C₄	—	188	2	no	—
158	C₂×C₄	—	188	2	no	—
159	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
160	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
162	C₂×C₄	D₄×D₄₇	1	2	yes	—
163	C₂×C₄	#10625354	1	2	yes	—
164	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₂²	—	376	2	no	—
167	C₂²	—	376	2	no	—
168	C₂²	—	188	2	no	—
169	C₂²	—	188	2	no	—
170	C₂²	—	188	2	no	—
171	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
172	C₂²	#13093999	1	2	yes	—
173	C₂²	#13094046	1	2	yes	—
174	C₂²	#13094052	1	1	yes	characteristic
175	C₂²	#13094057	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	188	2	no	—
179	C₄	—	188	4	no	—
180	C₄	#13093986	1	2	yes	—
181	C₄	#13094013	1	1	yes	characteristic
182	C₄	#13094020	1	1	yes	characteristic
183	C₂	—	376	2	no	—
184	C₂	#15689024	1	1	yes	characteristic
185	C₂	#15689120	1	1	yes	characteristic
186	C₂	#15689125	1	1	yes	characteristic
187	C₂	#15689339	1	2	yes	—
188	C₂	#15689346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

C₂×#13093949

C₂

yes

maximal, complemented

C₂×#13093946

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

#13094012

—

#13094012

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

C₂×(C₂×D₄₇)⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093949

—

#13093946

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#464

yes

—

D₄₇

—

C₂×#124

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic, complemented

C₂×#103

—

C₂×#102

—

C₄×C₄⋊C₄

—

100

C₄₇

#1983

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂²×D₄

—

103

C₂²×C₈

C₂×D₄₇

yes

—

104

#124

—

105

#124

—

106

C₂×C₄⋊C₄

—

107

C₂×C₂²⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×D₄₇

yes

characteristic

111

C₄⋊C₈

—

112

#103

—

113

#102

—

114

#95

—

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₁₈₈

yes

characteristic

127

C₂²×C₄

D₁₈₈

yes

characteristic

128

C₂²×C₄

C₂²×D₄₇

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

188

—

137

C₄²

C₄×D₄₇

yes

—

138

C₄²

C₂²⋊D₄₇

yes

—

139

C₂³

—

140

C₂³

—

188

—

141

C₂³

—

188

—

142

C₂³

—

188

—

143

C₂³

—

188

—

144

C₂³

C₂×D₁₈₈

yes

characteristic

145

Q₈

—

146

Q₈

—

188

—

147

Q₈

—

188

—

148

D₄

—

149

D₄

—

188

—

150

D₄

—

188

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

188

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

160

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

161

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

162

C₂×C₄

D₄×D₄₇

yes

—

163

C₂×C₄

#10625354

yes

—

164

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

165

C₈

—

166

C₂²

—

376

—

167

C₂²

—

376

—

168

C₂²

—

188

—

169

C₂²

—

188

—

170

C₂²

—

188

—

171

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

172

C₂²

#13093999

yes

—

173

C₂²

#13094046

yes

—

174

C₂²

#13094052

yes

characteristic

175

C₂²

#13094057

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

188

—

179

C₄

—

188

—

180

C₄

#13093986

yes

—

181

C₄

#13094013

yes

characteristic

182

C₄

#13094020

yes

characteristic

183

C₂

—

376

—

184

C₂

#15689024

yes

characteristic

185

C₂

#15689120

yes

characteristic

186

C₂

#15689125

yes

characteristic

187

C₂

#15689339

yes

—

188

C₂

#15689346

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#1983
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

#1983

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×#13093949	C₂	semidirect
4. C₄×C₄⋊Dic₄₇	C₂	semidirect
5. C₂×C₄⋊C₈×C₄₇	C₂	semidirect
9. #13094012	C₄	semidirect
96. C₂×C₄⋊C₈	D₄₇	semidirect
100. C₄₇	#1983	semidirect

H ≅ G/N

type

2. C₂×#13093949

C₂

semidirect

4. C₄×C₄⋊Dic₄₇

C₂

semidirect

5. C₂×C₄⋊C₈×C₄₇

C₂

semidirect

9. #13094012

C₄

semidirect

96. C₂×C₄⋊C₈

D₄₇

semidirect

100. C₄₇

#1983

semidirect

Subgroups — #20244318

Conjugacy classes of subgroups

Complements