Group #20244398 — subgroups

Subgroups — #20244398

Order 6016.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094003	C₂	1	2	yes	maximal
3	C₂×#13093949	C₂	1	2	yes	maximal, complemented
4	#15688886	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	2	yes	—
7	C₂×Dic₁₈₈	—	2	4	no	—
8	C₂×Dic₁₈₈	—	2	2	no	—
9	#13094003	—	2	4	no	—
10	#13094003	—	2	4	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	—	2	2	no	—
13	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
14	#13093949	—	2	4	no	—
15	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₄₇	—	2	2	no	—
19	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	—	2	2	no	—
21	C₂²×C₄×C₄₇	—	2	2	no	—
22	C₂²×C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄×C₄₇	—	2	1	no	—
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₂²×Dic₄₇	—	2	2	no	—
28	C₂×Dic₉₄	—	2	2	no	—
29	C₂×Dic₉₄	—	2	2	no	—
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×Dic₉₄	—	2	2	no	—
32	C₂×Dic₉₄	—	2	2	no	—
33	C₂×Dic₉₄	D₄	1	4	yes	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	D₄	1	2	yes	—
38	C₂²⋊Dic₄₇	—	2	8	no	—
39	C₄⋊Dic₄₇	—	2	2	no	—
40	C₄⋊Dic₄₇	D₄	1	4	yes	—
41	Dic₄₇⋊C₄	—	2	8	no	—
42	Dic₁₈₈	—	2	8	no	—
43	C₂³×C₄₇	—	2	2	no	—
44	C₂³×C₄₇	—	2	1	no	—
45	C₂³×C₄₇	—	2	1	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	Q₁₆	1	2	yes	complemented
48	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	—	4	2	no	—
62	C₂×Dic₄₇	—	4	2	no	—
63	C₂×Dic₄₇	—	4	2	no	—
64	Dic₉₄	—	2	4	no	—
65	Dic₉₄	—	2	4	no	—
66	Dic₉₄	—	4	2	no	—
67	Dic₉₄	—	4	2	no	—
68	C₈×C₄₇	—	2	2	no	—
69	C₈×C₄₇	—	4	1	no	—
70	C₂²×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	—	4	1	no	—
75	C₂²×C₄₇	#120	1	2	yes	—
76	C₂²×C₄₇	#121	1	1	yes	characteristic
77	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂²×C₄₇	#135	1	1	yes	characteristic
79	C₂²×C₄₇	#137	1	2	yes	—
80	C₄×C₄₇	—	4	1	no	—
81	C₄×C₄₇	#120	1	1	yes	characteristic
82	C₄×C₄₇	#121	1	2	yes	—
83	C₄×C₄₇	#127	1	1	yes	characteristic
84	Dic₄₇	—	4	4	no	—
85	Dic₄₇	—	4	4	no	—
86	#2001	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	4	2	no	—
88	C₂×C₄₇	#392	1	1	yes	characteristic
89	C₂×C₄₇	#451	1	2	yes	—
90	C₂×C₄₇	#452	1	2	yes	—
91	C₂×C₄₇	#467	1	1	yes	characteristic
92	C₂×C₄₇	#470	1	1	yes	characteristic
93	C₂²×Q₁₆	—	47	1	no	—
94	C₂×C₂²⋊Q₈	—	47	2	no	—
95	C₂×#103	—	47	2	no	—
96	C₂×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
97	#340	—	47	1	no	—
98	C₄₇	#2001	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	47	2	no	—
100	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
101	C₂×Q₁₆	—	94	4	no	—
102	C₂×Q₁₆	—	94	2	no	—
103	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
104	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
105	C₂²⋊Q₈	—	94	4	no	—
106	C₂²⋊Q₈	—	94	4	no	—
107	C₂×C₄⋊C₄	—	47	2	no	—
108	C₂×C₄⋊C₄	—	94	2	no	—
109	C₂×C₂²⋊C₄	—	94	2	no	—
110	#103	—	94	4	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂⁴	D₁₈₈	1	1	yes	characteristic
113	C₂×Q₈	—	47	4	no	—
114	C₂×Q₈	—	94	2	no	—
115	C₂×Q₈	—	94	2	no	—
116	C₂×Q₈	—	94	2	no	—
117	C₂×Q₈	—	94	2	no	—
118	C₂×Q₈	—	94	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	94	2	no	—
124	C₂²×C₄	—	94	2	no	—
125	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	Q₁₆	—	94	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
132	C₄⋊C₄	—	47	4	no	—
133	C₄⋊C₄	—	94	8	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₂²⋊C₄	—	94	8	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	2	no	—
140	C₂³	Dic₁₈₈	1	2	yes	complemented
141	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
142	Q₈	—	94	4	no	—
143	Q₈	—	94	4	no	—
144	Q₈	—	188	2	no	—
145	Q₈	—	188	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	94	4	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	—	188	2	no	—
154	C₂×C₄	—	188	2	no	—
155	C₂×C₄	—	188	2	no	—
156	C₂×C₄	—	188	2	no	—
157	C₂×C₄	#10625350	1	1	yes	characteristic
158	C₂×C₄	D₄×D₄₇	1	2	yes	—
159	C₂×C₄	D₄×D₄₇	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	1	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	#13093994	1	2	yes	—
169	C₂²	#13093998	1	1	yes	characteristic
170	C₂²	C₂×Dic₁₈₈	1	1	yes	characteristic
171	C₂²	#13094046	1	1	yes	characteristic
172	C₂²	#13094057	1	2	yes	—
173	C₄	—	4	1	no	—
174	C₄	—	188	4	no	—
175	C₄	—	188	4	no	—
176	C₄	#13093992	1	1	yes	characteristic
177	C₄	#13093997	1	2	yes	—
178	C₄	#13094020	1	1	yes	characteristic
179	C₂	—	4	2	no	—
180	C₂	#15689032	1	1	yes	characteristic
181	C₂	#15689281	1	2	yes	—
182	C₂	#15689287	1	2	yes	—
183	C₂	#15689364	1	1	yes	characteristic
184	C₂	#15689381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₈₈

C₂

yes

maximal, characteristic, complemented

C₂×#13094003

C₂

yes

maximal

C₂×#13093949

C₂

yes

maximal, complemented

#15688886

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094003

—

#13094003

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₂²⋊Dic₄₇

—

#13093949

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

Dic₄₇⋊C₄

—

Dic₁₈₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

Q₁₆

yes

complemented

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

—

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

characteristic

C₄×C₄₇

#121

yes

—

C₄×C₄₇

#127

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2001

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#392

yes

characteristic

C₂×C₄₇

#451

yes

—

C₂×C₄₇

#452

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×Q₁₆

—

C₂×C₂²⋊Q₈

—

C₂×#103

—

C₂×C₂²⋊C₈

D₄₇

yes

characteristic

#340

—

C₄₇

#2001

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂³×C₄

C₂×D₄₇

yes

characteristic

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂²×C₈

C₂×D₄₇

yes

characteristic

104

C₂²×C₈

C₂×D₄₇

yes

characteristic

105

C₂²⋊Q₈

—

106

C₂²⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₂²⋊C₄

—

110

#103

—

111

C₂²⋊C₈

—

112

C₂⁴

D₁₈₈

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

D₁₈₈

yes

characteristic

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₂²⋊D₄₇

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

Dic₁₈₈

yes

complemented

141

C₂³

C₂×D₁₈₈

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

188

—

145

Q₈

—

188

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

188

—

154

C₂×C₄

—

188

—

155

C₂×C₄

—

188

—

156

C₂×C₄

—

188

—

157

C₂×C₄

#10625350

yes

characteristic

158

C₂×C₄

D₄×D₄₇

yes

—

159

C₂×C₄

D₄×D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#13093994

yes

—

169

C₂²

#13093998

yes

characteristic

170

C₂²

C₂×Dic₁₈₈

yes

characteristic

171

C₂²

#13094046

yes

characteristic

172

C₂²

#13094057

yes

—

173

C₄

—

174

C₄

—

188

—

175

C₄

—

188

—

176

C₄

#13093992

yes

characteristic

177

C₄

#13093997

yes

—

178

C₄

#13094020

yes

characteristic

179

C₂

—

180

C₂

#15689032

yes

characteristic

181

C₂

#15689281

yes

—

182

C₂

#15689287

yes

—

183

C₂

#15689364

yes

characteristic

184

C₂

#15689381

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#2001
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₈₈

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₂²⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₂²⋊C₈

D₄₇

hypocenter

C₄₇

#2001

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂²×Dic₁₈₈	C₂	semidirect
3. C₂×#13093949	C₂	semidirect
4. #15688886	C₂	semidirect
47. C₂³×C₄₇	Q₁₆	semidirect
98. C₄₇	#2001	semidirect
140. C₂³	Dic₁₈₈	semidirect

H ≅ G/N

type

1. C₂²×Dic₁₈₈

C₂

semidirect

3. C₂×#13093949

C₂

semidirect

4. #15688886

C₂

semidirect

47. C₂³×C₄₇

Q₁₆

semidirect

98. C₄₇

#2001

semidirect

140. C₂³

Dic₁₈₈

semidirect

Subgroups — #20244398

Conjugacy classes of subgroups

Complements