Group #20247028 — subgroups

Subgroups — #20247028

Order 6016.

800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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×#13094022	C₂	1	2	yes	maximal, complemented
2	C₂×#13094003	C₂	1	1	yes	maximal, characteristic
3	#15689042	C₂	1	4	yes	maximal, complemented
4	#15689030	C₂	1	2	yes	maximal, complemented
5	#15688996	C₂	1	4	yes	maximal
6	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	#13094022	—	2	8	no	—
10	#13094003	—	2	4	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
17	C₄₇⋊#95	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂²×Q₈×C₄₇	—	2	8	no	—
20	C₂²×Q₈×C₄₇	—	2	4	no	—
21	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
22	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	—	2	32	no	—
27	C₂×Q₈×C₄₇	—	2	8	no	—
28	C₂×Q₈×C₄₇	—	2	4	no	—
29	C₂×Q₈×C₄₇	—	2	4	no	—
30	C₂×Q₈×C₄₇	—	2	4	no	—
31	C₂×Q₈×C₄₇	D₄	1	8	yes	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	8	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	D₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₄₇	C₂³	1	2	yes	—
40	C₂²×Dic₄₇	C₂³	1	2	yes	—
41	C₂²×Dic₄₇	C₂³	1	4	yes	—
42	C₂×Dic₉₄	—	2	2	no	—
43	C₂×Dic₉₄	—	2	4	no	—
44	C₂²⋊Dic₄₇	—	2	4	no	—
45	C₂²⋊Dic₄₇	Q₈	1	8	yes	complemented
46	C₄⋊Dic₄₇	—	2	2	no	—
47	Dic₄₇⋊C₄	—	2	4	no	—
48	Dic₄₇⋊C₄	—	2	16	no	—
49	C₄×Dic₄₇	—	2	4	no	—
50	C₂³×C₄₇	—	2	2	no	—
51	C₂³×C₄₇	—	2	2	no	—
52	C₂³×C₄₇	C₂×D₄	1	2	yes	—
53	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
54	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
55	Q₈×C₄₇	—	2	16	no	—
56	Q₈×C₄₇	—	2	16	no	—
57	C₂×C₄×C₄₇	—	2	16	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	8	no	—
62	C₂×C₄×C₄₇	C₂×D₄	1	8	yes	—
63	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	C₂×Q₈	1	8	yes	complemented
73	Dic₉₄	—	4	4	no	—
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	2	8	no	—
77	C₂²×C₄₇	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
80	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
81	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₄₇	—	2	8	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	#120	1	4	yes	—
85	Dic₄₇	—	2	8	no	—
86	Dic₄₇	—	4	2	no	—
87	Dic₄₇	—	4	2	no	—
88	#2434	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	—	2	4	no	—
90	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
91	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄₇	#536	1	1	yes	characteristic
93	C₂×C₄₇	D₄×Q₈	1	4	yes	—
94	C₂³×Q₈	D₄₇	1	1	yes	characteristic
95	C₂×C₄⋊Q₈	—	47	2	no	—
96	C₂×C₂²⋊Q₈	—	47	1	no	—
97	#395	—	47	4	no	—
98	#391	—	47	2	no	—
99	#385	—	47	4	no	—
100	C₄×C₂²⋊C₄	—	47	1	no	—
101	C₄₇	#2434	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₈	—	2	8	no	—
103	C₂²×Q₈	—	2	4	no	—
104	C₂²×Q₈	—	47	1	no	—
105	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
106	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
107	C₂³×C₄	C₂×D₄₇	1	2	yes	—
108	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
109	C₄⋊Q₈	—	94	8	no	—
110	C₂²⋊Q₈	—	94	4	no	—
111	C₂×C₄⋊C₄	—	47	2	no	—
112	C₂×C₄⋊C₄	—	47	1	no	—
113	C₂×C₄⋊C₄	—	47	8	no	—
114	C₂×C₂²⋊C₄	—	47	2	no	—
115	C₂×C₂²⋊C₄	—	47	2	no	—
116	C₂×C₄²	—	47	2	no	—
117	#95	—	47	2	no	—
118	#95	—	47	8	no	—
119	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
120	C₂×Q₈	—	2	32	no	—
121	C₂×Q₈	—	2	8	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	94	2	no	—
126	C₂×Q₈	—	94	4	no	—
127	C₂×Q₈	C₂²⋊D₄₇	1	8	yes	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	47	4	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	—	47	2	no	—
134	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
135	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
136	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₄⋊C₄	—	94	4	no	—
139	C₄⋊C₄	—	94	16	no	—
140	C₄⋊C₄	—	94	2	no	—
141	C₂²⋊C₄	—	47	8	no	—
142	C₂²⋊C₄	—	94	4	no	—
143	C₄²	—	94	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	Q₈×D₄₇	1	4	yes	—
147	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
148	C₂³	C₂³×D₄₇	1	1	yes	characteristic
149	Q₈	—	2	16	no	—
150	Q₈	—	2	16	no	—
151	Q₈	—	188	4	no	—
152	C₂×C₄	—	2	16	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	47	8	no	—
158	C₂×C₄	—	94	4	no	—
159	C₂×C₄	—	94	4	no	—
160	C₂×C₄	—	94	2	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	94	4	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
165	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	C₂²	—	2	8	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#13094022	1	4	yes	complemented
172	C₂²	C₂²×C₂²⋊D₄₇	1	2	yes	—
173	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
175	C₂²	#13094072	1	2	yes	—
176	C₄	—	2	8	no	—
177	C₄	—	2	2	no	—
178	C₄	—	94	8	no	—
179	C₄	—	188	2	no	—
180	C₄	—	188	2	no	—
181	C₄	#13093993	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	C₂×#13093993	1	1	yes	characteristic
184	C₂	C₂×#13094022	1	1	yes	characteristic
185	C₂	#15689697	1	1	yes	characteristic
186	C₂	Q₈×C₂²⋊D₄₇	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094022

C₂

yes

maximal, complemented

C₂×#13094003

C₂

yes

maximal, characteristic

#15689042

C₂

yes

maximal, complemented

#15689030

C₂

yes

maximal, complemented

#15688996

C₂

yes

maximal

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic

C₂³×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

#13094022

—

#13094003

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

Q₈

yes

complemented

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

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

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

complemented

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄⋊Q₈

yes

complemented

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2434

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#120

yes

characteristic

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

#536

yes

characteristic

C₂×C₄₇

D₄×Q₈

yes

—

C₂³×Q₈

D₄₇

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

#395

—

#391

—

#385

—

100

C₄×C₂²⋊C₄

—

101

C₄₇

#2434

yes

characteristic, complemented, Sylow

102

C₂²×Q₈

—

103

C₂²×Q₈

—

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂×D₄₇

yes

—

106

C₂²×Q₈

C₂×D₄₇

yes

—

107

C₂³×C₄

C₂×D₄₇

yes

—

108

C₂³×C₄

C₂×D₄₇

yes

characteristic

109

C₄⋊Q₈

—

110

C₂²⋊Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₄²

—

117

#95

—

118

#95

—

119

C₂⁴

C₂²×D₄₇

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

C₂²⋊D₄₇

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂²⋊D₄₇

yes

—

135

C₂²×C₄

C₂²×D₄₇

yes

—

136

C₂²×C₄

C₂²×D₄₇

yes

characteristic

137

C₂²×C₄

C₂²×D₄₇

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₂³

—

145

C₂³

—

146

C₂³

Q₈×D₄₇

yes

—

147

C₂³

C₂×C₂²⋊D₄₇

yes

—

148

C₂³

C₂³×D₄₇

yes

characteristic

149

Q₈

—

150

Q₈

—

151

Q₈

—

188

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

165

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

166

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

167

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#13094022

yes

complemented

172

C₂²

C₂²×C₂²⋊D₄₇

yes

—

173

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

174

C₂²

C₂×Q₈×D₄₇

yes

—

175

C₂²

#13094072

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

188

—

180

C₄

—

188

—

181

C₄

#13093993

yes

—

182

C₂

—

183

C₂

C₂×#13093993

yes

characteristic

184

C₂

C₂×#13094022

yes

characteristic

185

C₂

#15689697

yes

characteristic

186

C₂

Q₈×C₂²⋊D₄₇

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂³×Q₈×C₄₇	C₂
hypercenter	C₂³×Q₈	D₄₇
hypocenter	C₄₇	#2434
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂³×Q₈×C₄₇

C₂

hypercenter

C₂³×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×#13094022	C₂	semidirect
3. #15689042	C₂	semidirect
4. #15689030	C₂	semidirect
45. C₂²⋊Dic₄₇	Q₈	semidirect
72. C₂×Dic₄₇	C₂×Q₈	semidirect
77. C₂²×C₄₇	C₄⋊Q₈	semidirect
101. C₄₇	#2434	semidirect
171. C₂²	#13094022	semidirect

H ≅ G/N

type

1. C₂×#13094022

C₂

semidirect

3. #15689042

C₂

semidirect

4. #15689030

C₂

semidirect

45. C₂²⋊Dic₄₇

Q₈

semidirect

72. C₂×Dic₄₇

C₂×Q₈

semidirect

77. C₂²×C₄₇

C₄⋊Q₈

semidirect

101. C₄₇

#2434

semidirect

171. C₂²

#13094022

semidirect

Subgroups — #20247028

Conjugacy classes of subgroups

Complements