Group #20248384 — subgroups

Subgroups — #20248384

Order 6016.

604 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13094013	C₂	1	2	yes	maximal
2	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#15689049	C₂	1	4	yes	maximal, complemented
4	#15689030	C₂	1	2	yes	maximal, complemented
5	#15688980	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
10	#13094013	—	2	8	no	—
11	C₄×D₁₈₈	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	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	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×D₄₇	C₂³	1	2	yes	—
25	C₂×Q₈×C₄₇	—	2	4	no	—
26	C₂×Q₈×C₄₇	—	2	4	no	—
27	C₂×Q₈×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₄₇	C₂³	1	2	yes	—
33	C₂²×Dic₄₇	C₂³	1	4	yes	—
34	C₂×D₁₈₈	—	2	2	no	—
35	C₂×D₁₈₈	—	2	2	no	—
36	C₂×D₁₈₈	D₄	1	4	yes	—
37	C₂×C₄×D₄₇	—	2	4	no	—
38	C₂×C₄×D₄₇	—	2	8	no	—
39	C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₄²×C₄₇	D₄	1	4	yes	—
42	(C₂×D₄₇)⋊C₄	—	2	16	no	—
43	(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	C₄⋊Dic₄₇	—	2	2	no	—
45	C₄⋊Dic₄₇	—	2	4	no	—
46	C₄×Dic₄₇	—	2	4	no	—
47	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×D₄₇	—	2	2	no	—
49	C₂²×D₄₇	—	2	4	no	—
50	C₂²×D₄₇	—	2	2	no	—
51	C₂²×D₄₇	—	2	4	no	—
52	C₂²×D₄₇	C₂×D₄	1	4	yes	—
53	Q₈×C₄₇	—	4	8	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	#41	1	2	yes	—
63	C₂×C₄×C₄₇	#41	1	4	yes	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	2	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	—	2	4	no	—
69	D₁₈₈	—	2	8	no	—
70	C₄×D₄₇	—	4	8	no	—
71	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×#41	1	2	yes	—
75	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
76	C₂×D₄₇	—	2	4	no	—
77	C₂×D₄₇	—	2	8	no	—
78	C₂×D₄₇	—	4	2	no	—
79	C₂×D₄₇	—	4	2	no	—
80	C₂×D₄₇	—	4	4	no	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	4	4	no	—
83	C₄×C₄₇	#121	1	4	yes	—
84	C₄×C₄₇	#124	1	4	yes	complemented
85	Dic₄₇	—	4	2	no	—
86	Dic₄₇	—	4	4	no	—
87	#2547	—	47	1	no	maximal, Sylow
88	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
90	C₂×C₄₇	#536	1	1	yes	characteristic
91	C₂×C₄₇	#547	1	2	yes	—
92	C₂×C₄₇	#556	1	2	yes	—
93	D₄₇	—	4	4	no	—
94	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
95	C₂×#124	—	47	2	no	—
96	C₂×C₄×D₄	—	47	1	no	—
97	#397	—	47	4	no	—
98	#391	—	47	2	no	—
99	#380	—	47	4	no	—
100	C₄×C₄⋊C₄	—	47	1	no	—
101	C₄₇	#2547	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
103	C₂²×D₄	—	47	1	no	—
104	C₂³×C₄	—	47	2	no	—
105	C₄⋊Q₈	—	2	4	no	—
106	#124	—	94	8	no	—
107	C₄×D₄	—	94	4	no	—
108	C₂×C₄⋊C₄	—	47	1	no	—
109	C₂×C₄⋊C₄	—	47	2	no	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
112	C₂×C₂²⋊C₄	—	47	2	no	—
113	C₂×C₂²⋊C₄	—	47	8	no	—
114	C₂×C₄²	—	47	2	no	—
115	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
116	#95	—	47	2	no	—
117	#95	—	47	8	no	—
118	C₂⁴	—	47	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×D₄	—	47	4	no	—
123	C₂×D₄	—	94	2	no	—
124	C₂×D₄	—	94	2	no	—
125	C₂²×C₄	—	47	2	no	—
126	C₂²×C₄	—	47	4	no	—
127	C₂²×C₄	—	94	8	no	—
128	C₂²×C₄	—	94	4	no	—
129	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
130	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	94	4	no	—
136	C₄⋊C₄	—	94	2	no	—
137	C₂²⋊C₄	—	94	16	no	—
138	C₂²⋊C₄	—	94	4	no	—
139	C₄²	—	94	4	no	—
140	C₄²	C₂²⋊D₄₇	1	4	yes	—
141	C₂³	—	47	4	no	—
142	C₂³	—	94	2	no	—
143	C₂³	—	94	2	no	—
144	C₂³	—	94	4	no	—
145	C₂³	—	94	4	no	—
146	C₂³	C₂³×D₄₇	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	D₄	—	94	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	94	4	no	—
158	C₂×C₄	—	188	8	no	—
159	C₂×C₄	D₄×D₄₇	1	2	yes	—
160	C₂×C₄	#10625352	1	2	yes	—
161	C₂×C₄	#10625354	1	4	yes	—
162	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
164	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
165	C₂²	—	94	8	no	—
166	C₂²	—	94	4	no	—
167	C₂²	—	188	4	no	—
168	C₂²	—	188	2	no	—
169	C₂²	—	188	2	no	—
170	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
171	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂²	C₂×#10625352	1	1	yes	characteristic
173	C₂²	C₂×#10625354	1	2	yes	—
174	C₂²	#13094072	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	4	4	no	—
177	C₄	—	188	4	no	—
178	C₄	—	188	2	no	—
179	C₄	#13093995	1	4	yes	—
180	C₄	#13094013	1	4	yes	complemented
181	C₂	—	188	4	no	—
182	C₂	C₂×#13093995	1	1	yes	characteristic
183	C₂	C₂×#13094013	1	1	yes	characteristic
184	C₂	#15689697	1	1	yes	characteristic
185	C₂	#15689776	1	2	yes	—
186	C₂	#15689847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094013

C₂

yes

maximal

C₂×C₄×D₁₈₈

C₂

yes

maximal, characteristic

#15689049

C₂

yes

maximal, complemented

#15689030

C₂

yes

maximal, complemented

#15688980

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

#13094013

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

Q₈×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×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

#124

yes

complemented

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#556

yes

—

D₄₇

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

C₂×#124

—

C₂×C₄×D₄

—

#397

—

#391

—

#380

—

100

C₄×C₄⋊C₄

—

101

C₄₇

#2547

yes

characteristic, complemented, Sylow

102

C₂²×Q₈

C₂×D₄₇

yes

—

103

C₂²×D₄

—

104

C₂³×C₄

—

105

C₄⋊Q₈

—

106

#124

—

107

C₄×D₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₄₇

yes

characteristic

116

#95

—

117

#95

—

118

C₂⁴

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂²×D₄₇

yes

—

130

C₂²×C₄

C₂²×D₄₇

yes

characteristic

131

C₂²×C₄

C₂²×D₄₇

yes

characteristic

132

C₂²×C₄

C₂²×D₄₇

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₄²

C₂²⋊D₄₇

yes

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂³×D₄₇

yes

characteristic

147

Q₈

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

188

—

159

C₂×C₄

D₄×D₄₇

yes

—

160

C₂×C₄

#10625352

yes

—

161

C₂×C₄

#10625354

yes

—

162

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

163

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

164

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

188

—

168

C₂²

—

188

—

169

C₂²

—

188

—

170

C₂²

C₂×D₄×D₄₇

yes

characteristic

171

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

172

C₂²

C₂×#10625352

yes

characteristic

173

C₂²

C₂×#10625354

yes

—

174

C₂²

#13094072

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

188

—

178

C₄

—

188

—

179

C₄

#13093995

yes

—

180

C₄

#13094013

yes

complemented

181

C₂

—

188

—

182

C₂

C₂×#13093995

yes

characteristic

183

C₂

C₂×#13094013

yes

characteristic

184

C₂

#15689697

yes

characteristic

185

C₂

#15689776

yes

—

186

C₂

#15689847

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2547
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #15689049	C₂	semidirect
4. #15689030	C₂	semidirect
5. #15688980	C₂	semidirect
6. C₄×C₄⋊Dic₄₇	C₂	semidirect
7. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
84. C₄×C₄₇	#124	semidirect
94. C₂×C₄⋊Q₈	D₄₇	semidirect
101. C₄₇	#2547	semidirect
180. C₄	#13094013	semidirect

H ≅ G/N

type

3. #15689049

C₂

semidirect

4. #15689030

C₂

semidirect

5. #15688980

C₂

semidirect

6. C₄×C₄⋊Dic₄₇

C₂

semidirect

7. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

84. C₄×C₄₇

#124

semidirect

94. C₂×C₄⋊Q₈

D₄₇

semidirect

101. C₄₇

#2547

semidirect

180. C₄

#13094013

semidirect

Subgroups — #20248384

Conjugacy classes of subgroups

Complements