Group #20249179 — subgroups

Subgroups — #20249179

Order 6016.

548 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13093998	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689060	C₂	1	2	yes	maximal, complemented
4	#15689050	C₂	1	2	yes	maximal, complemented
5	#15689039	C₂	1	4	yes	maximal, complemented
6	#15689024	C₂	1	2	yes	maximal, complemented
7	#15689022	C₂	1	2	yes	maximal, complemented
8	C₄₇×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
11	#13093998	—	2	4	no	—
12	#13093983	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
20	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
22	C₄₇⋊#95	C₂²	1	4	yes	—
23	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
26	#95×C₄₇	C₂²	1	2	yes	—
27	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
28	C₂³×D₄₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	D₄	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₄₇	—	2	4	no	—
35	C₂×C₂²⋊D₄₇	—	4	4	no	—
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Dic₄₇	C₂³	1	2	yes	—
39	C₂×D₁₈₈	—	2	2	no	—
40	C₂×D₁₈₈	—	2	4	no	—
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂²⋊C₄×C₄₇	—	4	2	no	—
43	C₂²⋊Dic₄₇	—	2	4	no	—
44	(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	(C₂×D₄₇)⋊C₄	—	4	4	no	—
46	(C₂×D₄₇)⋊C₄	—	4	4	no	—
47	C₄⋊Dic₄₇	—	2	2	no	—
48	C₄⋊Dic₄₇	—	2	2	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₄×Dic₄₇	—	2	4	no	—
52	C₂³×C₄₇	—	2	2	no	—
53	C₂³×C₄₇	—	2	1	no	—
54	C₂³×C₄₇	—	2	2	no	—
55	C₂³×C₄₇	—	2	1	no	—
56	C₂³×C₄₇	C₂×D₄	1	2	yes	—
57	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×D₄₇	—	2	2	no	—
59	C₂²×D₄₇	—	2	4	no	—
60	C₂²×D₄₇	—	2	2	no	—
61	C₂²×D₄₇	—	2	4	no	—
62	C₂²×D₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	8	no	—
65	C₂×C₄×C₄₇	—	2	4	no	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₇	#41	1	4	yes	—
71	C₂²⋊D₄₇	—	4	8	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	2	no	—
76	C₂×Dic₄₇	—	2	4	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	C₂×Dic₄₇	—	2	2	no	—
79	C₂×Dic₄₇	—	2	2	no	—
80	D₁₈₈	—	4	4	no	—
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	—	2	2	no	—
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	—	4	2	no	—
86	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#41	1	2	yes	—
88	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
89	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂×D₄₇	—	4	2	no	—
92	C₂×D₄₇	—	4	2	no	—
93	C₂×D₄₇	—	4	2	no	—
94	C₂×D₄₇	—	4	4	no	—
95	C₂×D₄₇	—	4	4	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	4	4	no	—
98	C₄×C₄₇	#123	1	4	yes	—
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	—	4	2	no	—
101	Dic₄₇	—	4	2	no	—
102	#2624	—	47	1	no	maximal, Sylow
103	C₂×C₄₇	—	4	2	no	—
104	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
105	C₂×C₄₇	#535	1	1	yes	characteristic
106	C₂×C₄₇	#537	1	1	yes	characteristic
107	C₂×C₄₇	#540	1	2	yes	—
108	C₂×C₄₇	#542	1	2	yes	—
109	D₄₇	—	8	2	no	—
110	C₂×#121	—	47	1	no	—
111	C₂×#117	—	47	1	no	—
112	#399	—	47	2	no	—
113	#397	—	47	2	no	—
114	#394	—	47	4	no	—
115	#390	—	47	2	no	—
116	#389	—	47	2	no	—
117	#380	D₄₇	1	1	yes	characteristic, complemented
118	C₄₇	#2624	1	1	yes	characteristic, complemented, Sylow
119	C₂²×D₄	—	47	2	no	—
120	C₂²×D₄	—	47	1	no	—
121	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
122	#121	—	94	4	no	—
123	#117	—	94	4	no	—
124	C₂×C₄⋊C₄	—	47	1	no	—
125	C₂×C₄⋊C₄	—	47	1	no	—
126	C₂×C₄⋊C₄	—	47	1	no	—
127	C₂×C₄⋊C₄	—	47	4	no	—
128	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
129	C₂×C₂²⋊C₄	—	47	2	no	—
130	C₂×C₂²⋊C₄	—	47	2	no	—
131	C₂×C₂²⋊C₄	—	47	4	no	—
132	C₂×C₂²⋊C₄	—	47	4	no	—
133	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
134	C₂×C₄²	—	47	2	no	—
135	#95	—	47	4	no	—
136	#95	C₂×D₄₇	1	2	yes	—
137	C₂⁴	—	47	2	no	—
138	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×D₄	—	94	4	no	—
140	C₂×D₄	—	94	2	no	—
141	C₂×D₄	—	94	4	no	—
142	C₂×D₄	—	188	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	47	2	no	—
145	C₂²×C₄	—	47	2	no	—
146	C₂²×C₄	—	47	2	no	—
147	C₂²×C₄	D₁₈₈	1	4	yes	—
148	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
149	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	94	2	no	—
153	C₄⋊C₄	—	94	2	no	—
154	C₄⋊C₄	—	94	2	no	—
155	C₄⋊C₄	—	188	4	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	94	4	no	—
158	C₂²⋊C₄	—	94	4	no	—
159	C₂²⋊C₄	—	188	4	no	—
160	C₂²⋊C₄	—	188	4	no	—
161	C₄²	—	94	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	1	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	94	2	no	—
167	C₂³	—	94	4	no	—
168	C₂³	—	94	2	no	—
169	C₂³	—	94	4	no	—
170	C₂³	—	94	2	no	—
171	C₂³	C₂×D₁₈₈	1	2	yes	—
172	C₂³	C₂³×D₄₇	1	1	yes	characteristic
173	D₄	—	188	8	no	—
174	D₄	—	188	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	94	2	no	—
182	C₂×C₄	—	94	2	no	—
183	C₂×C₄	—	94	2	no	—
184	C₂×C₄	—	94	2	no	—
185	C₂×C₄	—	94	2	no	—
186	C₂×C₄	—	94	4	no	—
187	C₂×C₄	—	94	2	no	—
188	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
189	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
190	C₂×C₄	#10625352	1	4	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	188	4	no	—
197	C₂²	—	188	2	no	—
198	C₂²	—	188	2	no	—
199	C₂²	—	188	2	no	—
200	C₂²	—	188	4	no	—
201	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
202	C₂²	C₂×#10625352	1	2	yes	—
203	C₂²	#13094069	1	1	yes	characteristic
204	C₂²	#13094070	1	2	yes	—
205	C₂²	#13094072	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	4	4	no	—
208	C₄	—	188	2	no	—
209	C₄	—	188	2	no	—
210	C₄	—	188	2	no	—
211	C₄	#13094008	1	4	yes	—
212	C₂	—	4	2	no	—
213	C₂	—	376	2	no	—
214	C₂	C₂×#13094008	1	1	yes	characteristic
215	C₂	#15689692	1	1	yes	characteristic
216	C₂	#15689704	1	1	yes	characteristic
217	C₂	#15689726	1	2	yes	—
218	C₂	#15689746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093998

C₂

yes

maximal, characteristic

C₂×#13093983

C₂

yes

maximal, characteristic, complemented

#15689060

C₂

yes

maximal, complemented

#15689050

C₂

yes

maximal, complemented

#15689039

C₂

yes

maximal, complemented

#15689024

C₂

yes

maximal, complemented

#15689022

C₂

yes

maximal, complemented

C₄₇×#380

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₂²⋊D₄₇

C₂²

yes

—

#13093998

—

#13093983

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

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

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

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

#41

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

100

Dic₄₇

—

101

Dic₄₇

—

102

#2624

—

maximal, Sylow

103

C₂×C₄₇

—

104

C₂×C₄₇

C₂×#123

yes

characteristic

105

C₂×C₄₇

#535

yes

characteristic

106

C₂×C₄₇

#537

yes

characteristic

107

C₂×C₄₇

#540

yes

—

108

C₂×C₄₇

#542

yes

—

109

D₄₇

—

110

C₂×#121

—

111

C₂×#117

—

112

#399

—

113

#397

—

114

#394

—

115

#390

—

116

#389

—

117

#380

D₄₇

yes

characteristic, complemented

118

C₄₇

#2624

yes

characteristic, complemented, Sylow

119

C₂²×D₄

—

120

C₂²×D₄

—

121

C₂³×C₄

C₂×D₄₇

yes

characteristic

122

#121

—

123

#117

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

—

133

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

134

C₂×C₄²

—

135

#95

—

136

#95

C₂×D₄₇

yes

—

137

C₂⁴

—

138

C₂⁴

C₂²×D₄₇

yes

characteristic

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

188

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

D₁₈₈

yes

—

148

C₂²×C₄

C₂²×D₄₇

yes

—

149

C₂²×C₄

C₂²×D₄₇

yes

characteristic

150

C₂²×C₄

C₂²×D₄₇

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

188

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

188

—

160

C₂²⋊C₄

—

188

—

161

C₄²

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

C₂×D₁₈₈

yes

—

172

C₂³

C₂³×D₄₇

yes

characteristic

173

D₄

—

188

—

174

D₄

—

188

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×D₁₈₈

yes

—

189

C₂×C₄

C₂×D₁₈₈

yes

—

190

C₂×C₄

#10625352

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

188

—

197

C₂²

—

188

—

198

C₂²

—

188

—

199

C₂²

—

188

—

200

C₂²

—

188

—

201

C₂²

C₂²×D₁₈₈

yes

characteristic

202

C₂²

C₂×#10625352

yes

—

203

C₂²

#13094069

yes

characteristic

204

C₂²

#13094070

yes

—

205

C₂²

#13094072

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

188

—

209

C₄

—

188

—

210

C₄

—

188

—

211

C₄

#13094008

yes

—

212

C₂

—

213

C₂

—

376

—

214

C₂

C₂×#13094008

yes

characteristic

215

C₂

#15689692

yes

characteristic

216

C₂

#15689704

yes

characteristic

217

C₂

#15689726

yes

—

218

C₂

#15689746

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#380	C₂
hypercenter	#380	D₄₇
hypocenter	C₄₇	#2624
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #380

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#380

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #380

N	H ≅ G/N	type
2. C₂×#13093983	C₂	semidirect
3. #15689060	C₂	semidirect
4. #15689050	C₂	semidirect
5. #15689039	C₂	semidirect
6. #15689024	C₂	semidirect
7. #15689022	C₂	semidirect
8. C₄₇×#380	C₂	semidirect
117. #380	D₄₇	semidirect
118. C₄₇	#2624	semidirect

H ≅ G/N

type

2. C₂×#13093983

C₂

semidirect

3. #15689060

C₂

semidirect

4. #15689050

C₂

semidirect

5. #15689039

C₂

semidirect

6. #15689024

C₂

semidirect

7. #15689022

C₂

semidirect

8. C₄₇×#380

C₂

semidirect

117. #380

D₄₇

semidirect

118. C₄₇

#2624

semidirect

Subgroups — #20249179

Conjugacy classes of subgroups

Complements