Group #20251699 — subgroups

Subgroups — #20251699

Order 6016.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689022	C₂	1	2	yes	maximal, complemented
4	#15689008	C₂	1	4	yes	maximal, complemented
5	#15688997	C₂	1	4	yes	maximal
6	#15688980	C₂	1	2	yes	maximal
7	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	#13094016	—	2	4	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₄²×D₄₇	—	2	4	no	—
18	C₄₇⋊#95	C₂²	1	4	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	#95×C₄₇	C₂²	1	4	yes	—
23	C₂³×D₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂³	1	4	yes	—
28	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	C₂³	1	2	yes	—
30	C₂×C₄×D₄₇	—	2	4	no	—
31	C₂×C₄×D₄₇	—	2	4	no	—
32	C₂×C₄×D₄₇	—	2	2	no	—
33	C₂×C₄×D₄₇	—	2	2	no	—
34	C₂×C₄×D₄₇	—	2	4	no	—
35	C₂×C₄×D₄₇	Q₈	1	4	yes	—
36	C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₄²×C₄₇	D₄	1	4	yes	—
38	(C₂×D₄₇)⋊C₄	—	2	8	no	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	8	no	—
41	Dic₄₇⋊C₄	—	2	8	no	—
42	C₄×Dic₄₇	—	2	4	no	—
43	C₄×Dic₄₇	—	2	2	no	—
44	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	1	no	—
46	C₂²×D₄₇	—	2	2	no	—
47	C₂²×D₄₇	—	2	1	no	—
48	C₂²×D₄₇	C₂×Q₈	1	2	yes	—
49	C₂²×D₄₇	#41	1	4	yes	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₄₇	#41	1	2	yes	—
59	C₂×Dic₄₇	—	2	1	no	—
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	2	2	no	—
62	C₂×Dic₄₇	—	2	1	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
67	C₂×Dic₄₇	#41	1	4	yes	—
68	C₄×D₄₇	—	2	16	no	—
69	C₄×D₄₇	—	2	8	no	—
70	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×#41	1	2	yes	—
73	C₂²×C₄₇	C₂×#41	1	2	yes	—
74	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
75	C₂×D₄₇	—	2	2	no	—
76	C₂×D₄₇	—	2	4	no	—
77	C₂×D₄₇	—	2	4	no	—
78	C₂×D₄₇	—	2	2	no	—
79	C₂×D₄₇	#125	1	4	yes	complemented
80	C₄×C₄₇	—	2	4	no	—
81	C₄×C₄₇	—	4	4	no	—
82	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
83	Dic₄₇	—	2	2	no	—
84	Dic₄₇	—	2	4	no	—
85	Dic₄₇	—	4	4	no	—
86	#2851	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄₇	#525	1	2	yes	—
89	C₂×C₄₇	C₂×#125	1	1	yes	characteristic
90	C₂×C₄₇	#529	1	2	yes	—
91	C₂×C₄₇	#533	1	1	yes	characteristic
92	D₄₇	—	2	4	no	—
93	C₂×#125	—	47	1	no	—
94	C₂²×C₄²	—	47	1	no	—
95	#389	—	47	2	no	—
96	#387	—	47	4	no	—
97	#385	—	47	4	no	—
98	#382	D₄₇	1	1	yes	characteristic, complemented
99	#380	—	47	2	no	—
100	C₄₇	#2851	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	—	47	1	no	—
102	C₂³×C₄	—	47	2	no	—
103	#125	—	94	4	no	—
104	C₂×C₄⋊C₄	—	47	4	no	—
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	—	47	4	no	—
107	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
108	C₂×C₂²⋊C₄	—	47	4	no	—
109	C₂×C₄²	—	47	2	no	—
110	C₂×C₄²	—	47	1	no	—
111	C₂×C₄²	—	94	4	no	—
112	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
113	#95	—	47	4	no	—
114	#95	—	47	4	no	—
115	#95	C₂×D₄₇	1	4	yes	—
116	C₂⁴	—	47	1	no	—
117	C₂²×C₄	—	47	1	no	—
118	C₂²×C₄	—	47	2	no	—
119	C₂²×C₄	—	47	4	no	—
120	C₂²×C₄	—	47	4	no	—
121	C₂²×C₄	—	94	4	no	—
122	C₂²×C₄	—	94	4	no	—
123	C₂²×C₄	—	94	4	no	—
124	C₂²×C₄	—	94	2	no	—
125	C₂²×C₄	—	94	2	no	—
126	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
127	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	94	8	no	—
131	C₄⋊C₄	—	94	8	no	—
132	C₄⋊C₄	—	94	4	no	—
133	C₂²⋊C₄	—	94	8	no	—
134	C₄²	—	94	4	no	—
135	C₄²	—	94	2	no	—
136	C₄²	D₁₈₈	1	4	yes	—
137	C₂³	—	47	2	no	—
138	C₂³	—	47	4	no	—
139	C₂³	—	94	1	no	—
140	C₂³	—	94	2	no	—
141	C₂³	—	94	1	no	—
142	C₂³	C₂³×D₄₇	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	47	2	no	—
149	C₂×C₄	—	47	4	no	—
150	C₂×C₄	—	94	16	no	—
151	C₂×C₄	—	94	1	no	—
152	C₂×C₄	—	94	2	no	—
153	C₂×C₄	—	94	2	no	—
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	—	94	8	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	94	4	no	—
158	C₂×C₄	—	94	4	no	—
159	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
160	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
161	C₂×C₄	#10625352	1	2	yes	—
162	C₂×C₄	Q₈×D₄₇	1	2	yes	—
163	C₂²	—	47	4	no	—
164	C₂²	—	94	2	no	—
165	C₂²	—	94	4	no	—
166	C₂²	—	94	4	no	—
167	C₂²	—	94	2	no	—
168	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
169	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
170	C₂²	#41×D₄₇	1	2	yes	—
171	C₂²	#41×D₄₇	1	2	yes	—
172	C₂²	C₂×#10625352	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	4	4	no	—
175	C₄	—	94	2	no	—
176	C₄	—	94	4	no	—
177	C₄	—	188	4	no	—
178	C₄	#13094004	1	4	yes	—
179	C₂	—	94	4	no	—
180	C₂	C₂×#13094004	1	1	yes	characteristic
181	C₂	#15689635	1	2	yes	—
182	C₂	#125×D₄₇	1	1	yes	characteristic
183	C₂	#15689652	1	2	yes	—
184	C₂	#15689679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15689022

C₂

yes

maximal, complemented

#15689008

C₂

yes

maximal, complemented

#15688997

C₂

yes

maximal

#15688980

C₂

yes

maximal

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

#13094016

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

Q₈

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×Q₈

yes

—

C₂²×D₄₇

#41

yes

—

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×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

#41

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#125

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2851

—

maximal, Sylow

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

#525

yes

—

C₂×C₄₇

C₂×#125

yes

characteristic

C₂×C₄₇

#529

yes

—

C₂×C₄₇

#533

yes

characteristic

D₄₇

—

C₂×#125

—

C₂²×C₄²

—

—

—

—

D₄₇

yes

characteristic, complemented

#380

—

100

C₄₇

#2851

yes

characteristic, complemented, Sylow

101

C₂³×C₄

—

102

C₂³×C₄

—

103

#125

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×D₄₇

yes

characteristic

113

#95

—

114

#95

—

115

#95

C₂×D₄₇

yes

—

116

C₂⁴

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₄₇

yes

—

127

C₂²×C₄

C₂²×D₄₇

yes

—

128

C₂²×C₄

C₂²×D₄₇

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

D₁₈₈

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂³×D₄₇

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

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

—

159

C₂×C₄

C₂×D₁₈₈

yes

—

160

C₂×C₄

C₂×D₁₈₈

yes

—

161

C₂×C₄

#10625352

yes

—

162

C₂×C₄

Q₈×D₄₇

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂²×D₁₈₈

yes

characteristic

169

C₂²

C₂×Q₈×D₄₇

yes

characteristic

170

C₂²

#41×D₄₇

yes

—

171

C₂²

#41×D₄₇

yes

—

172

C₂²

C₂×#10625352

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

188

—

178

C₄

#13094004

yes

—

179

C₂

—

180

C₂

C₂×#13094004

yes

characteristic

181

C₂

#15689635

yes

—

182

C₂

#125×D₄₇

yes

characteristic

183

C₂

#15689652

yes

—

184

C₂

#15689679

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#382	C₂
hypercenter	#382	D₄₇
hypocenter	C₄₇	#2851
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#382

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
1. C₂×#13094016	C₂	semidirect
3. #15689022	C₂	semidirect
4. #15689008	C₂	semidirect
7. C₄₇×#382	C₂	semidirect
79. C₂×D₄₇	#125	semidirect
98. #382	D₄₇	semidirect
100. C₄₇	#2851	semidirect

H ≅ G/N

type

1. C₂×#13094016

C₂

semidirect

3. #15689022

C₂

semidirect

4. #15689008

C₂

semidirect

7. C₄₇×#382

C₂

semidirect

79. C₂×D₄₇

#125

semidirect

98. #382

D₄₇

semidirect

100. C₄₇

#2851

semidirect

Subgroups — #20251699

Conjugacy classes of subgroups

Complements