Group #20247881 — subgroups

Subgroups — #20247881

Order 6016.

596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689048	C₂	1	4	yes	maximal, complemented
3	#15689026	C₂	1	2	yes	maximal, complemented
4	#15689003	C₂	1	4	yes	maximal, complemented
5	#15688988	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#121×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	—	2	4	no	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	8	yes	—
17	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
18	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	#121×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂⁴×C₄₇	C₂³	1	2	yes	—
24	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₄₇	—	2	4	no	—
26	C₂×D₄×C₄₇	—	2	8	no	—
27	C₂×D₄×C₄₇	—	2	2	no	—
28	C₂×D₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	D₄	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₇	—	2	4	no	—
35	C₂²×Dic₄₇	—	4	4	no	—
36	C₂²×Dic₄₇	C₂³	1	4	yes	—
37	C₂²×Dic₄₇	C₂³	1	4	yes	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₂²⋊Dic₄₇	—	2	4	no	—
41	C₂²⋊Dic₄₇	—	4	8	no	—
42	C₄⋊Dic₄₇	—	2	4	no	—
43	C₄⋊Dic₄₇	—	2	4	no	—
44	C₄×Dic₄₇	—	2	4	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	C₂³×C₄₇	—	2	4	no	—
49	C₂³×C₄₇	—	2	4	no	—
50	C₂³×C₄₇	—	2	4	no	—
51	C₂³×C₄₇	C₂×D₄	1	2	yes	—
52	C₂³×C₄₇	#41	1	4	yes	—
53	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
54	D₄×C₄₇	—	4	8	no	—
55	D₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	8	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	#41	1	4	yes	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	8	no	—
68	C₂×Dic₄₇	—	4	8	no	—
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	8	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	—	4	2	no	—
75	C₂²×C₄₇	—	4	8	no	—
76	C₂²×C₄₇	#124	1	4	yes	—
77	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×#41	1	2	yes	—
79	C₂²×C₄₇	C₂×#41	1	2	yes	—
80	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	4	2	no	—
83	C₄×C₄₇	#123	1	4	yes	—
84	Dic₄₇	—	4	4	no	—
85	Dic₄₇	—	4	4	no	—
86	#2504	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	2	4	no	—
88	C₂×C₄₇	—	8	2	no	—
89	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
91	C₂×C₄₇	#535	1	1	yes	characteristic
92	C₂×C₄₇	#553	1	4	yes	—
93	C₂×#121	D₄₇	1	1	yes	characteristic
94	C₂²×C₄⋊C₄	—	47	1	no	—
95	#397	—	47	4	no	—
96	#390	—	47	2	no	—
97	#386	—	47	4	no	—
98	#382	—	47	2	no	—
99	C₄×C₂²⋊C₄	—	47	1	no	—
100	C₄₇	#2504	1	1	yes	characteristic, complemented, Sylow
101	C₂²×D₄	C₂×D₄₇	1	2	yes	—
102	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
103	C₂³×C₄	—	47	2	no	—
104	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
105	#121	—	2	4	no	—
106	C₂×C₄⋊C₄	—	47	2	no	—
107	C₂×C₄⋊C₄	—	47	2	no	—
108	C₂×C₄⋊C₄	—	94	4	no	—
109	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	47	2	no	—
111	C₂×C₂²⋊C₄	—	47	8	no	—
112	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
113	C₂×C₄²	—	47	2	no	—
114	#95	—	47	2	no	—
115	#95	—	47	8	no	—
116	C₂⁴	C₂²×D₄₇	1	2	yes	—
117	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×D₄	—	2	8	no	—
119	C₂×D₄	—	2	4	no	—
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	47	4	no	—
124	C₂²×C₄	—	47	4	no	—
125	C₂²×C₄	—	94	4	no	—
126	C₂²×C₄	—	188	4	no	—
127	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
128	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
129	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	94	4	no	—
133	C₄⋊C₄	—	94	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₂²⋊C₄	—	188	8	no	—
137	C₄²	—	94	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	—	2	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	#10625352	1	4	yes	—
145	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
146	C₂³	C₂³×D₄₇	1	1	yes	characteristic
147	D₄	—	4	8	no	—
148	D₄	—	4	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	94	8	no	—
158	C₂×C₄	—	188	8	no	—
159	C₂×C₄	#10625352	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
162	C₂²	—	2	8	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	8	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	#13094011	1	4	yes	—
170	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂²	C₂×#10625352	1	2	yes	—
172	C₂²	C₂×#10625352	1	2	yes	—
173	C₂²	#13094070	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	188	4	no	—
177	C₄	—	188	4	no	—
178	C₄	#13094005	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	—	8	2	no	—
181	C₂	C₂×#13094005	1	1	yes	characteristic
182	C₂	C₂×#13094011	1	1	yes	characteristic
183	C₂	#15689687	1	1	yes	characteristic
184	C₂	#15689825	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

#15689048

C₂

yes

maximal, complemented

#15689026

C₂

yes

maximal, complemented

#15689003

C₂

yes

maximal, complemented

#15688988

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#121×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

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×D₄×C₄₇

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#121×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×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

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

D₄×C₄₇

—

D₄×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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

#2504

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#121

D₄₇

yes

characteristic

C₂²×C₄⋊C₄

—

—

—

—

—

C₄×C₂²⋊C₄

—

100

C₄₇

#2504

yes

characteristic, complemented, Sylow

101

C₂²×D₄

C₂×D₄₇

yes

—

102

C₂²×D₄

C₂×D₄₇

yes

characteristic

103

C₂³×C₄

—

104

C₂³×C₄

C₂×D₄₇

yes

characteristic

105

#121

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

113

C₂×C₄²

—

114

#95

—

115

#95

—

116

C₂⁴

C₂²×D₄₇

yes

—

117

C₂⁴

C₂²×D₄₇

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

188

—

127

C₂²×C₄

C₂²⋊D₄₇

yes

—

128

C₂²×C₄

C₂²×D₄₇

yes

—

129

C₂²×C₄

C₂²×D₄₇

yes

characteristic

130

C₂²×C₄

C₂²×D₄₇

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

188

—

137

C₄²

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

#10625352

yes

—

145

C₂³

C₂×C₂²⋊D₄₇

yes

—

146

C₂³

C₂³×D₄₇

yes

characteristic

147

D₄

—

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

#10625352

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

161

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#13094011

yes

—

170

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

171

C₂²

C₂×#10625352

yes

—

172

C₂²

C₂×#10625352

yes

—

173

C₂²

#13094070

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

188

—

177

C₄

—

188

—

178

C₄

#13094005

yes

—

179

C₂

—

180

C₂

—

181

C₂

C₂×#13094005

yes

characteristic

182

C₂

C₂×#13094011

yes

characteristic

183

C₂

#15689687

yes

characteristic

184

C₂

#15689825

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#121×C₄₇	C₂
hypercenter	C₂×#121	D₄₇
hypocenter	C₄₇	#2504
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#121×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₄₇	C₂	semidirect
2. #15689048	C₂	semidirect
3. #15689026	C₂	semidirect
4. #15689003	C₂	semidirect
5. #15688988	C₂	semidirect
6. C₄×C₂²⋊Dic₄₇	C₂	semidirect
100. C₄₇	#2504	semidirect

H ≅ G/N

type

1. C₂²×C₄⋊Dic₄₇

C₂

semidirect

2. #15689048

C₂

semidirect

3. #15689026

C₂

semidirect

4. #15689003

C₂

semidirect

5. #15688988

C₂

semidirect

6. C₄×C₂²⋊Dic₄₇

C₂

semidirect

100. C₄₇

#2504

semidirect

Subgroups — #20247881

Conjugacy classes of subgroups

Complements