Group #20241629 — subgroups

Subgroups — #20241629

Order 6016.

1168 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 229 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×C₂²⋊Dic₄₇	C₂	1	4	yes	maximal, complemented
2	C₂×C₄₇⋊#95	C₂	1	2	yes	maximal, complemented
3	C₂³×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₄₇	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
6	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
7	C₄₇⋊#95	—	2	4	no	—
8	C₂⁵×C₄₇	C₂²	1	2	yes	—
9	C₂²×D₄×C₄₇	—	2	8	no	—
10	C₂²×D₄×C₄₇	—	2	4	no	—
11	C₂²×D₄×C₄₇	C₄	1	4	yes	complemented
12	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄₇	—	2	4	no	—
14	C₂⁴×C₄₇	—	2	4	no	—
15	C₂⁴×C₄₇	—	2	8	no	—
16	C₂⁴×C₄₇	—	2	8	no	—
17	C₂⁴×C₄₇	C₂×C₄	1	4	yes	—
18	C₂⁴×C₄₇	D₄	1	8	yes	—
19	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₄₇	—	2	32	no	—
21	C₂×D₄×C₄₇	—	2	16	no	—
22	C₂×D₄×C₄₇	—	2	8	no	—
23	C₂²×C₄×C₄₇	—	2	4	no	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	D₄	1	4	yes	—
26	C₂²×Dic₄₇	—	2	4	no	—
27	C₂²×Dic₄₇	—	2	8	no	—
28	C₂²×Dic₄₇	—	2	8	no	—
29	C₂²×Dic₄₇	—	2	4	no	—
30	C₂²×Dic₄₇	D₄	1	8	yes	complemented
31	C₂²⋊Dic₄₇	—	2	32	no	—
32	C₂³×C₄₇	—	2	4	no	—
33	C₂³×C₄₇	—	2	4	no	—
34	C₂³×C₄₇	—	2	16	no	—
35	C₂³×C₄₇	—	2	8	no	—
36	C₂³×C₄₇	—	2	16	no	—
37	C₂³×C₄₇	—	2	16	no	—
38	C₂³×C₄₇	—	2	8	no	—
39	C₂³×C₄₇	—	4	4	no	—
40	C₂³×C₄₇	—	4	8	no	—
41	C₂³×C₄₇	—	4	8	no	—
42	C₂³×C₄₇	—	4	4	no	—
43	C₂³×C₄₇	—	4	4	no	—
44	C₂³×C₄₇	—	4	4	no	—
45	C₂³×C₄₇	C₂²⋊C₄	1	8	yes	complemented
46	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂³×C₄₇	C₂×D₄	1	4	yes	—
48	C₂³×C₄₇	C₂×D₄	1	4	yes	—
49	C₂³×C₄₇	C₂×D₄	1	2	yes	—
50	C₂³×C₄₇	#41	1	4	yes	—
51	D₄×C₄₇	—	2	32	no	—
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	complemented
55	C₂×Dic₄₇	—	2	16	no	—
56	C₂×Dic₄₇	—	2	8	no	—
57	C₂×Dic₄₇	—	4	4	no	—
58	C₂×Dic₄₇	—	4	8	no	—
59	C₂×Dic₄₇	—	4	4	no	—
60	C₂²×C₄₇	—	2	8	no	—
61	C₂²×C₄₇	—	2	16	no	—
62	C₂²×C₄₇	—	4	8	no	—
63	C₂²×C₄₇	—	4	8	no	—
64	C₂²×C₄₇	—	4	16	no	—
65	C₂²×C₄₇	—	4	16	no	—
66	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄₇	C₄×D₄	1	4	yes	—
69	C₂²×C₄₇	#120	1	4	yes	—
70	C₂²×C₄₇	#120	1	4	yes	—
71	C₂²×C₄₇	#121	1	4	yes	—
72	C₂²×C₄₇	#121	1	8	yes	—
73	C₂²×C₄₇	#123	1	4	yes	—
74	C₂²×C₄₇	#124	1	2	yes	—
75	C₂²×C₄₇	#127	1	2	yes	—
76	C₄×C₄₇	—	2	4	no	—
77	Dic₄₇	—	4	8	no	—
78	#1438	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	—	4	8	no	—
80	C₂×C₄₇	#379	1	1	yes	characteristic
81	C₂×C₄₇	#386	1	4	yes	—
82	C₂×C₄₇	#390	1	2	yes	—
83	C₂×C₄₇	#392	1	4	yes	—
84	C₂×C₄₇	#394	1	4	yes	—
85	C₂³×D₄	D₄₇	1	1	yes	characteristic
86	C₂²×C₂²⋊C₄	—	47	4	no	—
87	C₂×#95	—	47	2	no	—
88	C₄₇	#1438	1	1	yes	characteristic, complemented, Sylow
89	C₂⁵	C₂×D₄₇	1	2	yes	—
90	C₂²×D₄	—	2	8	no	—
91	C₂²×D₄	—	2	4	no	—
92	C₂²×D₄	Dic₄₇	1	4	yes	complemented
93	C₂³×C₄	—	47	4	no	—
94	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	94	16	no	—
96	C₂×C₂²⋊C₄	—	94	8	no	—
97	#95	—	94	4	no	—
98	C₂⁴	—	2	8	no	—
99	C₂⁴	—	2	8	no	—
100	C₂⁴	—	2	4	no	—
101	C₂⁴	—	2	4	no	—
102	C₂⁴	C₂×Dic₄₇	1	4	yes	—
103	C₂⁴	C₂²⋊D₄₇	1	8	yes	—
104	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	2	32	no	—
106	C₂×D₄	—	2	16	no	—
107	C₂×D₄	—	2	8	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	47	8	no	—
110	C₂²×C₄	—	94	8	no	—
111	C₂²×C₄	—	94	4	no	—
112	C₂²×C₄	—	94	4	no	—
113	C₂²×C₄	—	94	8	no	—
114	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
116	C₂²⋊C₄	—	94	32	no	—
117	C₂³	—	2	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	2	16	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	16	no	—
122	C₂³	—	2	16	no	—
123	C₂³	—	2	8	no	—
124	C₂³	—	4	4	no	—
125	C₂³	—	4	8	no	—
126	C₂³	—	4	8	no	—
127	C₂³	—	4	4	no	—
128	C₂³	—	4	4	no	—
129	C₂³	—	4	4	no	—
130	C₂³	C₂²⋊Dic₄₇	1	8	yes	complemented
131	C₂³	D₄×D₄₇	1	4	yes	—
132	C₂³	#10625352	1	4	yes	—
133	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
134	C₂³	C₂×C₂²⋊D₄₇	1	4	yes	—
135	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
136	D₄	—	2	32	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	94	16	no	—
140	C₂×C₄	—	94	8	no	—
141	C₂×C₄	—	188	4	no	—
142	C₂×C₄	—	188	8	no	—
143	C₂×C₄	—	188	4	no	—
144	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	complemented
145	C₂²	—	2	8	no	—
146	C₂²	—	2	16	no	—
147	C₂²	—	4	8	no	—
148	C₂²	—	4	8	no	—
149	C₂²	—	4	16	no	—
150	C₂²	—	4	16	no	—
151	C₂²	C₂×C₂²⋊Dic₄₇	1	2	yes	—
152	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂²	D₄×Dic₄₇	1	4	yes	—
154	C₂²	#13093992	1	4	yes	—
155	C₂²	#13093993	1	4	yes	—
156	C₂²	#13093995	1	4	yes	—
157	C₂²	#13093996	1	8	yes	—
158	C₂²	#13094005	1	4	yes	—
159	C₂²	#13094011	1	2	yes	—
160	C₂²	#13094020	1	2	yes	—
161	C₄	—	2	4	no	—
162	C₄	—	188	8	no	—
163	C₂	—	4	8	no	—
164	C₂	#15688979	1	1	yes	characteristic
165	C₂	#15689003	1	4	yes	—
166	C₂	#15689026	1	2	yes	—
167	C₂	#15689031	1	4	yes	—
168	C₂	#15689037	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊Dic₄₇

C₂

yes

maximal, complemented

C₂×C₄₇⋊#95

C₂

yes

maximal, complemented

C₂³×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₄₇⋊#95

—

C₂⁵×C₄₇

C₂²

yes

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₄

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂×C₄

yes

—

C₂⁴×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

complemented

C₂²⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

complemented

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

#41

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

complemented

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

#120

yes

—

C₂²×C₄₇

#120

yes

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#123

yes

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

#127

yes

—

C₄×C₄₇

—

Dic₄₇

—

#1438

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#379

yes

characteristic

C₂×C₄₇

#386

yes

—

C₂×C₄₇

#390

yes

—

C₂×C₄₇

#392

yes

—

C₂×C₄₇

#394

yes

—

C₂³×D₄

D₄₇

yes

characteristic

C₂²×C₂²⋊C₄

—

C₂×#95

—

C₄₇

#1438

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₄₇

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

Dic₄₇

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

—

100

C₂⁴

—

101

C₂⁴

—

102

C₂⁴

C₂×Dic₄₇

yes

—

103

C₂⁴

C₂²⋊D₄₇

yes

—

104

C₂⁴

C₂²×D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×Dic₄₇

yes

—

115

C₂²×C₄

C₂²⋊D₄₇

yes

—

116

C₂²⋊C₄

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂²⋊Dic₄₇

yes

complemented

131

C₂³

D₄×D₄₇

yes

—

132

C₂³

#10625352

yes

—

133

C₂³

C₂²×Dic₄₇

yes

characteristic

134

C₂³

C₂×C₂²⋊D₄₇

yes

—

135

C₂³

C₂×C₂²⋊D₄₇

yes

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

188

—

142

C₂×C₄

—

188

—

143

C₂×C₄

—

188

—

144

C₂×C₄

C₂²⋊Dic₄₇

yes

complemented

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×C₂²⋊Dic₄₇

yes

—

152

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

153

C₂²

D₄×Dic₄₇

yes

—

154

C₂²

#13093992

yes

—

155

C₂²

#13093993

yes

—

156

C₂²

#13093995

yes

—

157

C₂²

#13093996

yes

—

158

C₂²

#13094005

yes

—

159

C₂²

#13094011

yes

—

160

C₂²

#13094020

yes

—

161

C₄

—

162

C₄

—

188

—

163

C₂

—

164

C₂

#15688979

yes

characteristic

165

C₂

#15689003

yes

—

166

C₂

#15689026

yes

—

167

C₂

#15689031

yes

—

168

C₂

#15689037

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Fitting	C₂³×D₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂³
hypercenter	C₂³×D₄	D₄₇
hypocenter	C₄₇	#1438
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

subgroup

G/H

center, Frattini

C₂⁴

C₂²×D₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Fitting

C₂³×D₄×C₄₇

C₂

socle

C₂⁴×C₄₇

C₂³

hypercenter

C₂³×D₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂²×C₂²⋊Dic₄₇	C₂	semidirect
2. C₂×C₄₇⋊#95	C₂	semidirect
11. C₂²×D₄×C₄₇	C₄	semidirect
30. C₂²×Dic₄₇	D₄	semidirect
45. C₂³×C₄₇	C₂²⋊C₄	semidirect
54. C₂×C₄×C₄₇	C₂²⋊C₄	semidirect
88. C₄₇	#1438	semidirect
92. C₂²×D₄	Dic₄₇	semidirect
130. C₂³	C₂²⋊Dic₄₇	semidirect
144. C₂×C₄	C₂²⋊Dic₄₇	semidirect

H ≅ G/N

type

1. C₂²×C₂²⋊Dic₄₇

C₂

semidirect

2. C₂×C₄₇⋊#95

C₂

semidirect

11. C₂²×D₄×C₄₇

C₄

semidirect

30. C₂²×Dic₄₇

D₄

semidirect

45. C₂³×C₄₇

C₂²⋊C₄

semidirect

54. C₂×C₄×C₄₇

C₂²⋊C₄

semidirect

88. C₄₇

#1438

semidirect

92. C₂²×D₄

Dic₄₇

semidirect

130. C₂³

C₂²⋊Dic₄₇

semidirect

144. C₂×C₄

C₂²⋊Dic₄₇

semidirect

Subgroups — #20241629

Conjugacy classes of subgroups

Complements