Group #20241715 — subgroups

Subgroups — #20241715

Order 6016.

440 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 93 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	#15689087	C₂	1	2	yes	maximal, complemented
2	#15688847	C₂	1	4	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625329	—	2	2	no	—
5	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
7	C₂⁵×C₄₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₄₇	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
11	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
12	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂⁴×C₄₇	—	2	1	no	—
15	C₂⁴×C₄₇	—	2	1	no	—
16	C₂⁴×C₄₇	—	2	4	no	—
17	C₂⁴×C₄₇	—	4	2	no	—
18	C₂⁴×C₄₇	—	4	2	no	—
19	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂⁴×C₄₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄×C₄₇	—	4	2	no	—
24	C₂²×C₄×C₄₇	—	4	2	no	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	2	yes	—
28	C₂²×C₄×C₄₇	D₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²⋊C₄×C₄₇	—	2	4	no	—
31	C₂²⋊C₄×C₄₇	—	2	4	no	—
32	#10625329	—	4	4	no	—
33	C₂×C₄₇⋊C₈	—	2	4	no	—
34	C₂³×C₄₇	—	2	2	no	—
35	C₂³×C₄₇	—	2	2	no	—
36	C₂³×C₄₇	—	2	2	no	—
37	C₂³×C₄₇	—	2	4	no	—
38	C₂³×C₄₇	—	2	4	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	—	4	2	no	—
42	C₂³×C₄₇	—	4	2	no	—
43	C₂³×C₄₇	—	4	16	no	—
44	C₂³×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	4	2	no	—
46	C₂³×C₄₇	—	4	2	no	—
47	C₂³×C₄₇	—	4	2	no	—
48	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂³×C₄₇	M₁₆	1	4	yes	complemented
51	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₄₇⋊C₈	—	4	4	no	—
63	C₂²×C₄₇	—	2	8	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	—	4	4	no	—
72	C₂²×C₄₇	—	4	8	no	—
73	C₂²×C₄₇	—	4	8	no	—
74	C₂²×C₄₇	—	4	4	no	—
75	C₂²×C₄₇	—	4	2	no	—
76	C₂²×C₄₇	—	4	2	no	—
77	C₂²×C₄₇	#99	1	2	yes	—
78	C₂²×C₄₇	#100	1	2	yes	—
79	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	4	2	no	—
84	#1462	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	—	2	2	no	—
86	C₂×C₄₇	—	2	2	no	—
87	C₂×C₄₇	—	2	2	no	—
88	C₂×C₄₇	—	4	4	no	—
89	C₂×C₄₇	#407	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#99	1	1	yes	characteristic
91	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
92	C₂²×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
93	#407	—	47	2	no	—
94	#323	—	47	4	no	—
95	C₄₇	#1462	1	1	yes	characteristic, complemented, Sylow
96	C₂⁵	Dic₄₇	1	1	yes	characteristic
97	C₂³×C₄	Dic₄₇	1	1	yes	characteristic
98	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
99	C₂×M₁₆	—	94	2	no	—
100	C₂×C₂²⋊C₄	—	2	2	no	—
101	C₂×C₂²⋊C₄	—	2	2	no	—
102	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
103	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
104	C₂²⋊C₈	—	47	4	no	—
105	C₂²⋊C₈	—	94	2	no	—
106	C₂⁴	—	2	4	no	—
107	C₂⁴	—	2	1	no	—
108	C₂⁴	—	2	1	no	—
109	C₂⁴	—	4	2	no	—
110	C₂⁴	—	4	2	no	—
111	C₂⁴	C₂×Dic₄₇	1	1	yes	characteristic
112	C₂⁴	C₂×Dic₄₇	1	2	yes	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
118	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
120	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
121	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
122	M₁₆	—	188	4	no	—
123	C₂×C₈	—	94	4	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	2	4	no	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	16	no	—
136	C₂³	—	4	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	—	4	2	no	—
139	C₂³	—	4	2	no	—
140	C₂³	#10625329	1	4	yes	complemented
141	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
142	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
154	C₈	—	188	4	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	4	4	no	—
164	C₂²	—	4	8	no	—
165	C₂²	—	4	8	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	#13093939	1	2	yes	—
170	C₂²	#13093941	1	2	yes	—
171	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂²	C₂×#10625329	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	2	no	—
178	C₂	—	2	2	no	—
179	C₂	—	4	4	no	—
180	C₂	#15689087	1	1	yes	characteristic
181	C₂	C₂×#13093939	1	1	yes	characteristic
182	C₂	C₂×#13093941	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689087

C₂

yes

maximal, complemented

#15688847

C₂

yes

maximal, complemented

C₂²×C₂²⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂⁵×C₄₇

C₄

yes

characteristic

C₂³×C₄×C₄₇

C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

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

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

M₁₆

yes

complemented

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#99

yes

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1462

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#407

yes

characteristic

C₂×C₄₇

C₂×#99

yes

characteristic

C₂×C₄₇

C₂×#100

yes

characteristic

C₂²×C₂²⋊C₄

D₄₇

yes

characteristic

#407

—

#323

—

C₄₇

#1462

yes

characteristic, complemented, Sylow

C₂⁵

Dic₄₇

yes

characteristic

C₂³×C₄

Dic₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₂²⋊C₄

Dic₄₇

yes

—

103

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

104

C₂²⋊C₈

—

105

C₂²⋊C₈

—

106

C₂⁴

—

107

C₂⁴

—

108

C₂⁴

—

109

C₂⁴

—

110

C₂⁴

—

111

C₂⁴

C₂×Dic₄₇

yes

characteristic

112

C₂⁴

C₂×Dic₄₇

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×Dic₄₇

yes

—

118

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

119

C₂²×C₄

C₂²⋊D₄₇

yes

—

120

C₂²×C₄

C₂²⋊D₄₇

yes

—

121

C₂²×C₄

C₂²×D₄₇

yes

characteristic

122

M₁₆

—

188

—

123

C₂×C₈

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

#10625329

yes

complemented

141

C₂³

C₂²⋊Dic₄₇

yes

characteristic

142

C₂³

C₂²⋊Dic₄₇

yes

characteristic

143

C₂³

C₂²×Dic₄₇

yes

characteristic

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

C₂²⋊Dic₄₇

yes

—

153

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

154

C₈

—

188

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#13093939

yes

—

170

C₂²

#13093941

yes

—

171

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

172

C₂²

C₂×#10625329

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₂

—

177

C₂

—

178

C₂

—

179

C₂

—

180

C₂

#15689087

yes

characteristic

181

C₂

C₂×#13093939

yes

characteristic

182

C₂

C₂×#13093941

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂²×C₂²⋊C₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂²×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#1462
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂²×C₂²⋊C₄×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₂²×C₂²⋊C₄

D₄₇

hypocenter

C₄₇

#1462

LCS

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
1. #15689087	C₂	semidirect
2. #15688847	C₂	semidirect
6. C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
50. C₂³×C₄₇	M₁₆	semidirect
95. C₄₇	#1462	semidirect
140. C₂³	#10625329	semidirect

H ≅ G/N

type

1. #15689087

C₂

semidirect

2. #15688847

C₂

semidirect

6. C₂²⋊(C₄₇⋊C₈)

C₂²

semidirect

50. C₂³×C₄₇

M₁₆

semidirect

95. C₄₇

#1462

semidirect

140. C₂³

#10625329

semidirect

Subgroups — #20241715

Conjugacy classes of subgroups

Complements