Group #20241719 — subgroups

Subgroups — #20241719

Order 6016.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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×#13093980	C₂	1	1	yes	maximal, characteristic
2	#15689177	C₂	1	2	yes	maximal, complemented
3	#15688848	C₂	1	2	yes	maximal, complemented
4	#15688847	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#323	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₄₇	C₄	1	1	yes	characteristic
7	#13093980	—	2	2	no	—
8	#13093980	—	2	2	no	—
9	C₂×Dic₄₇⋊C₄	C₄	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	—	2	2	no	—
15	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	#13093934	—	2	2	no	—
17	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂²×Dic₄₇	—	2	1	no	—
24	C₂²×Dic₄₇	—	2	1	no	—
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂²⋊C₄×C₄₇	—	2	2	no	—
32	C₂²⋊Dic₄₇	—	2	2	no	—
33	Dic₄₇⋊C₄	—	2	2	no	—
34	Dic₄₇⋊C₄	—	2	2	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₄×Dic₄₇	—	2	1	no	—
37	C₄×Dic₄₇	D₄	1	2	yes	—
38	C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	—	2	1	no	—
41	C₂³×C₄₇	—	2	1	no	—
42	C₂³×C₄₇	—	4	1	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂³×C₄₇	M₁₆	1	2	yes	—
45	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×Dic₄₇	—	2	4	no	—
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	1	no	—
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	2	1	no	—
55	C₂×Dic₄₇	—	2	1	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	4	1	no	—
58	C₂×Dic₄₇	—	4	1	no	—
59	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×Dic₄₇	M₁₆	1	2	yes	—
63	C₈×C₄₇	—	4	2	no	—
64	C₄₇⋊C₈	—	4	2	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	—	4	2	no	—
69	C₂²×C₄₇	—	4	2	no	—
70	C₂²×C₄₇	—	4	1	no	—
71	C₂²×C₄₇	—	4	1	no	—
72	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	4	1	no	—
77	Dic₄₇	—	2	1	no	—
78	Dic₄₇	—	2	1	no	—
79	Dic₄₇	—	2	2	no	—
80	Dic₄₇	—	2	2	no	—
81	Dic₄₇	—	4	1	no	—
82	#1464	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	—	2	1	no	—
85	C₂×C₄₇	—	4	2	no	—
86	C₂×C₄₇	#407	1	1	yes	characteristic
87	C₂×C₄₇	#410	1	1	yes	characteristic
88	C₂×C₄₇	#413	1	1	yes	characteristic
89	C₂×#117	—	47	1	no	—
90	#432	—	47	2	no	—
91	#324	—	47	2	no	—
92	#323	—	47	1	no	—
93	#323	D₄₇	1	1	yes	characteristic
94	C₄₇	#1464	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₄	—	47	1	no	—
96	#117	—	94	2	no	—
97	#117	—	94	2	no	—
98	C₂×C₄⋊C₄	—	47	1	no	—
99	C₂×C₄⋊C₄	—	47	1	no	—
100	C₂×C₂²⋊C₄	—	47	1	no	—
101	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₄²	—	47	1	no	—
103	C₂×C₄²	—	47	1	no	—
104	C₄⋊C₈	—	94	2	no	—
105	C₂²⋊C₈	—	47	2	no	—
106	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
107	#97	—	94	2	no	—
108	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	—	47	1	no	—
111	C₂²×C₄	—	47	1	no	—
112	C₂²×C₄	—	47	1	no	—
113	C₂²×C₄	—	94	1	no	—
114	C₂²×C₄	—	94	1	no	—
115	C₂²×C₄	—	94	2	no	—
116	C₂²×C₄	—	94	2	no	—
117	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	94	2	no	—
121	C₄⋊C₄	—	94	2	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	94	2	no	—
125	C₄²	—	47	2	no	—
126	C₄²	—	47	2	no	—
127	C₄²	—	94	1	no	—
128	C₄²	—	94	1	no	—
129	C₂³	—	2	1	no	—
130	C₂³	—	2	1	no	—
131	C₂³	—	4	1	no	—
132	C₂³	—	4	1	no	—
133	C₂³	#10625324	1	2	yes	—
134	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	47	1	no	—
139	C₂×C₄	—	47	1	no	—
140	C₂×C₄	—	47	2	no	—
141	C₂×C₄	—	47	2	no	—
142	C₂×C₄	—	94	4	no	—
143	C₂×C₄	—	94	4	no	—
144	C₂×C₄	—	94	1	no	—
145	C₂×C₄	—	94	1	no	—
146	C₂×C₄	—	94	1	no	—
147	C₂×C₄	—	94	1	no	—
148	C₂×C₄	—	94	1	no	—
149	C₂×C₄	—	188	1	no	—
150	C₂×C₄	—	188	1	no	—
151	C₂×C₄	D₄×D₄₇	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	188	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
162	C₂²	M₁₆×D₄₇	1	1	yes	characteristic
163	C₂²	C₂×#10625324	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	94	1	no	—
167	C₄	—	94	1	no	—
168	C₄	—	94	2	no	—
169	C₄	—	94	2	no	—
170	C₄	—	188	1	no	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	#15689085	1	1	yes	characteristic
175	C₂	#15689095	1	1	yes	characteristic
176	C₂	#15689104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093980

C₂

yes

maximal, characteristic

#15689177

C₂

yes

maximal, complemented

#15688848

C₂

yes

maximal, complemented

#15688847

C₂

yes

maximal, characteristic

C₄₇×#323

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₄

yes

characteristic

#13093980

—

#13093980

—

C₂×Dic₄₇⋊C₄

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093934

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

M₁₆

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1464

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#1464

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

C₂×D₄₇

yes

characteristic

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

C₂×D₄₇

yes

—

107

#97

—

108

C₂⁴

C₄×D₄₇

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄×D₄₇

yes

characteristic

118

C₂²×C₄

C₂²×D₄₇

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#10625324

yes

—

134

C₂³

C₂×C₄×D₄₇

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

188

—

150

C₂×C₄

—

188

—

151

C₂×C₄

D₄×D₄₇

yes

—

152

C₈

—

153

C₈

—

188

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

162

C₂²

M₁₆×D₄₇

yes

characteristic

163

C₂²

C₂×#10625324

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

188

—

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#15689085

yes

characteristic

175

C₂

#15689095

yes

characteristic

176

C₂

#15689104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#323	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#323	D₄₇
hypocenter	C₄₇	#1464
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #323

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#323

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #323

N	H ≅ G/N	type
2. #15689177	C₂	semidirect
3. #15688848	C₂	semidirect
94. C₄₇	#1464	semidirect

H ≅ G/N

type

2. #15689177

C₂

semidirect

3. #15688848

C₂

semidirect

94. C₄₇

#1464

semidirect

Subgroups — #20241719

Conjugacy classes of subgroups

Complements