Group #20244160 — subgroups

Subgroups — #20244160

Order 6016.

200 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13093959	C₂	1	1	yes	maximal, characteristic
2	#15689146	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	#15688888	C₂	1	1	yes	maximal, characteristic
6	#15688887	C₂	1	1	yes	maximal, characteristic
7	C₂×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093959	—	2	2	no	—
13	Dic₄₇⋊C₈	—	2	1	no	—
14	Dic₄₇⋊C₈	—	2	1	no	—
15	Dic₄₇⋊C₈	C₄	1	2	yes	—
16	Dic₄₇⋊C₈	C₄	1	2	yes	—
17	#13093934	—	2	1	no	—
18	C₈×Dic₄₇	—	2	1	no	—
19	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#108×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	Q₈	1	1	yes	characteristic
25	C₂²×Dic₄₇	Q₈	1	1	yes	characteristic
26	M₁₆×C₄₇	—	2	1	no	—
27	M₁₆×C₄₇	—	2	1	no	—
28	M₁₆×C₄₇	—	2	1	no	—
29	M₁₆×C₄₇	D₄	1	2	yes	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₄₇	D₄	1	1	yes	characteristic
38	C₄×Dic₄₇	D₄	1	1	yes	characteristic
39	#10625329	—	2	1	no	—
40	#10625329	—	2	1	no	—
41	#10625329	—	2	1	no	—
42	#10625329	Q₈	1	2	yes	—
43	C₂×C₄₇⋊C₈	—	2	1	no	—
44	C₂×C₄₇⋊C₈	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	4	1	no	—
57	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₄₇	—	2	2	no	—
62	C₈×C₄₇	—	2	2	no	—
63	C₈×C₄₇	—	4	1	no	—
64	C₄₇⋊C₈	—	2	2	no	—
65	C₄₇⋊C₈	—	2	2	no	—
66	C₄₇⋊C₈	—	4	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₇	#125	1	1	yes	characteristic
71	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
73	C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₄₇	#121	1	1	yes	characteristic
75	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
76	Dic₄₇	—	2	1	no	—
77	Dic₄₇	—	2	1	no	—
78	Dic₄₇	—	2	1	no	—
79	Dic₄₇	#108	1	2	yes	complemented
80	#1950	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	—	2	1	no	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	#389	1	1	yes	characteristic
84	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
85	C₂×C₄₇	#430	1	1	yes	characteristic
86	C₂×#108	—	47	1	no	—
87	C₂×#108	D₄₇	1	1	yes	characteristic
88	#423	—	47	1	no	—
89	C₂×C₄⋊C₈	—	47	1	no	—
90	C₄×M₁₆	—	47	1	no	—
91	#341	—	47	1	no	—
92	#341	—	47	1	no	—
93	C₄₇	#1950	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	47	1	no	—
95	C₂×M₁₆	—	47	1	no	—
96	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
97	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
98	C₂²×C₈	—	47	1	no	—
99	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
100	C₂×C₄²	—	47	1	no	—
101	#108	—	2	2	no	—
102	#108	—	94	2	no	—
103	C₄⋊C₈	—	47	2	no	—
104	C₄⋊C₈	—	47	2	no	—
105	C₄⋊C₈	—	94	1	no	—
106	C₄⋊C₈	—	94	1	no	—
107	#97	—	94	1	no	—
108	C₄×C₈	—	94	1	no	—
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	—	47	1	no	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	47	2	no	—
116	M₁₆	—	94	1	no	—
117	M₁₆	—	94	1	no	—
118	M₁₆	—	94	1	no	—
119	M₁₆	C₂²⋊D₄₇	1	2	yes	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	47	2	no	—
125	C₂×C₈	—	94	1	no	—
126	C₂×C₈	—	94	1	no	—
127	C₂×C₈	—	94	1	no	—
128	C₂×C₈	—	94	1	no	—
129	C₂×C₈	C₄×D₄₇	1	2	yes	—
130	C₄²	—	47	1	no	—
131	C₄²	—	47	1	no	—
132	C₄²	—	47	1	no	—
133	C₄²	—	47	1	no	—
134	C₂³	Q₈×D₄₇	1	1	yes	characteristic
135	C₂×C₄	—	47	1	no	—
136	C₂×C₄	—	47	1	no	—
137	C₂×C₄	—	47	1	no	—
138	C₂×C₄	—	47	1	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	188	1	no	—
141	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₂×C₄	#10625350	1	1	yes	characteristic
143	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
144	C₂×C₄	#10625352	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	94	2	no	—
151	C₈	—	94	2	no	—
152	C₈	—	188	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
156	C₂²	#13094014	1	1	yes	characteristic
157	C₂²	#13094022	1	1	yes	characteristic
158	C₄	—	47	2	no	—
159	C₄	—	94	1	no	—
160	C₄	—	94	1	no	—
161	C₄	—	94	1	no	—
162	C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₄	#13093989	1	1	yes	characteristic
164	C₄	#13093996	1	1	yes	characteristic
165	C₄	#13094002	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#15689018	1	1	yes	characteristic
169	C₂	#108×D₄₇	1	1	yes	characteristic
170	C₂	#15689170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093959

C₂

yes

maximal, characteristic

#15689146

C₂

yes

maximal, characteristic

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, characteristic

M₁₆×Dic₄₇

C₂

yes

maximal, characteristic

#15688888

C₂

yes

maximal, characteristic

#15688887

C₂

yes

maximal, characteristic

C₂×#108×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093959

—

Dic₄₇⋊C₈

—

Dic₄₇⋊C₈

—

Dic₄₇⋊C₈

C₄

yes

—

Dic₄₇⋊C₈

C₄

yes

—

#13093934

—

C₈×Dic₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#108×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

Q₈

yes

characteristic

C₂²×Dic₄₇

Q₈

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

—

—

—

Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#125

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

C₄×Q₈

yes

characteristic

C₄×C₄₇

#121

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

#341

—

C₄₇

#1950

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

100

C₂×C₄²

—

101

#108

—

102

#108

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#97

—

108

C₄×C₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

C₂²⋊D₄₇

yes

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₄×D₄₇

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₂³

Q₈×D₄₇

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

188

—

141

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

142

C₂×C₄

#10625350

yes

characteristic

143

C₂×C₄

D₄×D₄₇

yes

characteristic

144

C₂×C₄

#10625352

yes

characteristic

145

C₂×C₄

Q₈×D₄₇

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

188

—

153

C₂²

—

154

C₂²

—

155

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

156

C₂²

#13094014

yes

characteristic

157

C₂²

#13094022

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₄×C₂²⋊D₄₇

yes

characteristic

163

C₄

#13093989

yes

characteristic

164

C₄

#13093996

yes

characteristic

165

C₄

#13094002

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#15689018

yes

characteristic

169

C₂

#108×D₄₇

yes

characteristic

170

C₂

#15689170

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	C₂×#108	D₄₇
hypocenter	C₄₇	#1950
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#108×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
79. Dic₄₇	#108	semidirect
93. C₄₇	#1950	semidirect

H ≅ G/N

type

79. Dic₄₇

#108

semidirect

93. C₄₇

#1950

semidirect

Subgroups — #20244160

Conjugacy classes of subgroups

Complements