Group #20245016 — subgroups

Subgroups — #20245016

Order 6016.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688907	C₂	1	4	yes	maximal, complemented
4	C₂×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#13093964	—	2	1	no	—
6	#13093964	—	2	1	no	—
7	#13093964	—	2	1	no	—
8	#13093964	C₂²	1	2	yes	complemented
9	C₂×C₄₇⋊C₁₆	—	2	1	no	—
10	C₂×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
11	C₂×#41×C₄₇	C₄	1	1	yes	characteristic
12	#131×C₄₇	—	2	1	no	—
13	#131×C₄₇	—	2	1	no	—
14	#131×C₄₇	C₄	1	2	yes	—
15	#131×C₄₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₄₇	—	2	1	no	—
17	C₂×M₁₆×C₄₇	C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	—	2	1	no	—
19	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	#41×C₄₇	—	2	1	no	—
21	#41×C₄₇	—	2	1	no	—
22	#41×C₄₇	C₈	1	2	yes	—
23	#41×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×Q₈×C₄₇	C₈	1	1	yes	characteristic
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	C₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	M₁₆×C₄₇	—	2	1	no	—
30	M₁₆×C₄₇	—	2	2	no	—
31	M₁₆×C₄₇	—	2	2	no	—
32	M₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₄₇⋊C₁₆	—	2	2	no	—
43	C₄₇⋊C₁₆	—	2	2	no	—
44	C₂³×C₄₇	—	2	1	no	—
45	C₂³×C₄₇	M₁₆	1	1	yes	characteristic
46	Q₈×C₄₇	—	2	1	no	—
47	Q₈×C₄₇	C₂×C₈	1	2	yes	—
48	D₄×C₄₇	—	2	2	no	—
49	D₄×C₄₇	—	2	2	no	—
50	D₄×C₄₇	—	2	1	no	—
51	D₄×C₄₇	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₈×C₄₇	—	2	2	no	—
62	C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄₇	—	2	2	no	—
67	C₂²×C₄₇	—	4	1	no	—
68	C₂²×C₄₇	—	4	2	no	—
69	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
77	#2117	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	—	2	1	no	—
81	C₂×C₄₇	—	4	2	no	—
82	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂×#131	D₄₇	1	1	yes	characteristic
84	C₂×M₃₂	—	47	1	no	—
85	C₂×M₃₂	—	47	1	no	—
86	#350	—	47	4	no	—
87	C₄₇	#2117	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	Dic₄₇	1	1	yes	characteristic
89	#131	—	2	1	no	—
90	#131	—	2	1	no	—
91	#131	Dic₄₇	1	2	yes	—
92	#131	C₂×D₄₇	1	2	yes	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	Dic₄₇	1	1	yes	characteristic
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
97	M₃₂	—	47	2	no	—
98	M₃₂	—	94	1	no	—
99	M₃₂	—	94	1	no	—
100	M₃₂	—	94	1	no	—
101	C₂×C₁₆	—	47	2	no	—
102	C₂×C₁₆	—	94	1	no	—
103	#41	—	2	1	no	—
104	#41	—	2	1	no	—
105	#41	C₄₇⋊C₈	1	2	yes	—
106	#41	C₂×Dic₄₇	1	2	yes	—
107	C₂×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₄₇⋊C₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	C₂×Dic₄₇	1	2	yes	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
125	C₁₆	—	94	2	no	—
126	C₁₆	—	94	2	no	—
127	C₂³	—	2	1	no	—
128	C₂³	#10625329	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
131	D₄	—	2	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	2	1	no	—
134	D₄	C₂×C₄₇⋊C₈	1	2	yes	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
140	C₂×C₄	#10625329	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
146	C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
147	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²	—	2	2	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
153	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
154	C₂²	C₂×#10625329	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
157	C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
158	C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	2	no	—
164	C₂	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093964

C₂

yes

maximal, characteristic, complemented

C₂×#13093964

C₂

yes

maximal, characteristic, complemented

#15688907

C₂

yes

maximal, complemented

C₂×#131×C₄₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

complemented

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂²

yes

complemented

C₂×#41×C₄₇

C₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

#131×C₄₇

C₄

yes

—

#131×C₄₇

C₂²

yes

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₄

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

#41×C₄₇

C₈

yes

—

#41×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

C₈

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

M₁₆

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

C₂×C₈

yes

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

C₂²×C₈

yes

characteristic

#2117

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×#131

D₄₇

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

#350

—

C₄₇

#2117

yes

characteristic, complemented, Sylow

C₂×#41

Dic₄₇

yes

—

—

Dic₄₇

yes

—

#131

C₂×D₄₇

yes

—

C₂×M₁₆

—

C₂×M₁₆

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#41

—

104

#41

—

105

#41

C₄₇⋊C₈

yes

—

106

#41

C₂×Dic₄₇

yes

—

107

C₂×Q₈

C₄₇⋊C₈

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

C₄₇⋊C₈

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

C₂×Dic₄₇

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×Dic₄₇

yes

characteristic

120

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

121

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

122

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

123

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

124

C₂×C₈

C₂²×D₄₇

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₂³

—

128

C₂³

#10625329

yes

characteristic

129

Q₈

—

130

Q₈

C₂×C₄₇⋊C₈

yes

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

C₂×C₄₇⋊C₈

yes

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄₇⋊C₈

yes

characteristic

139

C₂×C₄

C₂×C₄₇⋊C₈

yes

characteristic

140

C₂×C₄

#10625329

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

143

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

144

C₈

—

145

C₈

C₂²⋊Dic₄₇

yes

characteristic

146

C₈

C₂²⋊Dic₄₇

yes

characteristic

147

C₈

C₂×C₂²⋊D₄₇

yes

characteristic

148

C₈

C₂×C₂²⋊D₄₇

yes

characteristic

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

153

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

154

C₂²

C₂×#10625329

yes

characteristic

155

C₄

—

156

C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

157

C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

158

C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

159

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₂×#131×C₄₇	C₂
socle	C₂×C₄₇	C₂×C₂²⋊C₈
hypercenter	C₂×#131	D₄₇
hypocenter	C₄₇	#2117
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

subgroup

G/H

center

C₈

C₂×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₂×#131×C₄₇

C₂

socle

C₂×C₄₇

C₂×C₂²⋊C₈

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#13093964	C₂	semidirect
2. C₂×#13093964	C₂	semidirect
3. #15688907	C₂	semidirect
8. #13093964	C₂²	semidirect
10. C₂×C₄₇⋊C₁₆	C₂²	semidirect
87. C₄₇	#2117	semidirect

H ≅ G/N

type

1. C₂×#13093964

C₂

semidirect

2. C₂×#13093964

C₂

semidirect

3. #15688907

C₂

semidirect

8. #13093964

C₂²

semidirect

10. C₂×C₄₇⋊C₁₆

C₂²

semidirect

87. C₄₇

#2117

semidirect

Subgroups — #20245016

Conjugacy classes of subgroups

Complements