Group #20243152 — subgroups

Subgroups — #20243152

Order 6016.

412 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 153 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×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093934	C₂	1	2	yes	maximal, complemented
4	#15688877	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₄₇	C₄	1	2	yes	—
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₄₇	—	2	4	no	—
9	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	complemented
13	#13093934	—	2	4	no	—
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₄₇	C₄	1	4	yes	complemented
17	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	—	2	4	no	—
25	C₂²×Dic₄₇	—	2	4	no	—
26	C₂²×Dic₄₇	—	2	4	no	—
27	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
32	C₄×Dic₄₇	—	2	2	no	—
33	C₄×Dic₄₇	—	2	4	no	—
34	C₄×Dic₄₇	D₄	1	4	yes	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
38	C₂³×C₄₇	—	2	1	no	—
39	C₂³×C₄₇	—	2	1	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	C₄²	1	2	yes	—
42	C₂³×C₄₇	M₁₆	1	4	yes	—
43	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	C₄²	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×Dic₄₇	—	2	16	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	2	2	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂×Dic₄₇	M₁₆	1	4	yes	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	—	2	4	no	—
66	C₂²×C₄₇	#97	1	4	yes	complemented
67	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₇	#117	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
71	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₄×C₄₇	C₄×D₄	1	2	yes	—
75	Dic₄₇	—	2	4	no	—
76	Dic₄₇	—	2	4	no	—
77	#1749	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	4	no	—
79	C₂×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#97	1	1	yes	characteristic
81	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
82	C₂×C₄₇	#407	1	2	yes	—
83	C₂×C₄₇	#432	1	2	yes	—
84	C₂²×C₄²	—	47	1	no	—
85	C₂×C₂²⋊C₈	—	47	1	no	—
86	C₂×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
87	C₂×#97	—	47	2	no	—
88	#336	—	47	2	no	—
89	C₄₇	#1749	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	47	2	no	—
91	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
92	C₂²×C₈	—	47	2	no	—
93	C₂²×C₈	C₂×D₄₇	1	2	yes	—
94	C₂×C₄²	—	47	1	no	—
95	C₂×C₄²	—	47	2	no	—
96	C₂×C₄²	—	47	1	no	—
97	C₂×C₄²	—	94	4	no	—
98	C₂²⋊C₈	—	47	4	no	—
99	C₂²⋊C₈	Dic₄₇	1	4	yes	complemented
100	#97	—	94	4	no	—
101	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	47	2	no	—
105	C₂²×C₄	—	47	2	no	—
106	C₂²×C₄	—	94	4	no	—
107	C₂²×C₄	—	94	4	no	—
108	C₂²×C₄	—	94	4	no	—
109	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	C₄×D₄₇	1	2	yes	—
111	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	47	4	no	—
116	C₂×C₈	—	94	2	no	—
117	C₂×C₈	—	94	2	no	—
118	C₂×C₈	C₂×Dic₄₇	1	4	yes	complemented
119	C₄²	—	47	4	no	—
120	C₄²	—	94	4	no	—
121	C₄²	—	94	2	no	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	1	no	—
124	C₂³	—	2	2	no	—
125	C₂³	#10625324	1	4	yes	—
126	C₂³	C₄×Dic₄₇	1	2	yes	—
127	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	47	4	no	—
133	C₂×C₄	—	47	4	no	—
134	C₂×C₄	—	94	16	no	—
135	C₂×C₄	—	94	2	no	—
136	C₂×C₄	—	94	2	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	94	2	no	—
139	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
141	C₂×C₄	D₄×D₄₇	1	2	yes	—
142	C₂×C₄	#10625352	1	2	yes	—
143	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	94	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	4	no	—
149	C₂²	—	2	4	no	—
150	C₂²	#13093934	1	4	yes	complemented
151	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
152	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
153	C₂²	#13093980	1	1	yes	characteristic
154	C₂²	M₁₆×D₄₇	1	2	yes	—
155	C₂²	C₂×#10625324	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	94	4	no	—
158	C₄	—	94	4	no	—
159	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
160	C₄	D₄×Dic₄₇	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
163	C₂	C₂×#13093934	1	1	yes	characteristic
164	C₂	M₁₆×Dic₄₇	1	1	yes	characteristic
165	C₂	#15689085	1	2	yes	—
166	C₂	#15689177	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×Dic₄₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#13093934

C₂

yes

maximal, complemented

#15688877

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₄

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

C₄

yes

complemented

#13093934

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

C₄

yes

complemented

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄²

yes

—

C₂³×C₄₇

M₁₆

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#97

yes

complemented

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#117

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂²×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₄×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

#1749

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₄₇

C₂×#97

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

#407

yes

—

C₂×C₄₇

#432

yes

—

C₂²×C₄²

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₄₇

yes

characteristic

C₂×#97

—

#336

—

C₄₇

#1749

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₄₇

yes

complemented

100

#97

—

101

C₂⁴

C₄×D₄₇

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

C₄×D₄₇

yes

—

111

C₂²×C₄

C₂×Dic₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂×Dic₄₇

yes

complemented

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

#10625324

yes

—

126

C₂³

C₄×Dic₄₇

yes

—

127

C₂³

C₂×C₄×D₄₇

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₄×Dic₄₇

yes

—

140

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

141

C₂×C₄

D₄×D₄₇

yes

—

142

C₂×C₄

#10625352

yes

—

143

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#13093934

yes

complemented

151

C₂²

C₂×C₄×Dic₄₇

yes

characteristic

152

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

153

C₂²

#13093980

yes

characteristic

154

C₂²

M₁₆×D₄₇

yes

—

155

C₂²

C₂×#10625324

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

160

C₄

D₄×Dic₄₇

yes

—

161

C₂

—

162

C₂

C₂²⋊C₄×Dic₄₇

yes

characteristic

163

C₂

C₂×#13093934

yes

characteristic

164

C₂

M₁₆×Dic₄₇

yes

characteristic

165

C₂

#15689085

yes

—

166

C₂

#15689177

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Fitting	C₂×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#1749
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄²

Fitting

C₂×C₂²⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

hypercenter

C₂×C₂²⋊C₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. C₂×#13093934	C₂	semidirect
4. #15688877	C₂	semidirect
12. C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
16. C₂²⋊C₈×C₄₇	C₄	semidirect
31. C₂×C₈×C₄₇	C₂×C₄	semidirect
37. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
66. C₂²×C₄₇	#97	semidirect
89. C₄₇	#1749	semidirect
99. C₂²⋊C₈	Dic₄₇	semidirect
118. C₂×C₈	C₂×Dic₄₇	semidirect
150. C₂²	#13093934	semidirect

H ≅ G/N

type

3. C₂×#13093934

C₂

semidirect

4. #15688877

C₂

semidirect

12. C₂²⋊(C₄₇⋊C₈)

C₄

semidirect

16. C₂²⋊C₈×C₄₇

C₄

semidirect

31. C₂×C₈×C₄₇

C₂×C₄

semidirect

37. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

66. C₂²×C₄₇

#97

semidirect

89. C₄₇

#1749

semidirect

99. C₂²⋊C₈

Dic₄₇

semidirect

118. C₂×C₈

C₂×Dic₄₇

semidirect

150. C₂²

#13093934

semidirect

Subgroups — #20243152

Conjugacy classes of subgroups

Complements