Group #20244315 — subgroups

Subgroups — #20244315

Order 6016.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13094021	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093944	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₈₈	—	2	1	no	—
6	C₂²×D₁₈₈	—	2	1	no	—
7	C₂²×D₁₈₈	C₂²	1	2	yes	—
8	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#13094021	—	2	2	no	—
10	#13094021	C₄	1	4	yes	complemented
11	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
12	#13093944	—	2	8	no	—
13	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₇	—	2	1	no	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	#95×C₄₇	—	2	1	no	—
17	C₂³×D₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×D₁₈₈	—	2	4	no	—
23	C₂×D₁₈₈	—	2	4	no	—
24	C₂×D₁₈₈	—	2	4	no	—
25	C₂×D₁₈₈	—	2	4	no	—
26	C₂×D₁₈₈	—	2	2	no	—
27	C₂×D₁₈₈	—	4	4	no	—
28	C₂×D₁₈₈	—	4	2	no	—
29	C₂×D₁₈₈	—	4	2	no	—
30	C₂×D₁₈₈	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	D₄	1	4	yes	complemented
33	C₄²×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄₇⋊C₈	—	2	4	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×D₄₇	—	2	4	no	—
40	C₂²×D₄₇	—	4	4	no	—
41	C₂²×D₄₇	—	4	2	no	—
42	C₂²×D₄₇	—	4	2	no	—
43	C₂²×D₄₇	—	4	4	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₄₇	D₈	1	2	yes	—
51	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	2	yes	—
56	D₁₈₈	—	2	8	no	—
57	D₁₈₈	—	4	8	no	—
58	D₁₈₈	—	4	8	no	—
59	D₁₈₈	—	4	2	no	—
60	D₁₈₈	—	4	2	no	—
61	C₄₇⋊C₈	—	4	2	no	—
62	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄₇	#121	1	2	yes	—
64	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×D₄₇	—	4	4	no	—
68	C₂×D₄₇	—	4	4	no	—
69	C₂×D₄₇	—	4	4	no	—
70	C₂×D₄₇	—	8	4	no	—
71	C₂×D₄₇	—	8	4	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	#102	1	4	yes	complemented
75	C₄×C₄₇	C₄×D₄	1	2	yes	—
76	C₄×C₄₇	#124	1	1	yes	characteristic
77	C₄×C₄₇	#127	1	1	yes	characteristic
78	#1982	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	#390	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
81	C₂×C₄₇	#418	1	1	yes	characteristic
82	C₂×C₄₇	#459	1	2	yes	—
83	C₂×C₄₇	#460	1	2	yes	—
84	D₄₇	—	8	4	no	—
85	C₂×#127	—	47	1	no	—
86	C₂×C₄⋊C₈	—	47	1	no	—
87	C₂×#102	—	47	4	no	—
88	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
89	C₄₇	#1982	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	—	47	2	no	—
91	C₂²×D₄	—	94	1	no	—
92	C₂²×D₄	—	94	1	no	—
93	C₂²×C₈	—	47	2	no	—
94	#127	—	47	4	no	—
95	#127	—	94	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	—	94	2	no	—
100	#102	—	94	8	no	—
101	#95	—	2	1	no	—
102	C₂⁴	—	94	2	no	—
103	C₂×D₄	—	47	4	no	—
104	C₂×D₄	—	94	4	no	—
105	C₂×D₄	—	94	4	no	—
106	C₂×D₄	—	94	4	no	—
107	C₂×D₄	—	94	2	no	—
108	C₂×D₄	—	94	4	no	—
109	C₂×D₄	—	188	4	no	—
110	C₂×D₄	—	188	2	no	—
111	C₂×D₄	—	188	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	C₂×C₈	—	94	4	no	—
117	C₂×C₈	—	94	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	D₁₈₈	1	4	yes	complemented
120	C₄²	—	4	2	no	—
121	C₄²	C₄×D₄₇	1	2	yes	—
122	C₄²	D₁₈₈	1	2	yes	—
123	C₂³	—	94	4	no	—
124	C₂³	—	188	4	no	—
125	C₂³	—	188	2	no	—
126	C₂³	—	188	2	no	—
127	C₂³	—	188	4	no	—
128	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	D₄	—	94	8	no	—
130	D₄	—	188	8	no	—
131	D₄	—	188	8	no	—
132	D₄	—	188	2	no	—
133	D₄	—	188	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
140	C₂×C₄	#10625334	1	2	yes	—
141	C₂×C₄	#10625336	1	2	yes	—
142	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
144	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₄	#10625350	1	2	yes	—
146	C₈	—	188	2	no	—
147	C₂²	—	376	4	no	—
148	C₂²	—	376	4	no	—
149	C₂²	—	188	4	no	—
150	C₂²	—	188	4	no	—
151	C₂²	—	188	4	no	—
152	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
153	C₂²	#13093998	1	2	yes	—
154	C₂²	C₂×#10625334	1	1	yes	characteristic
155	C₂²	C₂×#10625336	1	1	yes	characteristic
156	C₂²	#13094049	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	#13093944	1	4	yes	complemented
160	C₄	C₄×D₁₈₈	1	2	yes	—
161	C₄	#13094012	1	1	yes	characteristic
162	C₄	#13094021	1	1	yes	characteristic
163	C₂	—	376	4	no	—
164	C₂	#15689025	1	1	yes	characteristic
165	C₂	C₂×#13093944	1	1	yes	characteristic
166	C₂	#15689127	1	1	yes	characteristic
167	C₂	#15689318	1	2	yes	—
168	C₂	#15689323	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#13093944

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094021

—

#13094021

C₄

yes

complemented

C₄⋊(C₄₇⋊C₈)

—

#13093944

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

complemented

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#102

yes

complemented

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#460

yes

—

D₄₇

—

C₂×#127

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₄×C₄⋊C₄

D₄₇

yes

characteristic, complemented

C₄₇

#1982

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

100

#102

—

101

#95

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

188

—

110

C₂×D₄

—

188

—

111

C₂×D₄

—

188

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

114

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

115

C₂²×C₄

C₂²×D₄₇

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₈₈

yes

complemented

120

C₄²

—

121

C₄²

C₄×D₄₇

yes

—

122

C₄²

D₁₈₈

yes

—

123

C₂³

—

124

C₂³

—

188

—

125

C₂³

—

188

—

126

C₂³

—

188

—

127

C₂³

—

188

—

128

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

129

D₄

—

130

D₄

—

188

—

131

D₄

—

188

—

132

D₄

—

188

—

133

D₄

—

188

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

140

C₂×C₄

#10625334

yes

—

141

C₂×C₄

#10625336

yes

—

142

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

143

C₂×C₄

C₂×D₁₈₈

yes

—

144

C₂×C₄

C₂×D₁₈₈

yes

characteristic

145

C₂×C₄

#10625350

yes

—

146

C₈

—

188

—

147

C₂²

—

376

—

148

C₂²

—

376

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

—

188

—

152

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

153

C₂²

#13093998

yes

—

154

C₂²

C₂×#10625334

yes

characteristic

155

C₂²

C₂×#10625336

yes

characteristic

156

C₂²

#13094049

yes

—

157

C₄

—

158

C₄

—

159

C₄

#13093944

yes

complemented

160

C₄

C₄×D₁₈₈

yes

—

161

C₄

#13094012

yes

characteristic

162

C₄

#13094021

yes

characteristic

163

C₂

—

376

—

164

C₂

#15689025

yes

characteristic

165

C₂

C₂×#13093944

yes

characteristic

166

C₂

#15689127

yes

characteristic

167

C₂

#15689318

yes

—

168

C₂

#15689323

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#1982
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄×C₄⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₄₇

hypocenter

C₄₇

#1982

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. C₂×#13093944	C₂	semidirect
4. C₄×C₄⋊C₄×C₄₇	C₂	semidirect
10. #13094021	C₄	semidirect
32. C₄⋊C₄×C₄₇	D₄	semidirect
74. C₄×C₄₇	#102	semidirect
88. C₄×C₄⋊C₄	D₄₇	semidirect
89. C₄₇	#1982	semidirect
119. C₄⋊C₄	D₁₈₈	semidirect
159. C₄	#13093944	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

3. C₂×#13093944

C₂

semidirect

4. C₄×C₄⋊C₄×C₄₇

C₂

semidirect

10. #13094021

C₄

semidirect

32. C₄⋊C₄×C₄₇

D₄

semidirect

74. C₄×C₄₇

#102

semidirect

88. C₄×C₄⋊C₄

D₄₇

semidirect

89. C₄₇

#1982

semidirect

119. C₄⋊C₄

D₁₈₈

semidirect

159. C₄

#13093944

semidirect

Subgroups — #20244315

Conjugacy classes of subgroups

Complements