Group #20242871 — subgroups

Subgroups — #20242871

Order 6016.

184 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 51 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	#15689499	C₂	1	1	yes	maximal, characteristic
2	#15689324	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689320	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689250	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688866	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625325	—	2	1	no	—
9	C₂×#10625335	—	2	1	no	—
10	C₂×#10625334	—	2	1	no	—
11	C₂×#10625334	—	2	1	no	—
12	#13094021	C₂²	1	1	yes	characteristic
13	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₄₇⋊C₈	—	2	1	no	—
16	#13093946	—	2	1	no	—
17	#13093945	—	2	1	no	—
18	#13093934	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	C₂×Dic₉₄	D₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#10625335	—	4	2	no	—
34	#10625334	—	4	2	no	—
35	#10625334	—	4	2	no	—
36	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
37	Dic₄₇⋊C₄	—	2	1	no	—
38	C₄×Dic₄₇	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	#10625325	—	4	2	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	4	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	D₄×C₄₇	—	4	1	no	—
47	D₄×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	4	1	no	—
49	D₄×C₄₇	—	8	1	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	2	1	no	—
55	D₁₈₈	—	4	1	no	—
56	D₁₈₈	—	4	1	no	—
57	D₁₈₈	—	4	1	no	—
58	D₁₈₈	—	4	1	no	—
59	D₁₈₈	—	8	1	no	—
60	Dic₉₄	—	2	1	no	—
61	Dic₉₄	—	2	1	no	—
62	C₈×C₄₇	—	4	1	no	—
63	C₈×C₄₇	—	4	1	no	—
64	C₄₇⋊C₈	—	2	1	no	—
65	C₄₇⋊C₈	—	4	1	no	—
66	C₄₇⋊C₈	SD₁₆	1	2	yes	complemented
67	C₂²×C₄₇	—	8	1	no	—
68	C₂²×C₄₇	—	8	1	no	—
69	C₂²×C₄₇	—	8	1	no	—
70	C₂²×C₄₇	#121	1	1	yes	characteristic
71	C₂×D₄₇	—	8	1	no	—
72	C₂×D₄₇	—	8	1	no	—
73	C₂×D₄₇	—	8	1	no	—
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Dic₄₇	—	4	1	no	—
80	Dic₄₇	—	4	1	no	—
81	#1688	—	47	1	no	maximal, Sylow
82	C₂×C₄₇	—	16	1	no	—
83	C₂×C₄₇	#460	1	1	yes	characteristic
84	C₂×C₄₇	#469	1	1	yes	characteristic
85	C₂×C₄₇	#472	1	1	yes	characteristic
86	D₄₇	—	16	1	no	—
87	#493	—	47	1	no	—
88	#460	—	47	1	no	—
89	#460	—	47	1	no	—
90	#446	—	47	1	no	—
91	#334	—	47	1	no	—
92	#331	—	47	1	no	—
93	#331	D₄₇	1	1	yes	characteristic, complemented
94	C₄₇	#1688	1	1	yes	characteristic, complemented, Sylow
95	C₂×SD₁₆	—	94	1	no	—
96	C₂×SD₁₆	—	94	1	no	—
97	C₂×D₈	—	94	1	no	—
98	C₂×D₈	—	94	1	no	—
99	#127	—	47	1	no	—
100	#127	C₂×D₄₇	1	1	yes	characteristic
101	C₄×Q₈	—	47	1	no	—
102	C₄⋊C₈	—	47	1	no	—
103	C₄⋊C₈	—	94	1	no	—
104	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
105	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
106	#102	—	94	1	no	—
107	#102	—	94	1	no	—
108	#97	—	94	1	no	—
109	C₄×C₈	—	47	1	no	—
110	C₂×Q₈	—	47	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	94	1	no	—
115	C₂×D₄	—	94	1	no	—
116	C₂×D₄	—	94	1	no	—
117	SD₁₆	—	188	2	no	—
118	SD₁₆	—	188	2	no	—
119	D₈	—	188	2	no	—
120	D₈	—	188	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	47	1	no	—
124	C₂×C₈	—	47	1	no	—
125	C₂×C₈	—	94	1	no	—
126	C₄⋊C₄	—	47	1	no	—
127	C₄⋊C₄	—	94	1	no	—
128	C₄²	—	94	1	no	—
129	C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	C₂³	—	4	1	no	—
131	C₂³	—	188	1	no	—
132	Q₈	—	94	1	no	—
133	Q₈	—	94	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	376	1	no	—
140	D₄	—	188	1	no	—
141	D₄	—	188	1	no	—
142	D₄	—	188	1	no	—
143	D₄	—	188	1	no	—
144	C₂×C₄	—	94	1	no	—
145	C₂×C₄	—	94	1	no	—
146	C₂×C₄	#10625350	1	1	yes	characteristic
147	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	47	2	no	—
152	C₈	—	94	1	no	—
153	C₈	—	188	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	376	1	no	—
158	C₂²	—	376	1	no	—
159	C₂²	—	376	1	no	—
160	C₂²	#13093997	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	188	1	no	—
163	C₄	—	188	1	no	—
164	C₄	SD₁₆×D₄₇	1	1	yes	characteristic
165	C₄	#13094048	1	1	yes	characteristic
166	C₄	#13094048	1	1	yes	characteristic
167	C₄	#13094051	1	1	yes	characteristic
168	C₂	—	16	1	no	—
169	C₂	—	752	1	no	—
170	C₂	#15689322	1	1	yes	characteristic
171	C₂	#15689375	1	1	yes	characteristic
172	C₂	#15689392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689499

C₂

yes

maximal, characteristic

#15689324

C₂

yes

maximal, characteristic, complemented

#15689320

C₂

yes

maximal, characteristic, complemented

#15689250

C₂

yes

maximal, characteristic, complemented

#15688873

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₉₄

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093946

—

#13093945

—

#13093934

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

—

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

#10625325

—

C₂³×C₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1688

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1688

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

—

100

#127

C₂×D₄₇

yes

characteristic

101

C₄×Q₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₄₇

yes

characteristic

105

C₄⋊C₈

C₂×D₄₇

yes

characteristic

106

#102

—

107

#102

—

108

#97

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

SD₁₆

—

188

—

118

SD₁₆

—

188

—

119

D₈

—

188

—

120

D₈

—

188

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

C₂²×D₄₇

yes

characteristic

130

C₂³

—

131

C₂³

—

188

—

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

376

—

140

D₄

—

188

—

141

D₄

—

188

—

142

D₄

—

188

—

143

D₄

—

188

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#10625350

yes

characteristic

147

C₂×C₄

D₄×D₄₇

yes

characteristic

148

C₂×C₄

D₄×D₄₇

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

188

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

376

—

158

C₂²

—

376

—

159

C₂²

—

376

—

160

C₂²

#13093997

yes

characteristic

161

C₄

—

162

C₄

—

188

—

163

C₄

—

188

—

164

C₄

SD₁₆×D₄₇

yes

characteristic

165

C₄

#13094048

yes

characteristic

166

C₄

#13094048

yes

characteristic

167

C₄

#13094051

yes

characteristic

168

C₂

—

169

C₂

—

752

—

170

C₂

#15689322

yes

characteristic

171

C₂

#15689375

yes

characteristic

172

C₂

#15689392

yes

characteristic

	subgroup	G/H
center	C₂²	#13093997
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#331	C₂
socle	C₂²×C₄₇	#121
hypercenter	#331	D₄₇
hypocenter	C₄₇	#1688
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#13093997

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#331

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #15689324	C₂	semidirect
3. #15689320	C₂	semidirect
4. #15689250	C₂	semidirect
5. #15688873	C₂	semidirect
6. #15688866	C₂	semidirect
7. C₄₇×#331	C₂	semidirect
66. C₄₇⋊C₈	SD₁₆	semidirect
93. #331	D₄₇	semidirect
94. C₄₇	#1688	semidirect

H ≅ G/N

type

2. #15689324

C₂

semidirect

3. #15689320

C₂

semidirect

4. #15689250

C₂

semidirect

5. #15688873

C₂

semidirect

6. #15688866

C₂

semidirect

7. C₄₇×#331

C₂

semidirect

66. C₄₇⋊C₈

SD₁₆

semidirect

93. #331

D₄₇

semidirect

94. C₄₇

#1688

semidirect

Subgroups — #20242871

Conjugacy classes of subgroups

Complements