Group #20244204 — subgroups

Subgroups — #20244204

Order 6016.

344 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13094012	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093949	C₂	1	2	yes	maximal, complemented
3	C₂×#13093946	C₂	1	2	yes	maximal
4	#15688993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
8	#13094012	—	2	2	no	—
9	#13094012	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₇	—	2	2	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
13	#13093949	—	2	4	no	—
14	#13093946	—	2	4	no	—
15	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	#97×C₄₇	—	2	2	no	—
18	C₂³×D₄₇	—	2	1	no	—
19	C₂²×C₄×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₇	—	2	1	no	—
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	2	2	no	—
25	C₂×D₁₈₈	—	2	1	no	—
26	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
27	C₂×Dic₉₄	—	2	2	no	—
28	C₂×Dic₉₄	—	2	2	no	—
29	C₂×Dic₉₄	—	2	1	no	—
30	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	D₄	1	4	yes	—
34	C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₄²×C₄₇	D₄	1	2	yes	—
36	(C₂×D₄₇)⋊C₄	—	2	8	no	—
37	C₄⋊Dic₄₇	—	2	2	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	—	4	4	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	2	no	—
42	C₂²×D₄₇	—	4	1	no	—
43	C₂²×D₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	4	2	no	—
45	C₂²×D₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	4	1	no	—
55	C₂×Dic₄₇	—	4	1	no	—
56	C₂×Dic₄₇	—	4	2	no	—
57	C₂×Dic₄₇	—	4	4	no	—
58	D₁₈₈	—	2	4	no	—
59	D₁₈₈	—	4	2	no	—
60	Dic₉₄	—	2	4	no	—
61	Dic₉₄	—	4	2	no	—
62	C₈×C₄₇	—	2	4	no	—
63	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄₇	#124	1	1	yes	characteristic
65	C₂²×C₄₇	#127	1	1	yes	characteristic
66	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²×C₄₇	#137	1	2	yes	—
68	C₂×D₄₇	—	4	2	no	—
69	C₂×D₄₇	—	4	4	no	—
70	C₂×D₄₇	—	8	2	no	—
71	C₂×D₄₇	—	8	1	no	—
72	C₂×D₄₇	—	8	1	no	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	C₄×D₄	1	2	yes	—
75	C₄×C₄₇	#121	1	2	yes	—
76	Dic₄₇	—	4	2	no	—
77	Dic₄₇	—	8	2	no	—
78	#1959	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	#390	1	1	yes	characteristic
80	C₂×C₄₇	#417	1	2	yes	—
81	C₂×C₄₇	#489	1	2	yes	—
82	C₂×C₄₇	#496	1	1	yes	characteristic
83	C₂×C₄₇	#497	1	1	yes	characteristic
84	D₄₇	—	8	2	no	—
85	C₂×#124	—	47	1	no	—
86	C₂×#103	—	47	2	no	—
87	C₂×#102	—	47	2	no	—
88	C₂×#97	D₄₇	1	1	yes	characteristic, complemented
89	#384	—	47	1	no	—
90	C₄₇	#1959	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	—	47	1	no	—
92	C₂²×D₄	—	47	1	no	—
93	C₂²×C₈	C₂×D₄₇	1	2	yes	—
94	#124	—	47	4	no	—
95	#124	—	94	2	no	—
96	C₂×C₄⋊C₄	—	47	2	no	—
97	C₂×C₄⋊C₄	—	94	2	no	—
98	C₂×C₂²⋊C₄	—	94	2	no	—
99	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
100	#103	—	94	4	no	—
101	#102	—	94	4	no	—
102	#97	—	2	2	no	—
103	C₂⁴	—	94	1	no	—
104	C₂×Q₈	—	47	2	no	—
105	C₂×Q₈	—	94	1	no	—
106	C₂×Q₈	—	94	2	no	—
107	C₂×Q₈	—	94	2	no	—
108	C₂×D₄	—	47	2	no	—
109	C₂×D₄	—	94	2	no	—
110	C₂×D₄	—	94	2	no	—
111	C₂×D₄	—	94	1	no	—
112	C₂²×C₄	—	94	1	no	—
113	C₂²×C₄	—	94	2	no	—
114	C₂²×C₄	D₁₈₈	1	2	yes	—
115	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	D₁₈₈	1	4	yes	—
119	C₄⋊C₄	—	94	2	no	—
120	C₄⋊C₄	—	94	2	no	—
121	C₄⋊C₄	—	188	4	no	—
122	C₂²⋊C₄	—	94	8	no	—
123	C₄²	C₄×D₄₇	1	2	yes	—
124	C₄²	C₂²⋊D₄₇	1	2	yes	—
125	C₂³	—	94	2	no	—
126	C₂³	—	188	1	no	—
127	C₂³	—	188	1	no	—
128	C₂³	—	188	2	no	—
129	C₂³	—	188	2	no	—
130	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
131	Q₈	—	94	4	no	—
132	Q₈	—	188	2	no	—
133	D₄	—	94	4	no	—
134	D₄	—	188	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	188	2	no	—
139	C₂×C₄	—	188	4	no	—
140	C₂×C₄	—	188	1	no	—
141	C₂×C₄	—	188	1	no	—
142	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
143	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
145	C₂×C₄	#10625350	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₂²	—	376	2	no	—
149	C₂²	—	376	1	no	—
150	C₂²	—	376	1	no	—
151	C₂²	—	188	2	no	—
152	C₂²	—	188	4	no	—
153	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
154	C₂²	#13094012	1	1	yes	characteristic
155	C₂²	#13094021	1	1	yes	characteristic
156	C₂²	#13094052	1	2	yes	—
157	C₂²	#13094057	1	2	yes	—
158	C₄	—	2	4	no	—
159	C₄	—	376	2	no	—
160	C₄	—	188	2	no	—
161	C₄	C₄×D₁₈₈	1	2	yes	—
162	C₄	#13093998	1	2	yes	—
163	C₂	—	376	2	no	—
164	C₂	#15689025	1	1	yes	characteristic
165	C₂	#15689125	1	2	yes	—
166	C₂	#15689483	1	2	yes	—
167	C₂	#15689512	1	1	yes	characteristic
168	C₂	#15689517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

C₂×#13093949

C₂

yes

maximal, complemented

C₂×#13093946

C₂

yes

maximal

#15688993

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

#13094012

—

#13094012

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093949

—

#13093946

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

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

C₂²⋊C₄

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#137

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#489

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×#124

—

C₂×#103

—

C₂×#102

—

C₂×#97

D₄₇

yes

characteristic, complemented

#384

—

C₄₇

#1959

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

C₂×D₄₇

yes

—

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

100

#103

—

101

#102

—

102

#97

—

103

C₂⁴

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₁₈₈

yes

—

115

C₂²×C₄

C₂²×D₄₇

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

D₁₈₈

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

188

—

122

C₂²⋊C₄

—

123

C₄²

C₄×D₄₇

yes

—

124

C₄²

C₂²⋊D₄₇

yes

—

125

C₂³

—

126

C₂³

—

188

—

127

C₂³

—

188

—

128

C₂³

—

188

—

129

C₂³

—

188

—

130

C₂³

C₂×D₁₈₈

yes

characteristic

131

Q₈

—

132

Q₈

—

188

—

133

D₄

—

134

D₄

—

188

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

188

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

—

188

—

142

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

143

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

144

C₂×C₄

C₂×D₁₈₈

yes

—

145

C₂×C₄

#10625350

yes

—

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₈

—

148

C₂²

—

376

—

149

C₂²

—

376

—

150

C₂²

—

376

—

151

C₂²

—

188

—

152

C₂²

—

188

—

153

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

154

C₂²

#13094012

yes

characteristic

155

C₂²

#13094021

yes

characteristic

156

C₂²

#13094052

yes

—

157

C₂²

#13094057

yes

—

158

C₄

—

159

C₄

—

376

—

160

C₄

—

188

—

161

C₄

C₄×D₁₈₈

yes

—

162

C₄

#13093998

yes

—

163

C₂

—

376

—

164

C₂

#15689025

yes

characteristic

165

C₂

#15689125

yes

—

166

C₂

#15689483

yes

—

167

C₂

#15689512

yes

characteristic

168

C₂

#15689517

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#97	D₄₇
hypocenter	C₄₇	#1959
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#97×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×#97

D₄₇

hypocenter

C₄₇

#1959

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#13093949	C₂	semidirect
4. #15688993	C₂	semidirect
5. C₂×#97×C₄₇	C₂	semidirect
9. #13094012	C₄	semidirect
88. C₂×#97	D₄₇	semidirect
90. C₄₇	#1959	semidirect

H ≅ G/N

type

2. C₂×#13093949

C₂

semidirect

4. #15688993

C₂

semidirect

5. C₂×#97×C₄₇

C₂

semidirect

9. #13094012

C₄

semidirect

88. C₂×#97

D₄₇

semidirect

90. C₄₇

#1959

semidirect

Subgroups — #20244204

Conjugacy classes of subgroups

Complements