Group #20243784 — subgroups

Subgroups — #20243784

Order 6016.

444 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13094020	C₂	1	1	yes	maximal, characteristic
2	#15689598	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689129	C₂	1	2	yes	maximal, complemented
4	#15688884	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊D₄₇	—	2	2	no	—
8	#13094020	—	2	2	no	—
9	#13094020	C₄	1	4	yes	complemented
10	D₄×Dic₄₇	—	2	4	no	—
11	#13093983	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#13093946	—	2	4	no	—
15	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
17	#100×C₄₇	—	2	2	no	—
18	C₂⁴×C₄₇	D₄	1	2	yes	—
19	C₂³×D₄₇	—	2	1	no	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₄₇	—	2	8	no	—
26	C₂×C₂²⋊D₄₇	—	4	8	no	—
27	C₂²×Dic₄₇	—	2	1	no	—
28	C₂²×Dic₄₇	—	2	4	no	—
29	C₂×D₁₈₈	—	2	2	no	—
30	C₂×D₁₈₈	—	2	2	no	—
31	C₂×D₁₈₈	—	2	1	no	—
32	C₂×D₁₈₈	C₂×C₄	1	2	yes	complemented
33	M₁₆×C₄₇	—	2	2	no	—
34	M₁₆×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂²⋊Dic₄₇	—	2	4	no	—
37	C₂²⋊Dic₄₇	—	2	2	no	—
38	C₄⋊Dic₄₇	D₄	1	4	yes	complemented
39	C₄×Dic₄₇	—	2	1	no	—
40	C₄×Dic₄₇	—	2	2	no	—
41	C₄×Dic₄₇	C₂×C₄	1	2	yes	complemented
42	C₂³×C₄₇	—	2	2	no	—
43	C₂³×C₄₇	—	2	2	no	—
44	C₂³×C₄₇	—	4	4	no	—
45	C₂³×C₄₇	C₂²⋊C₄	1	4	yes	—
46	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₄₇	—	2	2	no	—
48	C₂²×D₄₇	—	4	4	no	—
49	C₂²×D₄₇	—	4	1	no	—
50	C₂²×D₄₇	—	4	1	no	—
51	D₄×C₄₇	—	2	4	no	—
52	D₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#41	1	2	yes	—
57	C₂²⋊D₄₇	—	4	16	no	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	8	4	no	—
62	D₁₈₈	—	2	4	no	—
63	D₁₈₈	—	4	2	no	—
64	C₈×C₄₇	—	4	2	no	—
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	—	2	4	no	—
67	C₂²×C₄₇	—	4	4	no	—
68	C₂²×C₄₇	—	4	4	no	—
69	C₂²×C₄₇	—	8	2	no	—
70	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄₇	#120	1	1	yes	characteristic
72	C₂²×C₄₇	#123	1	1	yes	characteristic
73	C₂×D₄₇	—	4	4	no	—
74	C₂×D₄₇	—	4	2	no	—
75	C₂×D₄₇	—	8	2	no	—
76	C₂×D₄₇	—	8	2	no	—
77	C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₄×C₄₇	#121	1	2	yes	—
79	Dic₄₇	—	2	4	no	—
80	Dic₄₇	—	4	4	no	—
81	#1887	—	47	1	no	maximal, Sylow
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	—	4	4	no	—
85	C₂×C₄₇	#386	1	1	yes	characteristic
86	C₂×C₄₇	#453	1	2	yes	—
87	D₄₇	—	8	2	no	—
88	C₂×#127	—	47	1	no	—
89	#518	—	47	1	no	—
90	#418	—	47	2	no	—
91	C₂×#100	D₄₇	1	1	yes	characteristic, complemented
92	#339	—	47	2	no	—
93	C₄₇	#1887	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	—	47	1	no	—
95	C₂²×D₄	—	94	2	no	—
96	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
97	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
98	#127	—	47	4	no	—
99	#127	—	94	2	no	—
100	C₄×D₄	—	94	4	no	—
101	#117	—	47	2	no	—
102	C₂×C₂²⋊C₄	—	94	2	no	—
103	C₂×C₄²	—	47	1	no	—
104	#102	—	94	4	no	—
105	#100	—	2	2	no	—
106	C₂⁴	—	94	1	no	—
107	C₂⁴	D₁₈₈	1	2	yes	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	47	2	no	—
111	C₂×D₄	—	94	8	no	—
112	C₂×D₄	—	94	2	no	—
113	C₂×D₄	—	94	2	no	—
114	C₂×D₄	—	94	1	no	—
115	C₂×D₄	—	188	8	no	—
116	C₂×D₄	C₄×D₄₇	1	2	yes	—
117	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
118	C₂²×C₄	—	94	1	no	—
119	C₂²×C₄	—	94	4	no	—
120	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₄⋊C₄	—	47	4	no	—
125	C₂²⋊C₄	—	94	4	no	—
126	C₂²⋊C₄	—	94	2	no	—
127	C₄²	—	47	2	no	—
128	C₄²	—	94	2	no	—
129	C₄²	—	94	1	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	4	no	—
133	C₂³	—	94	2	no	—
134	C₂³	—	188	4	no	—
135	C₂³	—	188	1	no	—
136	C₂³	—	188	1	no	—
137	C₂³	(C₂×D₄₇)⋊C₄	1	4	yes	—
138	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
139	D₄	—	2	4	no	—
140	D₄	—	2	4	no	—
141	D₄	—	94	4	no	—
142	D₄	—	188	16	no	—
143	D₄	—	188	2	no	—
144	C₂×C₄	—	94	4	no	—
145	C₂×C₄	—	94	2	no	—
146	C₂×C₄	—	94	4	no	—
147	C₂×C₄	—	376	4	no	—
148	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
149	C₂×C₄	D₄×D₄₇	1	2	yes	—
150	C₂×C₄	#10625352	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₈	—	4	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	376	2	no	—
159	C₂²	—	376	2	no	—
160	C₂²	—	188	4	no	—
161	C₂²	—	188	2	no	—
162	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
163	C₂²	#13093994	1	1	yes	characteristic
164	C₂²	#13094008	1	1	yes	characteristic
165	C₄	—	94	4	no	—
166	C₄	—	188	4	no	—
167	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
168	C₄	#13093996	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	4	4	no	—
172	C₂	—	376	2	no	—
173	C₂	#15689000	1	1	yes	characteristic
174	C₂	#15689293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094020

C₂

yes

maximal, characteristic

#15689598

C₂

yes

maximal, characteristic, complemented

#15689129

C₂

yes

maximal, complemented

#15688884

C₂

yes

maximal, complemented

C₂×#100×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₂²⋊D₄₇

—

#13094020

—

#13094020

C₄

yes

complemented

D₄×Dic₄₇

—

#13093983

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093946

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

#100×C₄₇

—

C₂⁴×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

complemented

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

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

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

#1887

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#386

yes

characteristic

C₂×C₄₇

#453

yes

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#339

—

C₄₇

#1887

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

C₂×D₄₇

yes

—

#127

—

#127

—

100

C₄×D₄

—

101

#117

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

—

104

#102

—

105

#100

—

106

C₂⁴

—

107

C₂⁴

D₁₈₈

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

188

—

116

C₂×D₄

C₄×D₄₇

yes

—

117

C₂×D₄

C₂²⋊D₄₇

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²×D₄₇

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

188

—

135

C₂³

—

188

—

136

C₂³

—

188

—

137

C₂³

(C₂×D₄₇)⋊C₄

yes

—

138

C₂³

C₂×D₁₈₈

yes

characteristic

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

188

—

143

D₄

—

188

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

376

—

148

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

149

C₂×C₄

D₄×D₄₇

yes

—

150

C₂×C₄

#10625352

yes

—

151

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

376

—

159

C₂²

—

376

—

160

C₂²

—

188

—

161

C₂²

—

188

—

162

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

163

C₂²

#13093994

yes

characteristic

164

C₂²

#13094008

yes

characteristic

165

C₄

—

166

C₄

—

188

—

167

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

168

C₄

#13093996

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

—

376

—

173

C₂

#15689000

yes

characteristic

174

C₂

#15689293

yes

—

	subgroup	G/H
center	C₂²	#13093994
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#100×C₄₇	C₂
socle	C₂²×C₄₇	#120
hypercenter	C₂×#100	D₄₇
hypocenter	C₄₇	#1887
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

#13093994

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#100×C₄₇

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #15689598	C₂	semidirect
3. #15689129	C₂	semidirect
4. #15688884	C₂	semidirect
5. C₂×#100×C₄₇	C₂	semidirect
9. #13094020	C₄	semidirect
32. C₂×D₁₈₈	C₂×C₄	semidirect
38. C₄⋊Dic₄₇	D₄	semidirect
41. C₄×Dic₄₇	C₂×C₄	semidirect
91. C₂×#100	D₄₇	semidirect
93. C₄₇	#1887	semidirect

H ≅ G/N

type

2. #15689598

C₂

semidirect

3. #15689129

C₂

semidirect

4. #15688884

C₂

semidirect

5. C₂×#100×C₄₇

C₂

semidirect

9. #13094020

C₄

semidirect

32. C₂×D₁₈₈

C₂×C₄

semidirect

38. C₄⋊Dic₄₇

D₄

semidirect

41. C₄×Dic₄₇

C₂×C₄

semidirect

91. C₂×#100

D₄₇

semidirect

93. C₄₇

#1887

semidirect

Subgroups — #20243784

Conjugacy classes of subgroups

Complements