Group #20244434 — subgroups

Subgroups — #20244434

Order 6016.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13094055	C₂	1	2	yes	maximal, complemented
2	#15689704	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093952	C₂	1	2	yes	maximal, complemented
4	#15688892	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	2	yes	—
7	#13094055	—	2	4	no	—
8	#13094055	—	2	4	no	—
9	C₂×#10625337	—	2	2	no	—
10	C₂×#10625336	—	2	2	no	—
11	C₂×#10625329	C₂²	1	2	yes	—
12	#13094004	—	2	2	no	—
13	#13093999	—	2	2	no	—
14	#13093998	—	2	2	no	—
15	#13093952	—	2	4	no	—
16	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	#95×C₄₇	—	2	2	no	—
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	—	2	2	no	—
23	C₂×Q₈×C₄₇	D₄	1	2	yes	complemented
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	D₄	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₄₇	—	2	2	no	—
29	#10625350	—	2	2	no	—
30	#10625350	—	2	2	no	—
31	#10625350	—	2	2	no	—
32	#10625350	—	2	2	no	—
33	C₂×D₁₈₈	—	2	2	no	—
34	C₂×D₁₈₈	D₄	1	2	yes	—
35	C₂×C₄×D₄₇	—	2	2	no	—
36	C₂×Dic₉₄	D₄	1	2	yes	—
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄²×C₄₇	—	2	2	no	—
39	#10625337	—	2	8	no	—
40	#10625336	—	2	8	no	—
41	(C₂×D₄₇)⋊C₄	—	2	4	no	—
42	C₄⋊Dic₄₇	—	2	2	no	—
43	#10625329	—	2	2	no	—
44	#10625329	D₄	1	4	yes	complemented
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₄₇	—	2	2	no	—
48	Q₈×C₄₇	—	2	4	no	—
49	Q₈×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²⋊D₄₇	—	4	2	no	—
63	C₂²⋊D₄₇	—	4	2	no	—
64	C₂×Dic₄₇	—	2	2	no	—
65	D₁₈₈	—	2	2	no	—
66	D₁₈₈	—	2	2	no	—
67	D₁₈₈	—	8	2	no	—
68	C₄×D₄₇	—	4	2	no	—
69	C₄×D₄₇	—	4	2	no	—
70	Dic₉₄	—	2	2	no	—
71	Dic₉₄	—	2	2	no	—
72	C₄₇⋊C₈	—	2	4	no	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	#120	1	1	yes	characteristic
76	C₂²×C₄₇	#121	1	1	yes	characteristic
77	C₂²×C₄₇	#127	1	1	yes	characteristic
78	C₂×D₄₇	—	4	2	no	—
79	C₂×D₄₇	—	4	2	no	—
80	C₂×D₄₇	—	4	2	no	—
81	C₄×C₄₇	—	4	2	no	—
82	C₄×C₄₇	—	4	2	no	—
83	C₄×C₄₇	—	4	2	no	—
84	C₄×C₄₇	#120	1	2	yes	—
85	C₄×C₄₇	#121	1	2	yes	—
86	Dic₄₇	—	4	2	no	—
87	#2007	—	47	1	no	maximal, Sylow
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	—	2	1	no	—
90	C₂×C₄₇	#392	1	1	yes	characteristic
91	C₂×C₄₇	#455	1	1	yes	characteristic
92	C₂×C₄₇	#456	1	1	yes	characteristic
93	D₄₇	—	8	2	no	—
94	C₂×#137	—	47	2	no	—
95	#537	—	47	1	no	—
96	C₂×#104	—	47	2	no	—
97	#391	D₄₇	1	1	yes	characteristic, complemented
98	#343	—	47	1	no	—
99	C₄₇	#2007	1	1	yes	characteristic, complemented, Sylow
100	C₂×#41	—	47	2	no	—
101	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
102	#137	—	94	4	no	—
103	#137	—	94	4	no	—
104	C₂×Q₁₆	—	94	2	no	—
105	C₂×SD₁₆	—	94	2	no	—
106	C₂×M₁₆	—	47	2	no	—
107	C₂²⋊Q₈	—	94	2	no	—
108	#121	—	94	2	no	—
109	#121	—	94	2	no	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#104	—	94	4	no	—
113	#95	—	2	2	no	—
114	#41	—	94	2	no	—
115	#41	—	94	2	no	—
116	#41	—	94	2	no	—
117	#41	—	94	2	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	—	47	2	no	—
122	C₂×Q₈	D₁₈₈	1	2	yes	complemented
123	C₂×D₄	—	47	2	no	—
124	C₂×D₄	—	94	2	no	—
125	C₂×D₄	—	94	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	94	2	no	—
129	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
130	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	Q₁₆	—	94	8	no	—
132	SD₁₆	—	94	8	no	—
133	M₁₆	—	47	4	no	—
134	M₁₆	—	94	2	no	—
135	C₂×C₈	—	94	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	94	2	no	—
138	C₂²⋊C₄	—	94	4	no	—
139	C₄²	—	2	2	no	—
140	C₂³	—	94	2	no	—
141	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	Q₈	—	2	4	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	94	2	no	—
145	Q₈	—	94	2	no	—
146	D₄	—	94	2	no	—
147	D₄	—	94	2	no	—
148	D₄	—	376	2	no	—
149	D₄	—	188	2	no	—
150	D₄	—	188	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	94	2	no	—
160	C₂×C₄	—	188	2	no	—
161	C₂×C₄	—	188	2	no	—
162	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
163	C₂×C₄	#10625350	1	1	yes	characteristic
164	C₂×C₄	D₄×D₄₇	1	2	yes	—
165	C₂×C₄	D₄×D₄₇	1	2	yes	—
166	C₈	—	94	4	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	188	2	no	—
170	C₂²	—	188	2	no	—
171	C₂²	—	188	2	no	—
172	C₂²	#13093992	1	1	yes	characteristic
173	C₂²	#13093998	1	1	yes	characteristic
174	C₂²	#13094020	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	188	2	no	—
179	C₄	#13093994	1	2	yes	—
180	C₄	#13093997	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	376	2	no	—
184	C₂	#15689032	1	1	yes	characteristic
185	C₂	#15689300	1	1	yes	characteristic
186	C₂	#15689305	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094055

C₂

yes

maximal, complemented

#15689704

C₂

yes

maximal, characteristic

C₂×#13093952

C₂

yes

maximal, complemented

#15688892

C₂

yes

maximal, characteristic, complemented

C₄₇×#391

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

—

—

—

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625337

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

#10625329

D₄

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

#2007

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#343

—

C₄₇

#2007

yes

characteristic, complemented, Sylow

100

C₂×#41

—

101

C₂²×Q₈

C₂×D₄₇

yes

characteristic

102

#137

—

103

#137

—

104

C₂×Q₁₆

—

105

C₂×SD₁₆

—

106

C₂×M₁₆

—

107

C₂²⋊Q₈

—

108

#121

—

109

#121

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

111

C₂×C₄²

C₂×D₄₇

yes

characteristic

112

#104

—

113

#95

—

114

#41

—

115

#41

—

116

#41

—

117

#41

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

D₁₈₈

yes

complemented

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂²⋊D₄₇

yes

—

130

C₂²×C₄

C₂²×D₄₇

yes

characteristic

131

Q₁₆

—

132

SD₁₆

—

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₂³

—

141

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

D₄

—

147

D₄

—

148

D₄

—

376

—

149

D₄

—

188

—

150

D₄

—

188

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

188

—

161

C₂×C₄

—

188

—

162

C₂×C₄

C₂×D₁₈₈

yes

characteristic

163

C₂×C₄

#10625350

yes

characteristic

164

C₂×C₄

D₄×D₄₇

yes

—

165

C₂×C₄

D₄×D₄₇

yes

—

166

C₈

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

188

—

170

C₂²

—

188

—

171

C₂²

—

188

—

172

C₂²

#13093992

yes

characteristic

173

C₂²

#13093998

yes

characteristic

174

C₂²

#13094020

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

188

—

179

C₄

#13093994

yes

—

180

C₄

#13093997

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

376

—

184

C₂

#15689032

yes

characteristic

185

C₂

#15689300

yes

characteristic

186

C₂

#15689305

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂²×C₄₇	#120
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2007
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#13093992

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#391

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
1. C₂×#13094055	C₂	semidirect
3. C₂×#13093952	C₂	semidirect
4. #15688892	C₂	semidirect
5. C₄₇×#391	C₂	semidirect
23. C₂×Q₈×C₄₇	D₄	semidirect
44. #10625329	D₄	semidirect
97. #391	D₄₇	semidirect
99. C₄₇	#2007	semidirect
122. C₂×Q₈	D₁₈₈	semidirect

H ≅ G/N

type

1. C₂×#13094055

C₂

semidirect

3. C₂×#13093952

C₂

semidirect

4. #15688892

C₂

semidirect

5. C₄₇×#391

C₂

semidirect

23. C₂×Q₈×C₄₇

D₄

semidirect

44. #10625329

D₄

semidirect

97. #391

D₄₇

semidirect

99. C₄₇

#2007

semidirect

122. C₂×Q₈

D₁₈₈

semidirect

Subgroups — #20244434

Conjugacy classes of subgroups

Complements