Group #20255108 — subgroups

Subgroups — #20255108

Order 6016.

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

normal subgroups only

This group has 484 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×#13094047	C₂	1	2	yes	maximal, complemented
2	#15689776	C₂	1	2	yes	maximal, complemented
3	#15689361	C₂	1	2	yes	maximal, complemented
4	#15689325	C₂	1	2	yes	maximal, complemented
5	#15689284	C₂	1	4	yes	maximal, complemented
6	C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625354	C₂²	1	4	yes	—
10	#13094047	—	2	8	no	—
11	C₂×#10625336	C₂²	1	4	yes	—
12	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₄₇	C₂²	1	2	yes	—
14	#13094009	—	2	4	no	—
15	#13094000	C₂²	1	4	yes	—
16	#13093999	—	2	2	no	—
17	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
18	C₄×D₁₈₈	C₂²	1	2	yes	—
19	C₄⋊C₄×D₄₇	—	2	2	no	—
20	#13093958	C₂²	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	#13093948	C₂²	1	4	yes	—
23	#13093946	C₂²	1	2	yes	—
24	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
30	#10625354	—	2	8	no	—
31	#10625354	—	4	4	no	—
32	C₂×D₁₈₈	—	2	4	no	—
33	C₂×D₁₈₈	D₄	1	1	yes	characteristic
34	C₂×D₁₈₈	C₂³	1	2	yes	—
35	C₂×C₄×D₄₇	—	2	2	no	—
36	C₂×C₄×D₄₇	—	2	4	no	—
37	C₂×C₄×D₄₇	C₂³	1	2	yes	—
38	Q₁₆×C₄₇	—	2	4	no	—
39	Q₁₆×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#10625336	—	2	8	no	—
46	(C₂×D₄₇)⋊C₄	—	2	4	no	—
47	(C₂×D₄₇)⋊C₄	—	2	2	no	—
48	(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
49	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
50	C₄⋊Dic₄₇	C₂³	1	2	yes	—
51	Dic₄₇⋊C₄	—	2	4	no	—
52	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
53	D₃₇₆	D₄	1	4	yes	complemented
54	C₈×D₄₇	—	2	4	no	—
55	C₂²×D₄₇	—	2	2	no	—
56	C₂²×D₄₇	C₂×D₄	1	2	yes	—
57	Q₈×C₄₇	—	2	4	no	—
58	Q₈×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
65	D₁₈₈	—	2	2	no	—
66	D₁₈₈	—	2	2	no	—
67	D₁₈₈	—	4	8	no	—
68	D₁₈₈	C₂×D₄	1	4	yes	—
69	C₄×D₄₇	—	2	2	no	—
70	C₄×D₄₇	—	2	2	no	—
71	C₄×D₄₇	—	4	8	no	—
72	C₄×D₄₇	—	4	4	no	—
73	C₈×C₄₇	—	2	1	no	—
74	C₈×C₄₇	C₂×D₄	1	2	yes	—
75	C₄₇⋊C₈	—	2	2	no	—
76	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×D₄₇	—	2	2	no	—
78	C₂×D₄₇	—	2	4	no	—
79	C₂×D₄₇	—	2	2	no	—
80	C₂×D₄₇	—	2	2	no	—
81	C₂×D₄₇	—	4	2	no	—
82	C₂×D₄₇	—	4	2	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	4	4	no	—
85	C₄×C₄₇	#135	1	2	yes	complemented
86	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
88	Dic₄₇	—	2	2	no	—
89	Dic₄₇	—	4	2	no	—
90	#3283	—	47	1	no	maximal, Sylow
91	C₂×C₄₇	D₄²	1	1	yes	characteristic
92	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
93	C₂×C₄₇	#578	1	1	yes	characteristic
94	D₄₇	—	4	2	no	—
95	D₄₇	—	4	4	no	—
96	C₂×#135	—	47	2	no	—
97	#547	—	47	2	no	—
98	#494	D₄₇	1	1	yes	characteristic, complemented
99	#467	—	47	2	no	—
100	#461	—	47	2	no	—
101	#452	—	47	4	no	—
102	C₄×D₈	—	47	1	no	—
103	C₈×D₄	—	47	1	no	—
104	C₄₇	#3283	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	47	4	no	—
106	#135	—	94	8	no	—
107	C₂×Q₁₆	C₂×D₄₇	1	2	yes	—
108	C₂×Q₁₆	C₂×D₄₇	1	2	yes	—
109	C₂×SD₁₆	—	47	4	no	—
110	C₂×D₈	—	47	1	no	—
111	C₂²×C₈	—	47	2	no	—
112	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
113	#123	—	94	4	no	—
114	C₂²⋊Q₈	—	47	4	no	—
115	#121	—	94	2	no	—
116	C₄×D₄	—	47	1	no	—
117	C₄×D₄	—	47	2	no	—
118	C₂×C₄⋊C₄	—	94	2	no	—
119	#107	—	47	1	no	—
120	C₄⋊C₈	—	47	1	no	—
121	#103	—	47	4	no	—
122	#102	—	47	2	no	—
123	C₂²⋊C₈	—	47	2	no	—
124	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
125	#41	—	94	8	no	—
126	#41	—	188	4	no	—
127	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
128	C₂×D₄	—	47	1	no	—
129	C₂×D₄	—	47	2	no	—
130	C₂×D₄	—	94	4	no	—
131	C₂²×C₄	—	47	2	no	—
132	C₂²×C₄	—	94	4	no	—
133	C₂²×C₄	—	94	2	no	—
134	Q₁₆	—	2	4	no	—
135	Q₁₆	—	4	2	no	—
136	SD₁₆	—	94	8	no	—
137	D₈	—	47	4	no	—
138	C₂×C₈	—	47	2	no	—
139	C₂×C₈	—	94	4	no	—
140	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	47	1	no	—
145	C₄⋊C₄	—	47	2	no	—
146	C₄⋊C₄	—	94	4	no	—
147	C₂²⋊C₄	—	47	2	no	—
148	C₂²⋊C₄	—	94	4	no	—
149	C₂²⋊C₄	—	94	2	no	—
150	C₄²	C₂²×D₄₇	1	1	yes	characteristic
151	C₂³	—	47	2	no	—
152	C₂³	—	94	2	no	—
153	Q₈	—	2	4	no	—
154	Q₈	—	2	4	no	—
155	D₄	—	47	4	no	—
156	D₄	—	94	2	no	—
157	D₄	—	94	2	no	—
158	D₄	—	188	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	47	2	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	94	2	no	—
163	C₂×C₄	—	94	2	no	—
164	C₂×C₄	—	188	8	no	—
165	C₂×C₄	—	188	4	no	—
166	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
167	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
168	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	94	2	no	—
171	C₈	D₄×D₄₇	1	2	yes	—
172	C₂²	—	94	2	no	—
173	C₂²	—	94	4	no	—
174	C₂²	—	94	2	no	—
175	C₂²	—	94	2	no	—
176	C₂²	—	188	2	no	—
177	C₂²	—	188	2	no	—
178	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	94	2	no	—
182	C₄	—	188	2	no	—
183	C₄	#13094047	1	2	yes	complemented
184	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
185	C₄	#13094069	1	1	yes	characteristic
186	C₂	—	188	2	no	—
187	C₂	—	188	4	no	—
188	C₂	#15689758	1	1	yes	characteristic
189	C₂	C₂×#13094047	1	1	yes	characteristic
190	C₂	#15689936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094047

C₂

yes

maximal, complemented

#15689776

C₂

yes

maximal, complemented

#15689361

C₂

yes

maximal, complemented

#15689325

C₂

yes

maximal, complemented

#15689284

C₂

yes

maximal, complemented

C₄×D₃₇₆

C₂

yes

maximal, characteristic, complemented

C₈×D₁₈₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₂²

yes

—

#13094047

—

C₂×#10625336

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₂²

yes

—

#13094009

—

#13094000

C₂²

yes

—

#13093999

—

C₄×D₁₈₈

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

—

C₄⋊C₄×D₄₇

—

#13093958

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093948

C₂²

yes

—

#13093946

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

#10625354

—

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂³

yes

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

C₂³

yes

—

D₃₇₆

D₄

yes

complemented

C₈×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

C₂×D₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#135

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3283

—

maximal, Sylow

C₂×C₄₇

D₄²

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

D₄₇

—

C₂×#135

—

#547

—

#494

D₄₇

yes

characteristic, complemented

#467

—

100

#461

—

101

#452

—

102

C₄×D₈

—

103

C₈×D₄

—

104

C₄₇

#3283

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

#135

—

107

C₂×Q₁₆

C₂×D₄₇

yes

—

108

C₂×Q₁₆

C₂×D₄₇

yes

—

109

C₂×SD₁₆

—

110

C₂×D₈

—

111

C₂²×C₈

—

112

C₄⋊Q₈

C₂×D₄₇

yes

—

113

#123

—

114

C₂²⋊Q₈

—

115

#121

—

116

C₄×D₄

—

117

C₄×D₄

—

118

C₂×C₄⋊C₄

—

119

#107

—

120

C₄⋊C₈

—

121

#103

—

122

#102

—

123

C₂²⋊C₈

—

124

C₄×C₈

C₂×D₄₇

yes

characteristic

125

#41

—

126

#41

—

188

—

127

C₂×Q₈

C₂²×D₄₇

yes

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

Q₁₆

—

135

Q₁₆

—

136

SD₁₆

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₂²×D₄₇

yes

characteristic

141

C₂×C₈

C₂²×D₄₇

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

C₂²×D₄₇

yes

characteristic

151

C₂³

—

152

C₂³

—

153

Q₈

—

154

Q₈

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

188

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

188

—

165

C₂×C₄

—

188

—

166

C₂×C₄

D₄×D₄₇

yes

characteristic

167

C₂×C₄

D₄×D₄₇

yes

characteristic

168

C₂×C₄

C₂³×D₄₇

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

D₄×D₄₇

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

188

—

177

C₂²

—

188

—

178

C₂²

C₂×D₄×D₄₇

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

188

—

183

C₄

#13094047

yes

complemented

184

C₄

C₂×D₄×D₄₇

yes

characteristic

185

C₄

#13094069

yes

characteristic

186

C₂

—

188

—

187

C₂

—

188

—

188

C₂

#15689758

yes

characteristic

189

C₂

C₂×#13094047

yes

characteristic

190

C₂

#15689936

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#494	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#494	D₄₇
hypocenter	C₄₇	#3283
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#494

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#13094047	C₂	semidirect
2. #15689776	C₂	semidirect
3. #15689361	C₂	semidirect
4. #15689325	C₂	semidirect
5. #15689284	C₂	semidirect
6. C₄×D₃₇₆	C₂	semidirect
7. C₈×D₁₈₈	C₂	semidirect
8. C₄₇×#494	C₂	semidirect
53. D₃₇₆	D₄	semidirect
85. C₄×C₄₇	#135	semidirect
98. #494	D₄₇	semidirect
104. C₄₇	#3283	semidirect
183. C₄	#13094047	semidirect

H ≅ G/N

type

1. C₂×#13094047

C₂

semidirect

2. #15689776

C₂

semidirect

3. #15689361

C₂

semidirect

4. #15689325

C₂

semidirect

5. #15689284

C₂

semidirect

6. C₄×D₃₇₆

C₂

semidirect

7. C₈×D₁₈₈

C₂

semidirect

8. C₄₇×#494

C₂

semidirect

53. D₃₇₆

D₄

semidirect

85. C₄×C₄₇

#135

semidirect

98. #494

D₄₇

semidirect

104. C₄₇

#3283

semidirect

183. C₄

#13094047

semidirect

Subgroups — #20255108

Conjugacy classes of subgroups

Complements