Group #20247880 — subgroups

Subgroups — #20247880

Order 6016.

596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13093998	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689050	C₂	1	4	yes	maximal, complemented
4	#15689025	C₂	1	2	yes	maximal, complemented
5	#15689000	C₂	1	4	yes	maximal, complemented
6	#15688989	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
10	C₂³×Dic₄₇	C₂²	1	2	yes	—
11	#13093998	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
18	C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
22	#95×C₄₇	C₂²	1	2	yes	—
23	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
24	C₂³×D₄₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₇	—	2	4	no	—
26	C₂²×C₄×C₄₇	D₄	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊D₄₇	—	2	4	no	—
31	C₂×C₂²⋊D₄₇	—	2	8	no	—
32	C₂²×Dic₄₇	—	2	4	no	—
33	C₂²×Dic₄₇	—	4	4	no	—
34	C₂²×Dic₄₇	C₂³	1	2	yes	—
35	C₂²×Dic₄₇	C₂³	1	4	yes	—
36	C₂×D₁₈₈	—	2	2	no	—
37	C₂×D₁₈₈	—	2	4	no	—
38	C₂²⋊C₄×C₄₇	—	2	4	no	—
39	C₄²×C₄₇	—	2	4	no	—
40	(C₂×D₄₇)⋊C₄	—	2	4	no	—
41	(C₂×D₄₇)⋊C₄	—	4	8	no	—
42	C₄⋊Dic₄₇	—	2	4	no	—
43	C₄⋊Dic₄₇	—	2	4	no	—
44	C₄⋊Dic₄₇	—	2	2	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	C₂×D₄	1	2	yes	—
48	C₂³×C₄₇	#41	1	4	yes	—
49	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²×D₄₇	—	2	2	no	—
51	C₂²×D₄₇	—	2	4	no	—
52	C₂²×D₄₇	—	2	4	no	—
53	C₂²×D₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	4	yes	—
62	C₂²⋊D₄₇	—	4	8	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	2	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	8	no	—
67	C₂×Dic₄₇	—	4	8	no	—
68	D₁₈₈	—	4	4	no	—
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	8	no	—
72	C₂²×C₄₇	#124	1	4	yes	—
73	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×#41	1	2	yes	—
75	C₂²×C₄₇	C₂×#41	1	2	yes	—
76	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
77	C₂×D₄₇	—	4	2	no	—
78	C₂×D₄₇	—	4	2	no	—
79	C₂×D₄₇	—	4	2	no	—
80	C₂×D₄₇	—	4	8	no	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	4	4	no	—
83	C₄×C₄₇	#123	1	4	yes	—
84	Dic₄₇	—	4	2	no	—
85	Dic₄₇	—	4	4	no	—
86	#2504	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	2	4	no	—
88	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
90	C₂×C₄₇	#535	1	1	yes	characteristic
91	C₂×C₄₇	#553	1	4	yes	—
92	D₄₇	—	8	2	no	—
93	C₂×#121	—	47	1	no	—
94	C₂²×C₄⋊C₄	—	47	1	no	—
95	#397	—	47	4	no	—
96	#390	—	47	2	no	—
97	#386	—	47	4	no	—
98	#382	—	47	2	no	—
99	C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic, complemented
100	C₄₇	#2504	1	1	yes	characteristic, complemented, Sylow
101	C₂²×D₄	—	47	2	no	—
102	C₂²×D₄	—	47	1	no	—
103	C₂³×C₄	—	47	2	no	—
104	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
105	#121	—	94	4	no	—
106	C₂×C₄⋊C₄	—	47	2	no	—
107	C₂×C₄⋊C₄	—	47	2	no	—
108	C₂×C₄⋊C₄	—	47	1	no	—
109	C₂×C₄⋊C₄	—	94	4	no	—
110	C₂×C₂²⋊C₄	—	47	2	no	—
111	C₂×C₂²⋊C₄	—	47	8	no	—
112	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
113	C₂×C₄²	C₂×D₄₇	1	2	yes	—
114	#95	—	47	8	no	—
115	#95	C₂×D₄₇	1	2	yes	—
116	C₂⁴	—	47	2	no	—
117	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×D₄	—	94	4	no	—
119	C₂×D₄	—	94	8	no	—
120	C₂×D₄	—	94	2	no	—
121	C₂×D₄	—	94	4	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	47	2	no	—
124	C₂²×C₄	—	47	4	no	—
125	C₂²×C₄	—	94	4	no	—
126	C₂²×C₄	—	188	4	no	—
127	C₂²×C₄	D₁₈₈	1	4	yes	—
128	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
129	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₄⋊C₄	—	94	4	no	—
132	C₄⋊C₄	—	94	4	no	—
133	C₄⋊C₄	—	94	2	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₂²⋊C₄	—	188	8	no	—
137	C₄²	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	94	2	no	—
141	C₂³	—	94	4	no	—
142	C₂³	—	94	4	no	—
143	C₂³	—	94	4	no	—
144	C₂³	C₂×D₁₈₈	1	2	yes	—
145	C₂³	#10625350	1	4	yes	—
146	C₂³	C₂³×D₄₇	1	1	yes	characteristic
147	D₄	—	188	8	no	—
148	D₄	—	188	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	94	8	no	—
158	C₂×C₄	—	188	8	no	—
159	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
160	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
161	C₂×C₄	#10625352	1	4	yes	—
162	C₂²	—	2	8	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	188	8	no	—
166	C₂²	—	188	2	no	—
167	C₂²	—	188	2	no	—
168	C₂²	—	188	2	no	—
169	C₂²	#13094012	1	4	yes	—
170	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
171	C₂²	C₂×#10625352	1	2	yes	—
172	C₂²	C₂×#10625350	1	2	yes	—
173	C₂²	#13094070	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	188	2	no	—
177	C₄	—	188	4	no	—
178	C₄	#13094008	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	—	376	2	no	—
181	C₂	C₂×#13094008	1	1	yes	characteristic
182	C₂	C₂×#13094012	1	1	yes	characteristic
183	C₂	#15689692	1	1	yes	characteristic
184	C₂	#15689819	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093998

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

#15689050

C₂

yes

maximal, complemented

#15689025

C₂

yes

maximal, complemented

#15689000

C₂

yes

maximal, complemented

#15688989

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₂²⋊D₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂²

yes

—

#13093998

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

#2504

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂×#121

—

C₂²×C₄⋊C₄

—

—

—

—

—

C₄×C₂²⋊C₄

D₄₇

yes

characteristic, complemented

100

C₄₇

#2504

yes

characteristic, complemented, Sylow

101

C₂²×D₄

—

102

C₂²×D₄

—

103

C₂³×C₄

—

104

C₂³×C₄

C₂×D₄₇

yes

characteristic

105

#121

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

113

C₂×C₄²

C₂×D₄₇

yes

—

114

#95

—

115

#95

C₂×D₄₇

yes

—

116

C₂⁴

—

117

C₂⁴

C₂²×D₄₇

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

188

—

127

C₂²×C₄

D₁₈₈

yes

—

128

C₂²×C₄

C₂²×D₄₇

yes

—

129

C₂²×C₄

C₂²×D₄₇

yes

characteristic

130

C₂²×C₄

C₂²×D₄₇

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

188

—

137

C₄²

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×D₁₈₈

yes

—

145

C₂³

#10625350

yes

—

146

C₂³

C₂³×D₄₇

yes

characteristic

147

D₄

—

188

—

148

D₄

—

188

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

188

—

159

C₂×C₄

C₂×D₁₈₈

yes

—

160

C₂×C₄

C₂×D₁₈₈

yes

—

161

C₂×C₄

#10625352

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

188

—

166

C₂²

—

188

—

167

C₂²

—

188

—

168

C₂²

—

188

—

169

C₂²

#13094012

yes

—

170

C₂²

C₂²×D₁₈₈

yes

characteristic

171

C₂²

C₂×#10625352

yes

—

172

C₂²

C₂×#10625350

yes

—

173

C₂²

#13094070

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

188

—

177

C₄

—

188

—

178

C₄

#13094008

yes

—

179

C₂

—

180

C₂

—

376

—

181

C₂

C₂×#13094008

yes

characteristic

182

C₂

C₂×#13094012

yes

characteristic

183

C₂

#15689692

yes

characteristic

184

C₂

#15689819

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₄₇	C₂
hypercenter	C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#2504
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄×C₂²⋊C₄×C₄₇

C₂

hypercenter

C₄×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₄₇	C₂	semidirect
3. #15689050	C₂	semidirect
4. #15689025	C₂	semidirect
5. #15689000	C₂	semidirect
6. #15688989	C₂	semidirect
7. C₄×C₂²⋊C₄×C₄₇	C₂	semidirect
99. C₄×C₂²⋊C₄	D₄₇	semidirect
100. C₄₇	#2504	semidirect

H ≅ G/N

type

2. C₂²×C₄⋊Dic₄₇

C₂

semidirect

3. #15689050

C₂

semidirect

4. #15689025

C₂

semidirect

5. #15689000

C₂

semidirect

6. #15688989

C₂

semidirect

7. C₄×C₂²⋊C₄×C₄₇

C₂

semidirect

99. C₄×C₂²⋊C₄

D₄₇

semidirect

100. C₄₇

#2504

semidirect

Subgroups — #20247880

Conjugacy classes of subgroups

Complements