Group #20249109 — subgroups

Subgroups — #20249109

Order 6016.

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

normal subgroups only

This group has 782 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	C₂³×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094005	C₂	1	2	yes	maximal
3	#15689043	C₂	1	4	yes	maximal, complemented
4	#15689035	C₂	1	4	yes	maximal, complemented
5	#15688986	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688984	C₂	1	2	yes	maximal, complemented
7	C₂×#123×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	—	2	4	no	—
9	C₂²×Dic₉₄	—	2	8	no	—
10	C₂²×Dic₉₄	C₂²	1	4	yes	—
11	C₂³×Dic₄₇	C₂²	1	2	yes	—
12	#13094005	—	2	8	no	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	8	yes	—
17	C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	#123×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
25	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₄₇	—	2	2	no	—
27	C₂×D₄×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	D₄	1	4	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×Dic₄₇	—	2	4	no	—
34	C₂²×Dic₄₇	—	2	4	no	—
35	C₂²×Dic₄₇	D₄	1	8	yes	—
36	C₂²×Dic₄₇	C₂³	1	4	yes	—
37	C₂²×Dic₄₇	C₂³	1	4	yes	—
38	C₂×Dic₉₄	—	2	32	no	—
39	C₂×Dic₉₄	—	2	8	no	—
40	C₂×Dic₉₄	—	2	16	no	—
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂²⋊C₄×C₄₇	—	2	4	no	—
44	C₂²⋊Dic₄₇	—	2	4	no	—
45	C₂²⋊Dic₄₇	—	2	8	no	—
46	Dic₄₇⋊C₄	—	2	8	no	—
47	C₂³×C₄₇	—	2	4	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	1	no	—
50	C₂³×C₄₇	—	2	4	no	—
51	C₂³×C₄₇	C₂×D₄	1	4	yes	—
52	C₂³×C₄₇	C₂×D₄	1	2	yes	—
53	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
54	D₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	4	no	—
61	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
62	C₂×Dic₄₇	—	2	16	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	8	no	—
67	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
68	Dic₉₄	—	2	32	no	—
69	C₂²×C₄₇	—	2	4	no	—
70	C₂²×C₄₇	—	2	8	no	—
71	C₂²×C₄₇	—	4	1	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	4	no	—
74	C₂²×C₄₇	#120	1	4	yes	—
75	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
76	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₄₇	2₋¹⁺⁴	1	4	yes	—
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	—	4	1	no	—
80	C₄×C₄₇	—	4	2	no	—
81	Dic₄₇	—	2	8	no	—
82	Dic₄₇	—	4	4	no	—
83	#2614	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	—	2	4	no	—
85	C₂×C₄₇	—	8	1	no	—
86	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
87	C₂×C₄₇	#536	1	2	yes	—
88	C₂×C₄₇	#536	1	4	yes	—
89	C₂³×Q₈	—	47	1	no	—
90	C₂×#123	—	47	2	no	—
91	C₂×#123	D₄₇	1	1	yes	characteristic
92	#395	—	47	4	no	—
93	#393	—	47	4	no	—
94	#381	—	47	1	no	—
95	#381	—	47	2	no	—
96	C₄₇	#2614	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	47	4	no	—
98	C₂²×Q₈	—	94	8	no	—
99	C₂²×Q₈	—	94	4	no	—
100	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
101	C₂³×C₄	—	47	2	no	—
102	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
103	#123	—	2	4	no	—
104	#123	—	94	8	no	—
105	C₂×C₄⋊C₄	—	47	4	no	—
106	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
107	C₂×C₂²⋊C₄	—	47	2	no	—
108	C₂×C₂²⋊C₄	—	47	4	no	—
109	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
111	#95	—	47	8	no	—
112	#95	—	47	4	no	—
113	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
114	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×Q₈	—	94	16	no	—
116	C₂×Q₈	—	94	32	no	—
117	C₂×Q₈	—	94	8	no	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	47	8	no	—
122	C₂²×C₄	—	47	4	no	—
123	C₂²×C₄	—	47	4	no	—
124	C₂²×C₄	—	94	4	no	—
125	C₂²×C₄	—	94	4	no	—
126	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
127	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	94	8	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	94	8	no	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	4	no	—
140	C₂³	D₄×D₄₇	1	4	yes	—
141	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
142	C₂³	C₂³×D₄₇	1	1	yes	characteristic
143	Q₈	—	94	32	no	—
144	D₄	—	4	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	47	8	no	—
152	C₂×C₄	—	94	16	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	94	8	no	—
157	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
158	C₂²	—	2	4	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	4	no	—
163	C₂²	#13093992	1	4	yes	—
164	C₂²	C₂×D₄×D₄₇	1	2	yes	—
165	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂²	#13094071	1	4	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	94	8	no	—
171	C₄	—	188	4	no	—
172	C₂	—	2	4	no	—
173	C₂	—	8	1	no	—
174	C₂	C₂×#13093992	1	1	yes	characteristic
175	C₂	#15689694	1	2	yes	—
176	C₂	#15689695	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₉₄

C₂

yes

maximal, characteristic, complemented

C₂×#13094005

C₂

yes

maximal

#15689043

C₂

yes

maximal, complemented

#15689035

C₂

yes

maximal, complemented

#15688986

C₂

yes

maximal, characteristic, complemented

#15688984

C₂

yes

maximal, complemented

C₂×#123×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

C₂²

yes

—

C₂³×Dic₄₇

C₂²

yes

—

#13094005

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#123×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

D₄×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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2614

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#120

yes

characteristic

C₂×C₄₇

#536

yes

—

C₂×C₄₇

#536

yes

—

C₂³×Q₈

—

C₂×#123

—

C₂×#123

D₄₇

yes

—

—

—

—

C₄₇

#2614

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

100

C₂²×D₄

C₂×D₄₇

yes

characteristic

101

C₂³×C₄

—

102

C₂³×C₄

C₂×D₄₇

yes

characteristic

103

#123

—

104

#123

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

107

C₂×C₂²⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

C₂×D₄₇

yes

characteristic

110

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

111

#95

—

112

#95

—

113

C₂⁴

C₂²×D₄₇

yes

characteristic

114

C₂⁴

C₂²×D₄₇

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²⋊D₄₇

yes

—

127

C₂²×C₄

C₂²×D₄₇

yes

—

128

C₂²×C₄

C₂²×D₄₇

yes

characteristic

129

C₂²×C₄

C₂²×D₄₇

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

D₄×D₄₇

yes

—

141

C₂³

C₂×C₂²⋊D₄₇

yes

—

142

C₂³

C₂³×D₄₇

yes

characteristic

143

Q₈

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

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

C₂×C₂²⋊D₄₇

yes

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#13093992

yes

—

164

C₂²

C₂×D₄×D₄₇

yes

—

165

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

166

C₂²

#13094071

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

188

—

172

C₂

—

173

C₂

—

174

C₂

C₂×#13093992

yes

characteristic

175

C₂

#15689694

yes

—

176

C₂

#15689695

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#123×C₄₇	C₂
hypercenter	C₂×#123	D₄₇
hypocenter	C₄₇	#2614
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#123

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#123×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#123

N	H ≅ G/N	type
1. C₂³×Dic₉₄	C₂	semidirect
3. #15689043	C₂	semidirect
4. #15689035	C₂	semidirect
5. #15688986	C₂	semidirect
6. #15688984	C₂	semidirect
96. C₄₇	#2614	semidirect

H ≅ G/N

type

1. C₂³×Dic₉₄

C₂

semidirect

3. #15689043

C₂

semidirect

4. #15689035

C₂

semidirect

5. #15688986

C₂

semidirect

6. #15688984

C₂

semidirect

96. C₄₇

#2614

semidirect

Subgroups — #20249109

Conjugacy classes of subgroups

Complements