Group #20246012 — subgroups

Subgroups — #20246012

Order 6016.

600 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 269 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×#13093983	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093983	C₂	1	1	yes	maximal, characteristic
3	#15689019	C₂	1	4	yes	maximal, complemented
4	#15688996	C₂	1	4	yes	maximal
5	#15688992	C₂	1	2	yes	maximal, complemented
6	#15688988	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#13093983	—	2	4	no	—
9	#13093983	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	8	yes	—
19	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
20	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	4	no	—
26	C₂²×C₄×C₄₇	—	4	4	no	—
27	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₇	D₄	1	4	yes	—
29	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	C₂³	1	2	yes	—
34	C₂²×Dic₄₇	C₂³	1	2	yes	—
35	C₂²×Dic₄₇	C₂³	1	4	yes	—
36	C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₂²⋊Dic₄₇	—	2	4	no	—
39	C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₄₇	—	2	2	no	—
41	C₄⋊Dic₄₇	—	2	2	no	—
42	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	complemented
43	Dic₄₇⋊C₄	—	4	8	no	—
44	C₄×Dic₄₇	—	2	4	no	—
45	C₄×Dic₄₇	C₂×C₄	1	8	yes	complemented
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂³×C₄₇	C₂×D₄	1	2	yes	—
50	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	4	8	no	—
55	C₂×C₄×C₄₇	C₄⋊C₄	1	16	yes	complemented
56	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	2	no	—
65	C₂×Dic₄₇	—	2	2	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄₇	—	2	4	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	2	4	no	—
73	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
79	C₄×C₄₇	—	4	4	no	—
80	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	Dic₄₇	—	2	8	no	—
83	Dic₄₇	—	4	2	no	—
84	Dic₄₇	—	4	2	no	—
85	#2317	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	—	2	4	no	—
87	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄₇	#520	1	2	yes	—
89	C₂×C₄₇	#535	1	1	yes	characteristic
90	C₂×C₄₇	#536	1	1	yes	characteristic
91	C₂×C₄₇	#544	1	2	yes	—
92	C₂×#117	—	47	1	no	—
93	C₂×#117	—	47	1	no	—
94	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
95	#389	—	47	4	no	—
96	#385	—	47	4	no	—
97	#384	—	47	2	no	—
98	#382	—	47	2	no	—
99	C₄₇	#2317	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₄	C₂×D₄₇	1	2	yes	—
101	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
102	#117	—	47	8	no	—
103	#117	—	94	4	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	—	47	1	no	—
107	C₂×C₄⋊C₄	—	47	1	no	—
108	C₂×C₄⋊C₄	—	47	8	no	—
109	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
111	C₂×C₂²⋊C₄	—	47	2	no	—
112	C₂×C₂²⋊C₄	—	47	2	no	—
113	C₂×C₄²	—	47	2	no	—
114	C₂×C₄²	—	47	2	no	—
115	#95	—	47	8	no	—
116	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	47	4	no	—
120	C₂²×C₄	—	47	2	no	—
121	C₂²×C₄	—	47	2	no	—
122	C₂²×C₄	Dic₉₄	1	4	yes	—
123	C₂²×C₄	C₄×D₄₇	1	4	yes	—
124	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
125	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	47	8	no	—
131	C₄⋊C₄	—	94	2	no	—
132	C₄⋊C₄	—	94	2	no	—
133	C₄⋊C₄	—	188	8	no	—
134	C₂²⋊C₄	—	47	8	no	—
135	C₂²⋊C₄	—	94	4	no	—
136	C₄²	—	47	8	no	—
137	C₄²	—	94	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Dic₉₄	1	2	yes	—
141	C₂³	C₂×C₄×D₄₇	1	2	yes	—
142	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂³	C₂³×D₄₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	—	47	8	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	94	4	no	—
150	C₂×C₄	—	94	2	no	—
151	C₂×C₄	—	94	2	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	Dic₄₇⋊C₄	1	16	yes	complemented
155	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
156	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
157	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×Dic₄₇⋊C₄	1	4	yes	—
166	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
167	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
169	C₂²	#13094069	1	2	yes	—
170	C₂²	#13094072	1	2	yes	—
171	C₄	—	4	4	no	—
172	C₄	—	94	8	no	—
173	C₄	—	188	2	no	—
174	C₄	—	188	2	no	—
175	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
176	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
179	C₂	#15689604	1	2	yes	—
180	C₂	#15689690	1	1	yes	characteristic
181	C₂	#15689697	1	1	yes	characteristic
182	C₂	#15689754	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal, characteristic

C₂×#13093983

C₂

yes

maximal, characteristic

#15689019

C₂

yes

maximal, complemented

#15688996

C₂

yes

maximal

#15688992

C₂

yes

maximal, complemented

#15688988

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

#13093983

—

#13093983

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2317

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

#520

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#117

—

C₂×#117

—

C₂²×C₄⋊C₄

D₄₇

yes

—

—

—

—

C₄₇

#2317

yes

characteristic, complemented, Sylow

100

C₂³×C₄

C₂×D₄₇

yes

—

101

C₂³×C₄

C₂×D₄₇

yes

characteristic

102

#117

—

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

#95

—

116

C₂⁴

C₂²×D₄₇

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

Dic₉₄

yes

—

123

C₂²×C₄

C₄×D₄₇

yes

—

124

C₂²×C₄

C₂²⋊D₄₇

yes

—

125

C₂²×C₄

C₂²×D₄₇

yes

—

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

C₂²×C₄

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

188

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂×Dic₉₄

yes

—

141

C₂³

C₂×C₄×D₄₇

yes

—

142

C₂³

C₂×C₂²⋊D₄₇

yes

—

143

C₂³

C₂³×D₄₇

yes

characteristic

144

C₂×C₄

—

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

Dic₄₇⋊C₄

yes

complemented

155

C₂×C₄

C₂×Dic₉₄

yes

—

156

C₂×C₄

C₂×Dic₉₄

yes

—

157

C₂×C₄

C₂×C₄×D₄₇

yes

—

158

C₂×C₄

C₂×C₄×D₄₇

yes

—

159

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×Dic₄₇⋊C₄

yes

—

166

C₂²

C₂²×C₄×D₄₇

yes

characteristic

167

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

168

C₂²

C₂²×Dic₉₄

yes

characteristic

169

C₂²

#13094069

yes

—

170

C₂²

#13094072

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

188

—

174

C₄

—

188

—

175

C₄

C₂×Dic₄₇⋊C₄

yes

—

176

C₄

C₂×Dic₄₇⋊C₄

yes

—

177

C₂

—

178

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

179

C₂

#15689604

yes

—

180

C₂

#15689690

yes

characteristic

181

C₂

#15689697

yes

characteristic

182

C₂

#15689754

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2317
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #15689019	C₂	semidirect
5. #15688992	C₂	semidirect
6. #15688988	C₂	semidirect
9. #13093983	C₄	semidirect
42. C₄⋊Dic₄₇	C₂×C₄	semidirect
45. C₄×Dic₄₇	C₂×C₄	semidirect
55. C₂×C₄×C₄₇	C₄⋊C₄	semidirect
99. C₄₇	#2317	semidirect
154. C₂×C₄	Dic₄₇⋊C₄	semidirect

H ≅ G/N

type

3. #15689019

C₂

semidirect

5. #15688992

C₂

semidirect

6. #15688988

C₂

semidirect

9. #13093983

C₄

semidirect

42. C₄⋊Dic₄₇

C₂×C₄

semidirect

45. C₄×Dic₄₇

C₂×C₄

semidirect

55. C₂×C₄×C₄₇

C₄⋊C₄

semidirect

99. C₄₇

#2317

semidirect

154. C₂×C₄

Dic₄₇⋊C₄

semidirect

Subgroups — #20246012

Conjugacy classes of subgroups

Complements