Group #20248341 — subgroups

Subgroups — #20248341

Order 6016.

676 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094003	C₂	1	4	yes	maximal, complemented
3	#15689059	C₂	1	4	yes	maximal, complemented
4	#15689026	C₂	1	2	yes	maximal
5	#15688992	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	C₂²	1	2	yes	—
9	#13094024	—	2	4	no	—
10	#13094003	—	2	16	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
17	C₄₇⋊#95	C₂²	1	2	yes	—
18	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
20	C₄×D₄×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄₇	C₂³	1	2	yes	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	2	2	no	—
27	C₂×D₄×C₄₇	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	—	2	8	no	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₇	C₂³	1	4	yes	—
35	C₂²×Dic₄₇	C₂³	1	4	yes	—
36	C₂×Dic₉₄	—	2	8	no	—
37	C₂×Dic₉₄	—	2	4	no	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₄²×C₄₇	D₄	1	4	yes	—
41	C₂²⋊Dic₄₇	—	2	16	no	—
42	C₄⋊Dic₄₇	—	2	4	no	—
43	C₄⋊Dic₄₇	—	2	4	no	—
44	C₄⋊Dic₄₇	D₄	1	8	yes	complemented
45	Dic₄₇⋊C₄	—	2	16	no	—
46	C₄×Dic₄₇	—	2	4	no	—
47	C₂³×C₄₇	—	2	4	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	—	2	4	no	—
51	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
52	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
53	D₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	4	8	no	—
58	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	#41	1	2	yes	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	8	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
69	Dic₉₄	—	4	8	no	—
70	C₂²×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	—	2	8	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	—	4	4	no	—
75	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
76	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₄₇	—	2	2	no	—
81	C₄×C₄₇	—	4	2	no	—
82	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	complemented
83	C₄×C₄₇	#127	1	4	yes	—
84	Dic₄₇	—	2	8	no	—
85	Dic₄₇	—	4	4	no	—
86	#2542	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	4	4	no	—
88	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
90	C₂×C₄₇	#536	1	1	yes	characteristic
91	C₂×C₄₇	#547	1	2	yes	—
92	C₂×C₄₇	D₄×Q₈	1	2	yes	—
93	C₂×C₄⋊Q₈	—	47	1	no	—
94	C₂×C₂²⋊Q₈	—	47	4	no	—
95	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
96	#399	—	47	4	no	—
97	#390	—	47	2	no	—
98	#384	—	47	2	no	—
99	C₄×C₄⋊C₄	—	47	1	no	—
100	C₄₇	#2542	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	47	2	no	—
102	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₄₇	1	2	yes	—
104	C₄⋊Q₈	—	94	4	no	—
105	C₂²⋊Q₈	—	94	16	no	—
106	C₄×D₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	47	2	no	—
108	C₂×C₄⋊C₄	—	47	8	no	—
109	C₂×C₄⋊C₄	—	47	2	no	—
110	C₂×C₄⋊C₄	—	47	2	no	—
111	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	47	8	no	—
113	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
114	C₂×C₄²	—	47	2	no	—
115	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
116	#95	—	47	2	no	—
117	C₂⁴	C₂²×D₄₇	1	2	yes	—
118	C₂×Q₈	—	94	8	no	—
119	C₂×Q₈	—	94	4	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	Dic₉₄	1	4	yes	complemented
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	47	4	no	—
126	C₂²×C₄	—	47	4	no	—
127	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	47	8	no	—
133	C₄⋊C₄	—	94	16	no	—
134	C₄⋊C₄	—	94	4	no	—
135	C₄⋊C₄	—	94	4	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	94	16	no	—
138	C₄²	—	94	4	no	—
139	C₄²	C₂²⋊D₄₇	1	4	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	4	no	—
144	C₂³	C₂×Dic₉₄	1	4	yes	—
145	C₂³	C₂³×D₄₇	1	1	yes	characteristic
146	Q₈	—	188	8	no	—
147	D₄	—	2	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	47	8	no	—
153	C₂×C₄	—	94	4	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	8	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
158	C₂×C₄	#10625350	1	2	yes	—
159	C₂×C₄	D₄×D₄₇	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	4	4	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	C₂×D₄×D₄₇	1	2	yes	—
169	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
171	C₂²	C₂×#10625350	1	1	yes	characteristic
172	C₂²	#13094071	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	94	8	no	—
176	C₄	—	188	4	no	—
177	C₄	#13094003	1	4	yes	complemented
178	C₄	#13094020	1	4	yes	—
179	C₂	—	4	4	no	—
180	C₂	C₂×#13094003	1	1	yes	characteristic
181	C₂	C₂×#13094020	1	1	yes	characteristic
182	C₂	#15689694	1	1	yes	characteristic
183	C₂	#15689778	1	2	yes	—
184	C₂	D₄×Dic₉₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic, complemented

C₂×#13094003

C₂

yes

maximal, complemented

#15689059

C₂

yes

maximal, complemented

#15689026

C₂

yes

maximal

#15688992

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094024

—

#13094003

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

Q₈

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

D₄×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

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

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

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

complemented

C₄×C₄₇

#127

yes

—

Dic₄₇

—

Dic₄₇

—

#2542

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

D₄×Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

D₄₇

yes

—

—

—

C₄×C₄⋊C₄

—

100

C₄₇

#2542

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂²×D₄

C₂×D₄₇

yes

characteristic

103

C₂³×C₄

C₂×D₄₇

yes

—

104

C₄⋊Q₈

—

105

C₂²⋊Q₈

—

106

C₄×D₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₄₇

yes

characteristic

116

#95

—

117

C₂⁴

C₂²×D₄₇

yes

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

Dic₉₄

yes

complemented

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²×D₄₇

yes

—

128

C₂²×C₄

C₂²×D₄₇

yes

characteristic

129

C₂²×C₄

C₂²×D₄₇

yes

characteristic

130

C₂²×C₄

C₂²×D₄₇

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

C₂²⋊D₄₇

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×Dic₉₄

yes

—

145

C₂³

C₂³×D₄₇

yes

characteristic

146

Q₈

—

188

—

147

D₄

—

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×Dic₉₄

yes

—

158

C₂×C₄

#10625350

yes

—

159

C₂×C₄

D₄×D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

161

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

162

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×D₄×D₄₇

yes

—

169

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

170

C₂²

C₂²×Dic₉₄

yes

characteristic

171

C₂²

C₂×#10625350

yes

characteristic

172

C₂²

#13094071

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

188

—

177

C₄

#13094003

yes

complemented

178

C₄

#13094020

yes

—

179

C₂

—

180

C₂

C₂×#13094003

yes

characteristic

181

C₂

C₂×#13094020

yes

characteristic

182

C₂

#15689694

yes

characteristic

183

C₂

#15689778

yes

—

184

C₂

D₄×Dic₉₄

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×D₄×C₄₇	C₂
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#2542
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄×D₄×C₄₇

C₂

hypercenter

C₂×C₄×D₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×#13094024	C₂	semidirect
2. C₂×#13094003	C₂	semidirect
3. #15689059	C₂	semidirect
5. #15688992	C₂	semidirect
6. C₄×C₄⋊Dic₄₇	C₂	semidirect
27. C₂×D₄×C₄₇	Q₈	semidirect
44. C₄⋊Dic₄₇	D₄	semidirect
82. C₄×C₄₇	C₂²⋊Q₈	semidirect
100. C₄₇	#2542	semidirect
122. C₂×D₄	Dic₉₄	semidirect
177. C₄	#13094003	semidirect

H ≅ G/N

type

1. C₂×#13094024

C₂

semidirect

2. C₂×#13094003

C₂

semidirect

3. #15689059

C₂

semidirect

5. #15688992

C₂

semidirect

6. C₄×C₄⋊Dic₄₇

C₂

semidirect

27. C₂×D₄×C₄₇

Q₈

semidirect

44. C₄⋊Dic₄₇

D₄

semidirect

82. C₄×C₄₇

C₂²⋊Q₈

semidirect

100. C₄₇

#2542

semidirect

122. C₂×D₄

Dic₉₄

semidirect

177. C₄

#13094003

semidirect

Subgroups — #20248341

Conjugacy classes of subgroups

Complements