Group #20248297 — subgroups

Subgroups — #20248297

Order 6016.

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

normal subgroups only

This group has 604 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×#13094003	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689054	C₂	1	2	yes	maximal, complemented
4	#15689037	C₂	1	2	yes	maximal
5	#15689011	C₂	1	4	yes	maximal, complemented
6	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688983	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
10	#13094003	—	2	8	no	—
11	C₄×Dic₉₄	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
15	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
18	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	2	yes	—
21	C₄₇⋊#95	C₂²	1	2	yes	—
22	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
24	C₄×D₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₂⁴×C₄₇	C₂³	1	2	yes	—
29	C₂×D₄×C₄₇	—	2	4	no	—
30	C₂×D₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	4	no	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₄₇	C₂³	1	4	yes	—
39	C₂²×Dic₄₇	C₂³	1	2	yes	—
40	C₂²×Dic₄₇	C₂³	1	2	yes	—
41	C₂×Dic₉₄	—	2	2	no	—
42	C₂×Dic₉₄	—	2	2	no	—
43	C₂×Dic₉₄	D₄	1	4	yes	complemented
44	C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂²⋊C₄×C₄₇	—	2	4	no	—
46	C₄²×C₄₇	D₄	1	4	yes	—
47	C₂²⋊Dic₄₇	—	2	8	no	—
48	C₂²⋊Dic₄₇	—	2	8	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	C₄⋊Dic₄₇	—	2	4	no	—
51	Dic₄₇⋊C₄	—	2	4	no	—
52	Dic₄₇⋊C₄	—	2	8	no	—
53	C₄×Dic₄₇	—	2	4	no	—
54	C₂³×C₄₇	—	2	2	no	—
55	C₂³×C₄₇	—	2	2	no	—
56	C₂³×C₄₇	—	2	2	no	—
57	C₂³×C₄₇	—	2	2	no	—
58	C₂³×C₄₇	—	2	2	no	—
59	C₂³×C₄₇	—	2	2	no	—
60	C₂³×C₄₇	#41	1	4	yes	—
61	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
62	D₄×C₄₇	—	2	8	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	4	8	no	—
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₇	#41	1	2	yes	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	2	no	—
76	C₂×Dic₄₇	—	2	2	no	—
77	C₂×Dic₄₇	—	2	4	no	—
78	C₂×Dic₄₇	—	2	4	no	—
79	C₂×Dic₄₇	—	2	4	no	—
80	C₂×Dic₄₇	—	2	4	no	—
81	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
82	Dic₉₄	—	2	8	no	—
83	C₂²×C₄₇	—	2	4	no	—
84	C₂²×C₄₇	—	2	4	no	—
85	C₂²×C₄₇	—	2	4	no	—
86	C₂²×C₄₇	—	4	2	no	—
87	C₂²×C₄₇	—	4	2	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#41	1	2	yes	—
94	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	4	2	no	—
98	C₄×C₄₇	#121	1	4	yes	—
99	Dic₄₇	—	2	4	no	—
100	Dic₄₇	—	4	2	no	—
101	Dic₄₇	—	4	4	no	—
102	#2536	—	47	1	no	maximal, Sylow
103	C₂×C₄₇	—	4	4	no	—
104	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
105	C₂×C₄₇	#525	1	1	yes	characteristic
106	C₂×C₄₇	#536	1	1	yes	characteristic
107	C₂×C₄₇	#548	1	2	yes	—
108	C₂×C₄₇	#552	1	2	yes	—
109	C₂×C₂²⋊Q₈	—	47	2	no	—
110	C₂×C₄×Q₈	—	47	1	no	—
111	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
112	#398	—	47	2	no	—
113	#394	—	47	2	no	—
114	#388	—	47	4	no	—
115	#382	—	47	1	no	—
116	#380	—	47	2	no	—
117	C₄₇	#2536	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	47	1	no	—
119	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
120	C₂³×C₄	C₂×D₄₇	1	2	yes	—
121	C₂²⋊Q₈	—	94	8	no	—
122	C₄×Q₈	—	94	4	no	—
123	C₄×D₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	47	2	no	—
125	C₂×C₄⋊C₄	—	47	1	no	—
126	C₂×C₄⋊C₄	—	47	4	no	—
127	C₂×C₄⋊C₄	—	47	2	no	—
128	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	—	47	4	no	—
130	C₂×C₂²⋊C₄	—	47	4	no	—
131	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
132	C₂×C₄²	—	47	2	no	—
133	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	#95	—	47	4	no	—
135	#95	—	47	2	no	—
136	#95	—	47	2	no	—
137	C₂⁴	C₂²×D₄₇	1	2	yes	—
138	C₂×Q₈	—	47	4	no	—
139	C₂×Q₈	—	94	2	no	—
140	C₂×Q₈	—	94	2	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	47	2	no	—
147	C₂²×C₄	—	47	4	no	—
148	C₂²×C₄	—	47	2	no	—
149	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
150	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	94	8	no	—
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	94	4	no	—
157	C₄⋊C₄	—	94	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	94	8	no	—
160	C₂²⋊C₄	—	94	8	no	—
161	C₄²	—	94	4	no	—
162	C₄²	C₂²⋊D₄₇	1	4	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	#10625350	1	4	yes	—
170	C₂³	C₂³×D₄₇	1	1	yes	characteristic
171	Q₈	—	94	8	no	—
172	D₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	47	4	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	94	2	no	—
182	C₂×C₄	—	94	4	no	—
183	C₂×C₄	—	94	2	no	—
184	C₂×C₄	—	94	4	no	—
185	C₂×C₄	—	94	4	no	—
186	C₂×C₄	—	94	4	no	—
187	C₂×C₄	D₄×D₄₇	1	2	yes	—
188	C₂×C₄	#10625352	1	2	yes	—
189	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
191	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
200	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂²	C₂×#10625352	1	1	yes	characteristic
202	C₂²	C₂×#10625350	1	2	yes	—
203	C₂²	#13094071	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	94	4	no	—
208	C₄	—	188	4	no	—
209	C₄	—	188	2	no	—
210	C₄	#13093995	1	4	yes	—
211	C₂	—	4	4	no	—
212	C₂	C₂×#13093995	1	1	yes	characteristic
213	C₂	#15689634	1	1	yes	characteristic
214	C₂	#15689694	1	1	yes	characteristic
215	C₂	#15689782	1	2	yes	—
216	C₂	#15689813	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094003

C₂

yes

maximal, complemented

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

#15689054

C₂

yes

maximal, complemented

#15689037

C₂

yes

maximal

#15689011

C₂

yes

maximal, complemented

#15688989

C₂

yes

maximal, characteristic, complemented

#15688983

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

#13094003

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

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

—

C₂²×C₄×C₄₇

—

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

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

complemented

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#41

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×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

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

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

100

Dic₄₇

—

101

Dic₄₇

—

102

#2536

—

maximal, Sylow

103

C₂×C₄₇

—

104

C₂×C₄₇

C₂×#121

yes

characteristic

105

C₂×C₄₇

#525

yes

characteristic

106

C₂×C₄₇

#536

yes

characteristic

107

C₂×C₄₇

#548

yes

—

108

C₂×C₄₇

#552

yes

—

109

C₂×C₂²⋊Q₈

—

110

C₂×C₄×Q₈

—

111

C₂×C₄×D₄

D₄₇

yes

characteristic

112

#398

—

113

#394

—

114

#388

—

115

#382

—

116

#380

—

117

C₄₇

#2536

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×D₄

C₂×D₄₇

yes

characteristic

120

C₂³×C₄

C₂×D₄₇

yes

—

121

C₂²⋊Q₈

—

122

C₄×Q₈

—

123

C₄×D₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

132

C₂×C₄²

—

133

C₂×C₄²

C₂×D₄₇

yes

characteristic

134

#95

—

135

#95

—

136

#95

—

137

C₂⁴

C₂²×D₄₇

yes

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²×D₄₇

yes

—

150

C₂²×C₄

C₂²×D₄₇

yes

characteristic

151

C₂²×C₄

C₂²×D₄₇

yes

characteristic

152

C₂²×C₄

C₂²×D₄₇

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₄²

C₂²⋊D₄₇

yes

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#10625350

yes

—

170

C₂³

C₂³×D₄₇

yes

characteristic

171

Q₈

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

D₄×D₄₇

yes

—

188

C₂×C₄

#10625352

yes

—

189

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

190

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

191

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂×D₄×D₄₇

yes

characteristic

200

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

201

C₂²

C₂×#10625352

yes

characteristic

202

C₂²

C₂×#10625350

yes

—

203

C₂²

#13094071

yes

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

188

—

209

C₄

—

188

—

210

C₄

#13093995

yes

—

211

C₂

—

212

C₂

C₂×#13093995

yes

characteristic

213

C₂

#15689634

yes

characteristic

214

C₂

#15689694

yes

characteristic

215

C₂

#15689782

yes

—

216

C₂

#15689813

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	#2536
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×#13094003	C₂	semidirect
2. C₂×C₄×Dic₉₄	C₂	semidirect
3. #15689054	C₂	semidirect
5. #15689011	C₂	semidirect
6. #15688989	C₂	semidirect
7. #15688983	C₂	semidirect
43. C₂×Dic₉₄	D₄	semidirect
117. C₄₇	#2536	semidirect

H ≅ G/N

type

1. C₂×#13094003

C₂

semidirect

2. C₂×C₄×Dic₉₄

C₂

semidirect

3. #15689054

C₂

semidirect

5. #15689011

C₂

semidirect

6. #15688989

C₂

semidirect

7. #15688983

C₂

semidirect

43. C₂×Dic₉₄

D₄

semidirect

117. C₄₇

#2536

semidirect

Subgroups — #20248297

Conjugacy classes of subgroups

Complements