Group #20253189 — subgroups

Subgroups — #20253189

Order 6016.

376 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#15689427	C₂	1	4	yes	maximal, complemented
3	#15689415	C₂	1	2	yes	maximal, complemented
4	#15689405	C₂	1	2	yes	maximal, complemented
5	#15689166	C₂	1	2	yes	maximal
6	#15689147	C₂	1	1	yes	maximal, characteristic
7	#15689131	C₂	1	2	yes	maximal
8	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	2	yes	—
11	C₄×Dic₉₄	—	2	1	no	—
12	C₄×Dic₉₄	—	2	1	no	—
13	C₄×Dic₉₄	C₂²	1	2	yes	—
14	C₄×Dic₉₄	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₄₇	—	2	1	no	—
18	#13093957	C₂²	1	4	yes	—
19	#13093955	C₂²	1	4	yes	—
20	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
22	#13093947	C₂²	1	4	yes	—
23	#13093947	C₂²	1	4	yes	—
24	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	#125×C₄₇	C₂²	1	2	yes	—
27	C₂²⋊Q₈×C₄₇	—	2	1	no	—
28	C₂²⋊Q₈×C₄₇	—	2	1	no	—
29	C₄×Q₈×C₄₇	—	2	1	no	—
30	C₄×Q₈×C₄₇	—	2	1	no	—
31	#117×C₄₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	—	2	1	no	—
34	C₂×Q₈×C₄₇	—	2	1	no	—
35	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	1	no	—
39	C₂²×Dic₄₇	—	2	1	no	—
40	C₂×Dic₉₄	—	2	1	no	—
41	C₂×Dic₉₄	—	2	2	no	—
42	C₂×Dic₉₄	Q₈	1	4	yes	complemented
43	C₂×Dic₉₄	C₂³	1	2	yes	—
44	C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
49	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
50	C₂²⋊C₄×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊Dic₄₇	—	2	1	no	—
57	C₄⋊Dic₄₇	C₂³	1	2	yes	—
58	Dic₄₇⋊C₄	—	2	2	no	—
59	Dic₄₇⋊C₄	—	2	2	no	—
60	C₄×Dic₄₇	—	2	2	no	—
61	C₄×Dic₄₇	—	2	2	no	—
62	#10625329	—	2	4	no	—
63	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
64	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₄₇	—	4	2	no	—
66	Q₈×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	4	1	no	—
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₄₇	#41	1	2	yes	—
78	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₄₇	—	2	4	no	—
80	C₂×Dic₄₇	—	2	2	no	—
81	C₂×Dic₄₇	—	2	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	Dic₉₄	—	2	4	no	—
84	Dic₉₄	—	2	2	no	—
85	Dic₉₄	C₂×Q₈	1	4	yes	—
86	C₄₇⋊C₈	—	2	4	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	—	2	1	no	—
89	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
90	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	1	no	—
93	C₄×C₄₇	—	2	1	no	—
94	C₄×C₄₇	—	4	2	no	—
95	C₄×C₄₇	—	4	2	no	—
96	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
97	C₄×C₄₇	#137	1	2	yes	—
98	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	Dic₄₇	—	2	4	no	—
101	Dic₄₇	—	4	2	no	—
102	#3078	—	47	1	no	maximal, Sylow
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
106	C₂×C₄₇	#575	1	1	yes	characteristic
107	#533	D₄₇	1	1	yes	characteristic
108	C₂×C₄×Q₈	—	47	1	no	—
109	#479	—	47	4	no	—
110	#477	—	47	2	no	—
111	#475	—	47	2	no	—
112	#428	—	47	2	no	—
113	#423	—	47	1	no	—
114	#419	—	47	2	no	—
115	C₄₇	#3078	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	47	1	no	—
117	C₂×M₁₆	—	47	2	no	—
118	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
119	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
120	#125	C₂×D₄₇	1	2	yes	—
121	C₂²⋊Q₈	—	2	1	no	—
122	C₂²⋊Q₈	—	2	1	no	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	—	47	2	no	—
126	C₄×Q₈	—	47	2	no	—
127	C₄×Q₈	—	94	1	no	—
128	C₄×Q₈	—	94	1	no	—
129	#117	C₂×D₄₇	1	2	yes	—
130	C₂×C₄⋊C₄	—	47	1	no	—
131	C₂×C₄⋊C₄	—	94	1	no	—
132	C₂×C₄²	—	94	1	no	—
133	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	#107	—	47	4	no	—
135	#106	—	47	4	no	—
136	C₄⋊C₈	—	47	2	no	—
137	C₄⋊C₈	—	47	2	no	—
138	#103	—	47	4	no	—
139	#103	—	47	4	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	47	4	no	—
143	C₂×Q₈	—	47	2	no	—
144	C₂×Q₈	—	94	2	no	—
145	C₂×Q₈	—	94	1	no	—
146	C₂²×C₄	—	94	1	no	—
147	C₂²×C₄	—	94	1	no	—
148	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	M₁₆	—	94	4	no	—
152	C₂×C₈	—	47	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	47	2	no	—
158	C₄⋊C₄	—	94	2	no	—
159	C₄⋊C₄	—	94	2	no	—
160	C₄⋊C₄	—	94	1	no	—
161	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
162	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	94	2	no	—
166	C₄²	—	94	2	no	—
167	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
168	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
172	Q₈	—	4	2	no	—
173	Q₈	—	4	2	no	—
174	Q₈	—	47	4	no	—
175	Q₈	—	94	4	no	—
176	Q₈	—	94	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	94	4	no	—
182	C₂×C₄	—	94	2	no	—
183	C₂×C₄	—	94	2	no	—
184	C₂×C₄	—	188	2	no	—
185	C₂×C₄	Q₈×D₄₇	1	2	yes	—
186	C₂×C₄	#10625354	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
193	C₈	—	94	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#13094000	1	2	yes	—
197	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	94	4	no	—
204	C₄	—	188	2	no	—
205	C₄	#13094000	1	2	yes	—
206	C₄	#13094054	1	2	yes	—
207	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
208	C₄	C₂×#10625354	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	C₂×#13094000	1	1	yes	characteristic
211	C₂	C₂×#13094054	1	1	yes	characteristic
212	C₂	#15689921	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic

#15689427

C₂

yes

maximal, complemented

#15689415

C₂

yes

maximal, complemented

#15689405

C₂

yes

maximal, complemented

#15689166

C₂

yes

maximal

#15689147

C₂

yes

maximal, characteristic

#15689131

C₂

yes

maximal

C₄₇×#533

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093947

C₂²

yes

—

#13093947

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#125×C₄₇

C₂²

yes

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

Q₈

yes

complemented

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

C₂×Q₈

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

#137

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

100

Dic₄₇

—

101

Dic₄₇

—

102

#3078

—

maximal, Sylow

103

C₂×C₄₇

—

104

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

105

C₂×C₄₇

C₂×#137

yes

characteristic

106

C₂×C₄₇

#575

yes

characteristic

107

#533

D₄₇

yes

characteristic

108

C₂×C₄×Q₈

—

109

#479

—

110

#477

—

111

#475

—

112

#428

—

113

#423

—

114

#419

—

115

C₄₇

#3078

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

119

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

120

#125

C₂×D₄₇

yes

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

#117

C₂×D₄₇

yes

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

—

133

C₂×C₄²

C₂×D₄₇

yes

characteristic

134

#107

—

135

#106

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#103

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

149

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

150

C₂²×C₄

C₂²×D₄₇

yes

characteristic

151

M₁₆

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

C₂²×D₄₇

yes

—

162

C₄⋊C₄

C₂²×D₄₇

yes

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₂²⋊D₄₇

yes

characteristic

168

C₄²

C₂²⋊D₄₇

yes

characteristic

169

C₄²

C₂²×D₄₇

yes

characteristic

170

C₄²

C₂²×D₄₇

yes

characteristic

171

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

188

—

185

C₂×C₄

Q₈×D₄₇

yes

—

186

C₂×C₄

#10625354

yes

—

187

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

190

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

191

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

192

C₂×C₄

C₂³×D₄₇

yes

characteristic

193

C₈

—

194

C₂²

—

195

C₂²

—

196

C₂²

#13094000

yes

—

197

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

188

—

205

C₄

#13094000

yes

—

206

C₄

#13094054

yes

—

207

C₄

C₂×Q₈×D₄₇

yes

characteristic

208

C₄

C₂×#10625354

yes

characteristic

209

C₂

—

210

C₂

C₂×#13094000

yes

characteristic

211

C₂

C₂×#13094054

yes

characteristic

212

C₂

#15689921

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3078
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #15689427	C₂	semidirect
3. #15689415	C₂	semidirect
4. #15689405	C₂	semidirect
42. C₂×Dic₉₄	Q₈	semidirect
115. C₄₇	#3078	semidirect

H ≅ G/N

type

2. #15689427

C₂

semidirect

3. #15689415

C₂

semidirect

4. #15689405

C₂

semidirect

42. C₂×Dic₉₄

Q₈

semidirect

115. C₄₇

#3078

semidirect

Subgroups — #20253189

Conjugacy classes of subgroups

Complements