Group #26115364 — subgroups

Subgroups — #26115364

Order 12032.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#20243505	C₂	1	1	yes	maximal, characteristic
2	#20243241	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688869	C₂	1	4	yes	maximal
4	C₄₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093935	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	—	2	2	no	—
10	#15688869	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#384	—	2	1	no	—
14	C₄₇×#384	—	2	1	no	—
15	C₄₇×#384	C₄	1	4	yes	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
22	#13093935	—	4	2	no	—
23	C₂²×Q₈×C₄₇	—	2	2	no	—
24	C₄⋊Q₈×C₄₇	—	2	4	no	—
25	C₄⋊Q₈×C₄₇	—	2	4	no	—
26	C₄⋊Q₈×C₄₇	—	4	8	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₄₇	—	4	2	no	—
37	C₂×Q₈×C₄₇	—	4	2	no	—
38	C₂×Q₈×C₄₇	—	4	8	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₄₇	—	2	8	no	—
46	C₄⋊C₄×C₄₇	—	4	8	no	—
47	C₄⋊C₄×C₄₇	—	4	8	no	—
48	C₄⋊C₄×C₄₇	—	4	4	no	—
49	C₄²×C₄₇	—	2	4	no	—
50	C₄²×C₄₇	—	2	4	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	SD₁₆	1	2	yes	—
53	C₄²×C₄₇	Q₁₆	1	2	yes	—
54	C₄²×C₄₇	C₂×D₄	1	2	yes	—
55	C₄²×C₄₇	#41	1	2	yes	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₄₇	—	8	4	no	—
63	Q₈×C₄₇	—	8	4	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	—	2	2	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	—	4	8	no	—
71	C₂×C₄×C₄₇	#101	1	4	yes	—
72	C₂×C₄×C₄₇	#103	1	4	yes	—
73	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	#121	1	2	yes	—
75	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₄₇	#137	1	2	yes	—
81	C₄₇⋊C₈	—	4	4	no	—
82	C₄₇⋊C₈	—	8	2	no	—
83	#9703	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	#390	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#101	1	2	yes	—
86	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
87	C₂²×C₄₇	#418	1	2	yes	—
88	C₂²×C₄₇	#419	1	1	yes	characteristic
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	8	4	no	—
93	C₄×C₄₇	#461	1	2	yes	—
94	C₄×C₄₇	#462	1	2	yes	—
95	C₄×C₄₇	#490	1	2	yes	—
96	C₄×C₄₇	#496	1	2	yes	—
97	#2868	D₄₇	1	1	yes	characteristic
98	#1831	—	47	1	no	—
99	#1767	—	47	1	no	—
100	C₂×#332	—	47	4	no	—
101	C₂×C₄₇	#1549	1	2	yes	—
102	C₂×C₄₇	#1556	1	2	yes	—
103	C₂×C₄₇	#1961	1	1	yes	characteristic
104	C₂×C₄₇	#1966	1	1	yes	characteristic
105	C₂×C₄₇	#1981	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	—	2	2	no	—
107	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
108	C₂×C₄⋊C₈	—	47	2	no	—
109	C₂×C₄⋊C₈	—	47	2	no	—
110	C₂×#97	—	94	1	no	—
111	#384	—	2	1	no	—
112	#384	—	2	1	no	—
113	#384	Dic₄₇	1	4	yes	—
114	C₄³	C₂×D₄₇	1	1	yes	characteristic
115	#336	—	94	2	no	—
116	#336	—	94	1	no	—
117	#332	—	94	8	no	—
118	C₄₇	#9703	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	—	94	2	no	—
121	C₂²×C₈	—	94	2	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₈	—	47	4	no	—
135	C₄⋊C₈	—	94	4	no	—
136	C₄⋊C₈	—	94	2	no	—
137	#97	—	188	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂×C₈	—	94	4	no	—
148	C₂×C₈	—	188	2	no	—
149	C₂×C₈	—	188	4	no	—
150	C₂×C₈	—	188	2	no	—
151	C₂×C₈	—	188	2	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#10625336	1	2	yes	—
160	C₄²	#10625337	1	2	yes	—
161	C₄²	D₄×D₄₇	1	2	yes	—
162	C₄²	#10625352	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#13093943	1	4	yes	—
174	C₂×C₄	#13093948	1	4	yes	—
175	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
176	C₂×C₄	#13093995	1	2	yes	—
177	C₂×C₄	#13094011	1	1	yes	characteristic
178	C₂×C₄	#13094020	1	1	yes	characteristic
179	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
180	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
181	C₂×C₄	#13094050	1	4	yes	—
182	C₂×C₄	#13094055	1	2	yes	—
183	C₈	—	376	2	no	—
184	C₈	—	188	4	no	—
185	C₂²	#15689026	1	1	yes	characteristic
186	C₂²	C₂×#13093943	1	2	yes	—
187	C₂²	C₂×#13093948	1	1	yes	characteristic
188	C₂²	#15689128	1	2	yes	—
189	C₂²	#15689132	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	8	4	no	—
194	C₄	#15689325	1	2	yes	—
195	C₄	#15689330	1	2	yes	—
196	C₄	#15689487	1	2	yes	—
197	C₄	#15689510	1	2	yes	—
198	C₂	#20242073	1	2	yes	—
199	C₂	#20242105	1	2	yes	—
200	C₂	#20244213	1	1	yes	characteristic
201	C₂	#20244233	1	1	yes	characteristic
202	C₂	#20244312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal, characteristic

#20243241

C₂

yes

maximal, characteristic

C₂×#15688869

C₂

yes

maximal

C₄₇×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#384

—

C₄₇×#384

—

C₄₇×#384

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

#13093935

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#103

yes

characteristic

C₂²×C₄₇

#418

yes

—

C₂²×C₄₇

#419

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#461

yes

—

C₄×C₄₇

#462

yes

—

C₄×C₄₇

#490

yes

—

C₄×C₄₇

#496

yes

—

#2868

D₄₇

yes

characteristic

#1831

—

#1767

—

100

C₂×#332

—

101

C₂×C₄₇

#1549

yes

—

102

C₂×C₄₇

#1556

yes

—

103

C₂×C₄₇

#1961

yes

characteristic

104

C₂×C₄₇

#1966

yes

characteristic

105

C₂×C₄₇

#1981

yes

characteristic

106

C₂×C₄⋊Q₈

—

107

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

110

C₂×#97

—

111

#384

—

112

#384

—

113

#384

Dic₄₇

yes

—

114

C₄³

C₂×D₄₇

yes

characteristic

115

#336

—

116

#336

—

117

#332

—

118

C₄₇

#9703

yes

characteristic, complemented, Sylow

119

C₂²×Q₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

129

C₂×C₄²

C₂×Dic₄₇

yes

—

130

C₂×C₄²

C₂²⋊D₄₇

yes

—

131

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

132

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

133

C₂×C₄²

C₂²×D₄₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#97

—

188

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

143

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

144

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

188

—

149

C₂×C₈

—

188

—

150

C₂×C₈

—

188

—

151

C₂×C₈

—

188

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#10625336

yes

—

160

C₄²

#10625337

yes

—

161

C₄²

D₄×D₄₇

yes

—

162

C₄²

#10625352

yes

—

163

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#13093943

yes

—

174

C₂×C₄

#13093948

yes

—

175

C₂×C₄

D₄×Dic₄₇

yes

—

176

C₂×C₄

#13093995

yes

—

177

C₂×C₄

#13094011

yes

characteristic

178

C₂×C₄

#13094020

yes

characteristic

179

C₂×C₄

C₂×#10625336

yes

characteristic

180

C₂×C₄

C₂×#10625337

yes

characteristic

181

C₂×C₄

#13094050

yes

—

182

C₂×C₄

#13094055

yes

—

183

C₈

—

376

—

184

C₈

—

188

—

185

C₂²

#15689026

yes

characteristic

186

C₂²

C₂×#13093943

yes

—

187

C₂²

C₂×#13093948

yes

characteristic

188

C₂²

#15689128

yes

—

189

C₂²

#15689132

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#15689325

yes

—

195

C₄

#15689330

yes

—

196

C₄

#15689487

yes

—

197

C₄

#15689510

yes

—

198

C₂

#20242073

yes

—

199

C₂

#20242105

yes

—

200

C₂

#20244213

yes

characteristic

201

C₂

#20244233

yes

characteristic

202

C₂

#20244312

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2868	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2868	D₄₇
hypocenter	C₄₇	#9703
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2868

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
118. C₄₇	#9703	semidirect

H ≅ G/N

type

118. C₄₇

#9703

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243505	2
#20243241	2
C₂×#15688869	2
C₄₇×#2868	2
#9703	47

subgroup

index

C₄₇×#2868

Subgroups — #26115364

Conjugacy classes of subgroups

Complements

Maximal subgroups