Group #26120437 — subgroups

Subgroups — #26120437

Order 12032.

342 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20243562	C₂	1	1	yes	maximal, characteristic
2	#20242118	C₂	1	2	yes	maximal
3	#20242003	C₂	1	2	yes	maximal
4	#20241299	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#3255	C₂	1	1	yes	maximal, characteristic
6	#15689180	C₂²	1	2	yes	—
7	C₂×#13093959	—	2	2	no	—
8	#15689147	C₂²	1	1	yes	characteristic
9	C₂×#13093935	C₂²	1	1	yes	characteristic
10	#15688872	—	2	2	no	—
11	#15688869	—	2	2	no	—
12	#15688865	—	2	4	no	—
13	C₄₇×#533	C₂²	1	2	yes	—
14	C₄₇×#475	—	2	2	no	—
15	C₄₇×#470	—	2	2	no	—
16	C₄₇×#462	—	2	2	no	—
17	C₄₇×#423	C₂²	1	1	yes	characteristic
18	C₂×#10625329	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	1	no	—
20	#13093959	—	4	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	#13093935	—	2	1	no	—
25	#13093935	—	2	2	no	—
26	#13093935	D₄	1	2	yes	—
27	C₂×Q₁₆×C₄₇	—	2	2	no	—
28	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
31	#125×C₄₇	—	2	2	no	—
32	C₂²⋊Q₈×C₄₇	—	2	2	no	—
33	C₂²⋊Q₈×C₄₇	—	2	2	no	—
34	C₄×Q₈×C₄₇	—	2	2	no	—
35	C₄×Q₈×C₄₇	—	2	2	no	—
36	#117×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	#106×C₄₇	—	2	2	no	—
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	#103×C₄₇	—	2	4	no	—
41	C₂×Q₈×C₄₇	—	2	2	no	—
42	C₂×Q₈×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	2	yes	—
45	Q₁₆×C₄₇	—	8	2	no	—
46	M₁₆×C₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₄⋊C₄×C₄₇	—	4	2	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₂²⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	—	4	2	no	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	1	yes	characteristic
59	#10625329	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	2	2	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	4	1	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂³×C₄₇	#124	1	1	yes	characteristic
66	Q₈×C₄₇	—	4	2	no	—
67	Q₈×C₄₇	—	4	2	no	—
68	Q₈×C₄₇	—	8	2	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	—	4	2	no	—
74	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#121	1	2	yes	—
77	C₂×C₄×C₄₇	#123	1	2	yes	—
78	C₈×C₄₇	—	4	2	no	—
79	C₄₇⋊C₈	—	2	4	no	—
80	C₄₇⋊C₈	—	8	2	no	—
81	C₄₇⋊C₈	—	8	1	no	—
82	#10528	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	#394	1	1	yes	characteristic
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	8	2	no	—
90	C₄×C₄₇	—	8	2	no	—
91	C₄×C₄₇	#453	1	1	yes	characteristic
92	C₄×C₄₇	#456	1	1	yes	characteristic
93	#3255	D₄₇	1	1	yes	characteristic
94	#1849	—	47	1	no	—
95	#1559	—	47	2	no	—
96	#1531	—	47	2	no	—
97	#1307	—	47	1	no	—
98	C₂×C₄₇	—	4	1	no	—
99	C₂×C₄₇	#2039	1	1	yes	characteristic
100	C₂×C₄₇	#2049	1	2	yes	—
101	#533	C₂×D₄₇	1	2	yes	—
102	#475	—	2	2	no	—
103	#470	—	2	2	no	—
104	#462	—	2	2	no	—
105	#432	—	47	2	no	—
106	C₂×#108	—	94	2	no	—
107	#423	—	47	1	no	—
108	#423	C₂×D₄₇	1	1	yes	characteristic
109	C₂×#97	—	47	1	no	—
110	#333	—	94	2	no	—
111	#332	—	94	2	no	—
112	#330	—	94	4	no	—
113	C₄₇	#10528	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×M₁₆	—	94	1	no	—
116	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
117	C₂²×C₈	—	94	1	no	—
118	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
119	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
120	#125	—	2	2	no	—
121	C₂²⋊Q₈	—	2	2	no	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
127	#108	—	188	4	no	—
128	#106	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	94	2	no	—
131	C₄⋊C₈	—	94	2	no	—
132	#103	—	2	4	no	—
133	C₂²⋊C₈	—	94	2	no	—
134	#97	—	47	2	no	—
135	#97	—	94	2	no	—
136	#97	—	94	1	no	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂²×C₄	#10625352	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
141	Q₁₆	—	8	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	188	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	94	2	no	—
146	C₂×C₈	—	94	2	no	—
147	C₂×C₈	—	94	2	no	—
148	C₂×C₈	—	188	1	no	—
149	C₂×C₈	—	188	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	D₄×D₄₇	1	1	yes	characteristic
159	C₄²	#10625352	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂³	#13094011	1	1	yes	characteristic
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#13093993	1	1	yes	characteristic
171	C₂×C₄	#13093995	1	1	yes	characteristic
172	C₂×C₄	#13093996	1	2	yes	—
173	C₂×C₄	#13094005	1	2	yes	—
174	C₈	—	4	2	no	—
175	C₈	—	94	4	no	—
176	C₈	—	376	2	no	—
177	C₈	—	376	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	#15689037	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	2	no	—
185	C₄	—	8	2	no	—
186	C₄	#15689291	1	1	yes	characteristic
187	C₄	#15689306	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	#20244609	1	1	yes	characteristic
190	C₂	#20244665	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243562

C₂

yes

maximal, characteristic

#20242118

C₂

yes

maximal

#20242003

C₂

yes

maximal

#20241299

C₂

yes

maximal, characteristic

C₄₇×#3255

C₂

yes

maximal, characteristic

#15689180

C₂²

yes

—

C₂×#13093959

—

#15689147

C₂²

yes

characteristic

C₂×#13093935

C₂²

yes

—

—

—

C₄₇×#533

C₂²

yes

—

C₄₇×#475

—

C₄₇×#470

—

C₄₇×#462

—

C₄₇×#423

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

#13093959

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

D₄

yes

—

C₂×Q₁₆×C₄₇

—

C₂×M₁₆×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄⋊C₈×C₄₇

—

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

Q₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10528

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#394

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#2039

yes

characteristic

100

C₂×C₄₇

#2049

yes

—

101

#533

C₂×D₄₇

yes

—

102

#475

—

103

#470

—

104

#462

—

105

#432

—

106

C₂×#108

—

107

#423

—

108

#423

C₂×D₄₇

yes

characteristic

109

C₂×#97

—

110

#333

—

111

#332

—

112

#330

—

113

C₄₇

#10528

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂²⋊D₄₇

yes

—

117

C₂²×C₈

—

118

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

119

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

120

#125

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄²

C₂²×D₄₇

yes

characteristic

127

#108

—

188

—

128

#106

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

#97

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

#10625352

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

188

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

188

—

149

C₂×C₈

—

188

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₄²

D₄×D₄₇

yes

characteristic

159

C₄²

#10625352

yes

characteristic

160

C₄²

C₂×C₂²⋊D₄₇

yes

—

161

C₂³

#13094011

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13093993

yes

characteristic

171

C₂×C₄

#13093995

yes

characteristic

172

C₂×C₄

#13093996

yes

—

173

C₂×C₄

#13094005

yes

—

174

C₈

—

175

C₈

—

176

C₈

—

376

—

177

C₈

—

376

—

178

C₂²

—

179

C₂²

—

180

C₂²

#15689037

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#15689291

yes

characteristic

187

C₄

#15689306

yes

characteristic

188

C₂

—

189

C₂

#20244609

yes

characteristic

190

C₂

#20244665

yes

—

	subgroup	G/H
center	C₂²	#15689037
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3255	C₂
socle	C₂²×C₄₇	#394
prime radical	C₂³×C₄₇	#124
hypercenter	#3255	D₄₇
hypocenter	C₄₇	#10528
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

subgroup

G/H

center

C₂²

#15689037

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3255

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

N	H ≅ G/N	type
113. C₄₇	#10528	semidirect

H ≅ G/N

type

113. C₄₇

#10528

semidirect

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

subgroup	index
#20243562	2
#20242118	2
#20242003	2
#20241299	2
C₄₇×#3255	2
#10528	47

subgroup

index

C₄₇×#3255

Subgroups — #26120437

Conjugacy classes of subgroups

Complements

Maximal subgroups