Group #26113811 — subgroups

Subgroups — #26113811

Order 12032.

436 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20252494	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242102	C₂	1	2	yes	maximal
3	#20242026	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241260	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3138	C₂	1	1	yes	maximal, characteristic
6	#15689254	—	2	4	no	—
7	#15689184	—	2	2	no	—
8	#15689180	C₂²	1	2	yes	—
9	#15689180	C₂²	1	2	yes	—
10	#15688867	—	2	4	no	—
11	#15688865	—	2	2	no	—
12	C₄₇×#533	C₂²	1	1	yes	characteristic
13	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#490	—	2	2	no	—
15	C₄₇×#486	C₄	1	4	yes	—
16	C₄₇×#418	—	2	2	no	—
17	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093935	—	2	2	no	—
24	#13093935	—	2	2	no	—
25	#13093935	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×D₄×C₄₇	—	2	1	no	—
28	C₂²×D₄×C₄₇	—	2	2	no	—
29	C₂×M₁₆×C₄₇	—	2	1	no	—
30	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
31	#127×C₄₇	—	2	1	no	—
32	#127×C₄₇	—	2	2	no	—
33	#127×C₄₇	C₂×C₄	1	2	yes	—
34	#125×C₄₇	D₄	1	2	yes	—
35	C₂²⋊Q₈×C₄₇	—	2	2	no	—
36	C₄×Q₈×C₄₇	—	2	2	no	—
37	#117×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	#102×C₄₇	—	2	8	no	—
40	#97×C₄₇	—	2	1	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂⁴×C₄₇	—	4	1	no	—
43	C₂×Q₈×C₄₇	—	2	2	no	—
44	C₂×D₄×C₄₇	—	2	2	no	—
45	C₂×D₄×C₄₇	—	2	4	no	—
46	C₂×D₄×C₄₇	—	4	4	no	—
47	C₂×D₄×C₄₇	—	4	1	no	—
48	C₂×D₄×C₄₇	—	4	2	no	—
49	C₂×D₄×C₄₇	—	8	4	no	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	2	yes	—
52	M₁₆×C₄₇	—	4	2	no	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₂²⋊C₄×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂³×C₄₇	—	4	2	no	—
65	C₂³×C₄₇	—	8	2	no	—
66	C₂³×C₄₇	—	8	4	no	—
67	C₂³×C₄₇	#123	1	1	yes	characteristic
68	Q₈×C₄₇	—	8	2	no	—
69	D₄×C₄₇	—	4	4	no	—
70	D₄×C₄₇	—	8	4	no	—
71	D₄×C₄₇	—	8	4	no	—
72	D₄×C₄₇	—	8	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#121	1	2	yes	—
80	C₈×C₄₇	—	4	2	no	—
81	C₄₇⋊C₈	—	4	4	no	—
82	C₄₇⋊C₈	—	8	2	no	—
83	#9424	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	—	8	2	no	—
87	C₂²×C₄₇	—	8	4	no	—
88	C₂²×C₄₇	—	16	4	no	—
89	C₂²×C₄₇	#386	1	1	yes	characteristic
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	4	no	—
94	C₄×C₄₇	#454	1	2	yes	—
95	#3138	D₄₇	1	1	yes	characteristic
96	#2987	—	47	1	no	—
97	#1555	—	47	2	no	—
98	#1537	—	47	1	no	—
99	#1291	—	47	2	no	—
100	C₂×C₄₇	—	4	1	no	—
101	C₂×C₄₇	—	16	2	no	—
102	C₂×C₄₇	#1888	1	1	yes	characteristic
103	C₂×C₄₇	#1903	1	2	yes	—
104	#533	C₂×D₄₇	1	1	yes	characteristic
105	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
106	#490	—	2	2	no	—
107	#486	Dic₄₇	1	4	yes	—
108	#446	—	94	4	no	—
109	#433	—	94	2	no	—
110	#432	—	47	2	no	—
111	#432	—	47	2	no	—
112	#418	—	2	2	no	—
113	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
114	#331	—	94	4	no	—
115	#330	—	94	2	no	—
116	C₄₇	#9424	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	2	2	no	—
118	C₂²×D₄	—	2	1	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
121	#127	—	2	2	no	—
122	#127	—	2	1	no	—
123	#127	C₂×Dic₄₇	1	2	yes	—
124	#125	C₂²⋊D₄₇	1	2	yes	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	#117	C₂²⋊D₄₇	1	2	yes	—
128	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊C₈	—	47	4	no	—
130	C₄⋊C₈	—	94	4	no	—
131	C₄⋊C₈	—	94	2	no	—
132	#102	—	2	8	no	—
133	C₂²⋊C₈	—	94	2	no	—
134	C₂²⋊C₈	—	94	2	no	—
135	#97	—	2	1	no	—
136	#97	—	94	2	no	—
137	#97	—	94	2	no	—
138	#97	—	94	2	no	—
139	C₄×C₈	—	94	2	no	—
140	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	8	4	no	—
149	C₂²×C₄	#10625352	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
151	M₁₆	—	4	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	94	4	no	—
154	C₂×C₈	—	188	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	D₄×D₄₇	1	2	yes	—
161	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂³	—	4	2	no	—
164	C₂³	—	8	2	no	—
165	C₂³	—	8	4	no	—
166	C₂³	#13094005	1	1	yes	characteristic
167	Q₈	—	8	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	8	4	no	—
170	D₄	—	8	4	no	—
171	D₄	—	8	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
177	C₂×C₄	#13093992	1	1	yes	characteristic
178	C₂×C₄	#13093996	1	2	yes	—
179	C₈	—	4	2	no	—
180	C₈	—	376	2	no	—
181	C₈	—	188	4	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	#15689003	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	#15689295	1	2	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	16	2	no	—
195	C₂	#20243792	1	1	yes	characteristic
196	C₂	#20243870	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252494

C₂

yes

maximal, characteristic, complemented

#20242102

C₂

yes

maximal

#20242026

C₂

yes

maximal, characteristic, complemented

#20241260

C₂

yes

maximal, complemented

C₄₇×#3138

C₂

yes

maximal, characteristic

#15689254

—

#15689184

—

#15689180

C₂²

yes

—

#15689180

C₂²

yes

—

#15688867

—

#15688865

—

C₄₇×#533

C₂²

yes

characteristic

C₂×#127×C₄₇

C₂²

yes

characteristic

C₄₇×#490

—

C₄₇×#486

C₄

yes

—

C₄₇×#418

—

C₄×M₁₆×C₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

#127×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₂×C₄

yes

—

#125×C₄₇

D₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#102×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9424

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#386

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#454

yes

—

#3138

D₄₇

yes

—

—

—

—

100

C₂×C₄₇

—

101

C₂×C₄₇

—

102

C₂×C₄₇

#1888

yes

characteristic

103

C₂×C₄₇

#1903

yes

—

104

#533

C₂×D₄₇

yes

characteristic

105

C₂×#127

C₂×D₄₇

yes

characteristic

106

#490

—

107

#486

Dic₄₇

yes

—

108

#446

—

109

#433

—

110

#432

—

111

#432

—

112

#418

—

113

C₄×M₁₆

C₂×D₄₇

yes

characteristic

114

#331

—

115

#330

—

116

C₄₇

#9424

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

118

C₂²×D₄

—

119

C₂×M₁₆

—

120

C₄⋊Q₈

C₂×Dic₄₇

yes

—

121

#127

—

122

#127

—

123

#127

C₂×Dic₄₇

yes

—

124

#125

C₂²⋊D₄₇

yes

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

C₂²⋊D₄₇

yes

—

128

C₂×C₄²

C₂²×D₄₇

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#102

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

135

#97

—

136

#97

—

137

#97

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

C₂×Dic₄₇

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

#10625352

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

188

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂²⋊Dic₄₇

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

D₄×D₄₇

yes

—

161

C₄²

C₂²×Dic₄₇

yes

characteristic

162

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

#13094005

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

176

C₂×C₄

D₄×Dic₄₇

yes

—

177

C₂×C₄

#13093992

yes

characteristic

178

C₂×C₄

#13093996

yes

—

179

C₈

—

180

C₈

—

376

—

181

C₈

—

188

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#15689003

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#15689295

yes

—

193

C₂

—

194

C₂

—

195

C₂

#20243792

yes

characteristic

196

C₂

#20243870

yes

—

	subgroup	G/H
center	C₂²	#15689003
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3138	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3138	D₄₇
hypocenter	C₄₇	#9424
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

subgroup

G/H

center

C₂²

#15689003

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3138

C₂

socle

C₂²×C₄₇

#386

prime radical

C₂×#127×C₄₇

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

N	H ≅ G/N	type
1. #20252494	C₂	semidirect
3. #20242026	C₂	semidirect
4. #20241260	C₂	semidirect
116. C₄₇	#9424	semidirect

H ≅ G/N

type

1. #20252494

C₂

semidirect

3. #20242026

C₂

semidirect

4. #20241260

C₂

semidirect

116. C₄₇

#9424

semidirect

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

subgroup	index
#20252494	2
#20242102	2
#20242026	2
#20241260	2
C₄₇×#3138	2
#9424	47

subgroup

index

C₄₇×#3138

Subgroups — #26113811

Conjugacy classes of subgroups

Complements

Maximal subgroups