Group #26135927 — subgroups

Subgroups — #26135927

Order 12032.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15689521	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688946	C₂	1	2	yes	maximal
3	#20244136	C₂	1	1	yes	maximal, characteristic
4	#20241494	C₂	1	2	yes	maximal
5	C₄₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	—	2	1	no	—
7	#15689521	—	2	4	no	—
8	C₂×#13093960	C₂²	1	1	yes	characteristic
9	#15689161	C₂²	1	1	yes	characteristic
10	C₂×#13093956	—	2	1	no	—
11	C₂×#13093956	C₂²	1	2	yes	—
12	#15689149	—	2	1	no	—
13	#15688946	—	2	4	no	—
14	#15688883	—	2	1	no	—
15	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₉₄	—	4	1	no	—
18	C₂×#10625329	—	2	1	no	—
19	#13094024	—	2	4	no	—
20	#13094024	—	4	2	no	—
21	#13093983	—	2	1	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	4	1	no	—
24	#13093960	—	2	2	no	—
25	#13093958	Q₈	1	2	yes	complemented
26	#13093956	—	2	4	no	—
27	#13093956	—	2	2	no	—
28	#13093956	—	2	1	no	—
29	#13093956	D₄	1	4	yes	—
30	Dic₄₇⋊C₈	—	4	2	no	—
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	M₃₂×C₄₇	—	2	2	no	—
35	M₃₂×C₄₇	Q₈	1	4	yes	complemented
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₄₇	—	4	2	no	—
41	C₂×Dic₉₄	—	4	4	no	—
42	C₂×Dic₉₄	—	8	4	no	—
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
50	C₄²×C₄₇	—	2	1	no	—
51	C₄²×C₄₇	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₄₇	—	4	1	no	—
53	C₄⋊Dic₄₇	—	2	4	no	—
54	C₄⋊Dic₄₇	—	2	2	no	—
55	C₄⋊Dic₄₇	—	4	4	no	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	C₄×Dic₄₇	—	4	1	no	—
58	#10625329	—	4	1	no	—
59	#10625329	—	4	1	no	—
60	C₂×C₄₇⋊C₈	—	4	1	no	—
61	C₁₆×C₄₇	—	2	4	no	—
62	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
71	C₂×Dic₄₇	—	4	4	no	—
72	C₂×Dic₄₇	—	4	2	no	—
73	C₂×Dic₄₇	—	8	4	no	—
74	Dic₉₄	—	8	8	no	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
78	C₄₇⋊C₈	—	8	1	no	—
79	#13457	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	#450	1	1	yes	characteristic
82	C₂²×C₄₇	#476	1	1	yes	characteristic
83	C₂²×C₄₇	#501	1	1	yes	characteristic
84	C₄×C₄₇	—	2	4	no	—
85	C₄×C₄₇	#395	1	1	yes	characteristic
86	C₄×C₄₇	#448	1	1	yes	characteristic
87	C₄×C₄₇	#475	1	1	yes	characteristic
88	C₄×C₄₇	#498	1	1	yes	characteristic
89	Dic₄₇	—	8	4	no	—
90	Dic₄₇	—	8	2	no	—
91	C₂×#498	—	47	1	no	—
92	C₂×#365	—	47	2	no	—
93	#1945	—	47	1	no	—
94	#1386	—	47	2	no	—
95	#1376	D₄₇	1	1	yes	characteristic
96	C₂×C₄₇	—	2	1	no	—
97	C₂×C₄₇	—	2	1	no	—
98	C₂×C₄₇	#2058	1	1	yes	characteristic
99	C₂×C₄₇	#2230	1	2	yes	—
100	C₂×C₄⋊Q₈	—	94	1	no	—
101	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
102	#498	—	94	4	no	—
103	C₂×#108	—	47	1	no	—
104	#427	—	47	1	no	—
105	C₂×#106	—	47	2	no	—
106	C₂×#106	—	94	1	no	—
107	#424	—	94	1	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	#365	—	94	4	no	—
110	#339	—	94	1	no	—
111	C₄₇	#13457	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	188	1	no	—
113	C₂×M₁₆	—	94	1	no	—
114	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	94	4	no	—
117	C₄⋊Q₈	—	188	2	no	—
118	#117	—	94	1	no	—
119	C₂×C₄⋊C₄	—	94	2	no	—
120	C₂×C₄⋊C₄	—	188	1	no	—
121	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	Dic₉₄	1	4	yes	complemented
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	94	2	no	—
126	#107	—	47	2	no	—
127	#106	—	47	4	no	—
128	#106	—	94	4	no	—
129	#106	—	94	2	no	—
130	#106	—	94	1	no	—
131	C₄⋊C₈	—	188	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	376	4	no	—
134	C₂×Q₈	—	188	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	188	2	no	—
137	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
138	M₁₆	—	188	1	no	—
139	M₁₆	—	188	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	188	1	no	—
143	C₂×C₈	#10625325	1	2	yes	—
144	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
145	C₂×C₈	D₄×D₄₇	1	2	yes	—
146	C₂×C₈	#10625352	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
148	C₄⋊C₄	—	94	4	no	—
149	C₄⋊C₄	—	94	2	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄⋊C₄	—	188	2	no	—
152	C₂²⋊C₄	—	188	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	188	1	no	—
155	C₄²	#10625325	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	#13094052	1	1	yes	characteristic
158	Q₈	—	376	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	376	4	no	—
162	C₂×C₄	—	188	4	no	—
163	C₂×C₄	—	188	2	no	—
164	C₂×C₄	#13093994	1	1	yes	characteristic
165	C₂×C₄	#13094004	1	1	yes	characteristic
166	C₂×C₄	#13094024	1	1	yes	characteristic
167	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
168	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
169	C₂×C₄	#13094057	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	376	1	no	—
172	C₈	#13094001	1	2	yes	—
173	C₈	#13094023	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#15689276	1	1	yes	characteristic
176	C₂²	#15689414	1	1	yes	characteristic
177	C₂²	#15689532	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	376	4	no	—
180	C₄	—	376	2	no	—
181	C₄	#15689044	1	1	yes	characteristic
182	C₄	#15689264	1	1	yes	characteristic
183	C₄	#15689409	1	1	yes	characteristic
184	C₄	#15689521	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#20244722	1	1	yes	characteristic
188	C₂	#20245560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689521

C₂

yes

maximal, characteristic

C₂×#15688946

C₂

yes

maximal

#20244136

C₂

yes

maximal, characteristic

#20241494

C₂

yes

maximal

C₄₇×#1376

C₂

yes

maximal, characteristic

C₂×#13094024

—

#15689521

—

C₂×#13093960

C₂²

yes

characteristic

#15689161

C₂²

yes

characteristic

C₂×#13093956

—

C₂×#13093956

C₂²

yes

—

#15689149

—

#15688946

—

#15688883

—

C₂×M₃₂×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

—

—

—

—

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093960

—

#13093958

Q₈

yes

—

—

—

D₄

yes

—

Dic₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

Q₈

yes

complemented

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

#137

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

—

#13457

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2058

yes

characteristic

C₂×C₄₇

#2230

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×M₃₂

C₂×D₄₇

yes

—

102

#498

—

103

C₂×#108

—

104

#427

—

105

C₂×#106

—

106

C₂×#106

—

107

#424

—

108

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

109

#365

—

110

#339

—

111

C₄₇

#13457

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

188

—

113

C₂×M₁₆

—

114

C₂²×C₈

D₁₈₈

yes

characteristic

115

C₂²×C₈

C₂²×D₄₇

yes

characteristic

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

188

—

118

#117

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

188

—

121

C₂×C₄²

D₁₈₈

yes

characteristic

122

M₃₂

—

123

M₃₂

Dic₉₄

yes

complemented

124

C₂×C₁₆

—

125

#108

—

126

#107

—

127

#106

—

128

#106

—

129

#106

—

130

#106

—

131

C₄⋊C₈

—

188

—

132

C₄×C₈

—

133

C₂×Q₈

—

376

—

134

C₂×Q₈

—

188

—

135

C₂²×C₄

—

136

C₂²×C₄

—

188

—

137

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

138

M₁₆

—

188

—

139

M₁₆

—

188

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

188

—

143

C₂×C₈

#10625325

yes

—

144

C₂×C₈

C₂×Dic₉₄

yes

—

145

C₂×C₈

D₄×D₄₇

yes

—

146

C₂×C₈

#10625352

yes

characteristic

147

C₂×C₈

Q₈×D₄₇

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

188

—

151

C₄⋊C₄

—

188

—

152

C₂²⋊C₄

—

188

—

153

C₄²

—

154

C₄²

—

188

—

155

C₄²

#10625325

yes

—

156

C₁₆

—

157

C₂³

#13094052

yes

characteristic

158

Q₈

—

376

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

376

—

162

C₂×C₄

—

188

—

163

C₂×C₄

—

188

—

164

C₂×C₄

#13093994

yes

characteristic

165

C₂×C₄

#13094004

yes

characteristic

166

C₂×C₄

#13094024

yes

characteristic

167

C₂×C₄

C₂×#10625325

yes

characteristic

168

C₂×C₄

C₂×#10625325

yes

characteristic

169

C₂×C₄

#13094057

yes

characteristic

170

C₈

—

171

C₈

—

376

—

172

C₈

#13094001

yes

—

173

C₈

#13094023

yes

—

174

C₂²

—

175

C₂²

#15689276

yes

characteristic

176

C₂²

#15689414

yes

characteristic

177

C₂²

#15689532

yes

characteristic

178

C₄

—

179

C₄

—

376

—

180

C₄

—

376

—

181

C₄

#15689044

yes

characteristic

182

C₄

#15689264

yes

characteristic

183

C₄

#15689409

yes

characteristic

184

C₄

#15689521

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#20244722

yes

characteristic

188

C₂

#20245560

yes

—

	subgroup	G/H
center	C₂²	#15689414
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1376	C₂
socle	C₂²×C₄₇	#476
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
hypercenter	#1376	D₄₇
hypocenter	C₄₇	#13457
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#15689414

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1376

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
25. #13093958	Q₈	semidirect
35. M₃₂×C₄₇	Q₈	semidirect
111. C₄₇	#13457	semidirect
123. M₃₂	Dic₉₄	semidirect

H ≅ G/N

type

25. #13093958

Q₈

semidirect

35. M₃₂×C₄₇

Q₈

semidirect

111. C₄₇

#13457

semidirect

123. M₃₂

Dic₉₄

semidirect

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

subgroup	index
C₂×#15689521	2
C₂×#15688946	2
#20244136	2
#20241494	2
C₄₇×#1376	2
#13457	47

subgroup

index

C₄₇×#1376

Subgroups — #26135927

Conjugacy classes of subgroups

Complements

Maximal subgroups