Group #26131656 — subgroups

Subgroups — #26131656

Order 12032.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20245160	C₂	1	2	yes	maximal
2	M₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243269	C₂	1	2	yes	maximal
4	#20241492	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	#15689079	—	2	2	no	—
9	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688945	C₄	1	8	yes	complemented
11	#15688898	C₄	1	2	yes	complemented
12	C₁₆×Dic₄₇	C₄	1	2	yes	complemented
13	#15688882	—	2	2	no	—
14	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#427	C₂²	1	2	yes	—
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
19	#13093964	C₂×C₄	1	4	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	#13093960	—	2	2	no	—
22	#13093960	C₂×C₄	1	4	yes	—
23	#13093935	—	4	2	no	—
24	C₄×C₄₇⋊C₈	—	4	2	no	—
25	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₄₇	Q₈	1	1	yes	characteristic
28	C₈×Dic₄₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	#117×C₄₇	—	2	2	no	—
31	M₃₂×C₄₇	—	2	2	no	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	#107×C₄₇	—	2	2	no	—
34	#106×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂²⋊C₄×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	—	4	2	no	—
44	C₄×Dic₄₇	—	2	1	no	—
45	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
47	#10625329	—	2	4	no	—
48	#10625329	—	4	2	no	—
49	C₂×C₄₇⋊C₈	—	4	2	no	—
50	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
53	C₂×C₄₇⋊C₈	SD₁₆	1	2	yes	—
54	C₁₆×C₄₇	—	2	2	no	—
55	C₄₇⋊C₁₆	C₄²	1	4	yes	—
56	C₂³×C₄₇	#135	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	4	4	no	—
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
62	C₂×Dic₄₇	—	2	1	no	—
63	C₂×Dic₄₇	—	2	1	no	—
64	C₂×Dic₄₇	—	4	2	no	—
65	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₄₇	#117	1	1	yes	characteristic
67	C₈×C₄₇	C₄×Q₈	1	2	yes	—
68	C₄₇⋊C₈	—	4	4	no	—
69	C₄₇⋊C₈	#106	1	1	yes	characteristic
70	C₄₇⋊C₈	#106	1	2	yes	—
71	C₄₇⋊C₈	#106	1	1	yes	characteristic
72	#12668	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	4	1	no	—
74	C₂²×C₄₇	#427	1	1	yes	characteristic
75	C₂²×C₄₇	C₄×SD₁₆	1	2	yes	—
76	C₄×C₄₇	—	8	4	no	—
77	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
79	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
80	Dic₄₇	—	2	2	no	—
81	Dic₄₇	—	4	1	no	—
82	#2155	—	47	2	no	—
83	C₄×M₃₂	—	47	1	no	—
84	#1772	—	47	2	no	—
85	#1385	—	47	1	no	—
86	#1384	D₄₇	1	1	yes	characteristic
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	2	2	no	—
89	C₂×C₄₇	C₄×#106	1	1	yes	characteristic
90	C₂×M₃₂	—	47	1	no	—
91	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
92	C₂×#108	—	47	2	no	—
93	#427	C₂×D₄₇	1	2	yes	—
94	#405	—	94	2	no	—
95	C₂×C₄×C₈	—	47	1	no	—
96	#365	—	47	8	no	—
97	#346	—	47	2	no	—
98	C₄×C₁₆	—	47	2	no	—
99	#339	—	94	2	no	—
100	C₄₇	#12668	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	94	2	no	—
102	C₂²×C₈	—	47	1	no	—
103	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	C₂×C₄²	—	47	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	47	4	no	—
108	C₂×C₁₆	—	47	2	no	—
109	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
110	#108	—	47	4	no	—
111	#108	—	94	2	no	—
112	#107	—	2	2	no	—
113	#106	C₄×D₄₇	1	4	yes	—
114	#97	—	188	2	no	—
115	C₄×C₈	—	47	1	no	—
116	C₄×C₈	—	47	1	no	—
117	C₄×C₈	—	47	1	no	—
118	C₄×C₈	—	47	1	no	—
119	C₄×C₈	—	188	2	no	—
120	C₂²×C₄	—	94	1	no	—
121	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
122	M₁₆	—	94	4	no	—
123	M₁₆	—	188	2	no	—
124	C₂×C₈	—	47	1	no	—
125	C₂×C₈	—	47	1	no	—
126	C₂×C₈	—	47	2	no	—
127	C₂×C₈	—	47	2	no	—
128	C₂×C₈	—	188	2	no	—
129	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
131	C₂×C₈	#10625352	1	2	yes	—
132	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
133	C₄⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	—	47	1	no	—
137	C₄²	—	47	1	no	—
138	C₄²	—	94	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	47	4	no	—
141	C₂³	#13094043	1	1	yes	characteristic
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	94	1	no	—
144	C₂×C₄	—	94	1	no	—
145	C₂×C₄	—	188	2	no	—
146	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
148	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
149	C₂×C₄	#13094047	1	1	yes	characteristic
150	C₈	—	47	1	no	—
151	C₈	—	47	2	no	—
152	C₈	—	47	1	no	—
153	C₈	—	188	4	no	—
154	C₈	C₄²×D₄₇	1	1	yes	characteristic
155	C₈	#13093980	1	1	yes	characteristic
156	C₈	#13093989	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#15689160	1	1	yes	characteristic
159	C₂²	#15689214	1	2	yes	—
160	C₄	—	8	4	no	—
161	C₄	—	94	2	no	—
162	C₄	—	188	1	no	—
163	C₄	#15688974	1	1	yes	characteristic
164	C₄	#106×D₄₇	1	1	yes	characteristic
165	C₄	#15689208	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	—	2	1	no	—
168	C₂	#20243276	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245160

C₂

yes

maximal

M₃₂×Dic₄₇

C₂

yes

maximal, characteristic

#20243269

C₂

yes

maximal

#20241492

C₂

yes

maximal, characteristic

C₄₇×#1384

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

#15689079

—

C₂×C₈×Dic₄₇

C₂²

yes

characteristic

#15688945

C₄

yes

complemented

#15688898

C₄

yes

complemented

C₁₆×Dic₄₇

C₄

yes

complemented

#15688882

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

#13093964

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093960

—

#13093960

C₂×C₄

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₈×Dic₄₇

C₂×C₄

yes

characteristic

C₈×Dic₄₇

C₂×C₄

yes

characteristic

C₈×Dic₄₇

Q₈

yes

characteristic

C₈×Dic₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#107×C₄₇

—

#106×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×Dic₄₇

C₄⋊C₄

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

SD₁₆

yes

—

C₂×C₄₇⋊C₈

SD₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₄²

yes

—

C₂³×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

C₂×C₄²

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#106

yes

characteristic

C₄₇⋊C₈

#106

yes

—

C₄₇⋊C₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

#427

yes

characteristic

C₂²×C₄₇

C₄×SD₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₇

C₂×#106

yes

characteristic

C₄×C₄₇

C₄×SD₁₆

yes

—

Dic₄₇

—

Dic₄₇

—

#2155

—

C₄×M₃₂

—

#1772

—

#1385

—

#1384

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₄×#106

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₇

yes

characteristic

C₂×#108

—

#427

C₂×D₄₇

yes

—

#405

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

#339

—

100

C₄₇

#12668

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₄₇

yes

characteristic

104

#117

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₄₇

yes

—

110

#108

—

111

#108

—

112

#107

—

113

#106

C₄×D₄₇

yes

—

114

#97

—

188

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

188

—

120

C₂²×C₄

—

121

C₂²×C₄

D₄×D₄₇

yes

characteristic

122

M₁₆

—

123

M₁₆

—

188

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

188

—

129

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₈

C₂×C₄×D₄₇

yes

—

131

C₂×C₈

#10625352

yes

—

132

C₂×C₈

Q₈×D₄₇

yes

characteristic

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#13094043

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

188

—

146

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

147

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

148

C₂×C₄

SD₁₆×D₄₇

yes

—

149

C₂×C₄

#13094047

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

188

—

154

C₈

C₄²×D₄₇

yes

characteristic

155

C₈

#13093980

yes

characteristic

156

C₈

#13093989

yes

—

157

C₂²

—

158

C₂²

#15689160

yes

characteristic

159

C₂²

#15689214

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

188

—

163

C₄

#15688974

yes

characteristic

164

C₄

#106×D₄₇

yes

characteristic

165

C₄

#15689208

yes

—

166

C₂

—

167

C₂

—

168

C₂

#20243276

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1384	C₂
socle	C₂×C₄₇	C₄×#106
prime radical	C₂³×C₄₇	#135
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1384	D₄₇
hypocenter	C₄₇	#12668
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#106×D₄₇

commutator

C₈×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1384

C₂

C₂×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
10. #15688945	C₄	semidirect
11. #15688898	C₄	semidirect
12. C₁₆×Dic₄₇	C₄	semidirect
100. C₄₇	#12668	semidirect

H ≅ G/N

type

10. #15688945

C₄

semidirect

11. #15688898

C₄

semidirect

12. C₁₆×Dic₄₇

C₄

semidirect

100. C₄₇

#12668

semidirect

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

subgroup	index
#20245160	2
M₃₂×Dic₄₇	2
#20243269	2
#20241492	2
C₄₇×#1384	2
#12668	47

subgroup

index

M₃₂×Dic₄₇

C₄₇×#1384

Subgroups — #26131656

Conjugacy classes of subgroups

Complements

Maximal subgroups