Group #26050332 — subgroups

Subgroups — C₄₇×#10634 #26050332

Order 12032.

346 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₄₇×#1961	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1838	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#1545	C₂	1	2	yes	maximal
4	C₄₇×#1286	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1281	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#431	C₂²	1	2	yes	—
8	C₄₇×#423	C₂²	1	2	yes	—
9	C₂×#102×C₄₇	—	2	2	no	—
10	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#382	C₂²	1	1	yes	characteristic
12	C₄₇×#341	—	2	2	no	—
13	C₄₇×#331	—	2	4	no	—
14	C₂²×D₄×C₄₇	—	2	1	no	—
15	C₂²×D₄×C₄₇	—	2	2	no	—
16	C₂×M₁₆×C₄₇	—	2	2	no	—
17	C₂²×C₈×C₄₇	D₄	1	2	yes	—
18	#127×C₄₇	—	2	1	no	—
19	#127×C₄₇	—	2	2	no	—
20	#127×C₄₇	D₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₄₇	—	2	2	no	—
24	C₄⋊C₈×C₄₇	—	2	2	no	—
25	C₄⋊C₈×C₄₇	—	2	2	no	—
26	#102×C₄₇	—	4	4	no	—
27	C₂²⋊C₈×C₄₇	—	2	2	no	—
28	#97×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	—	2	2	no	—
30	#95×C₄₇	—	2	2	no	—
31	C₂⁴×C₄₇	—	4	1	no	—
32	C₂×D₄×C₄₇	—	2	2	no	—
33	C₂×D₄×C₄₇	—	2	4	no	—
34	C₂×D₄×C₄₇	—	4	1	no	—
35	C₂×D₄×C₄₇	—	4	2	no	—
36	C₂×D₄×C₄₇	—	4	2	no	—
37	C₂×D₄×C₄₇	—	4	2	no	—
38	C₂×D₄×C₄₇	—	8	4	no	—
39	C₂²×C₄×C₄₇	—	4	1	no	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	#41	1	2	yes	—
42	M₁₆×C₄₇	—	4	2	no	—
43	M₁₆×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₂×C₈×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	1	yes	characteristic
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₂³×C₄₇	—	4	2	no	—
54	C₂³×C₄₇	—	8	4	no	—
55	C₂³×C₄₇	—	8	1	no	—
56	C₂³×C₄₇	—	8	1	no	—
57	C₂³×C₄₇	#124	1	1	yes	characteristic
58	D₄×C₄₇	—	4	4	no	—
59	D₄×C₄₇	—	8	4	no	—
60	D₄×C₄₇	—	8	4	no	—
61	D₄×C₄₇	—	8	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	8	2	no	—
64	C₂×C₄×C₄₇	—	8	1	no	—
65	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#123	1	2	yes	—
67	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#126	1	2	yes	—
69	C₂×C₄×C₄₇	#135	1	2	yes	—
70	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₈×C₄₇	—	4	2	no	—
72	C₈×C₄₇	—	8	2	no	—
73	C₈×C₄₇	—	8	2	no	—
74	#10634	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	8	2	no	—
77	C₂²×C₄₇	—	8	4	no	—
78	C₂²×C₄₇	—	16	2	no	—
79	C₂²×C₄₇	—	16	2	no	—
80	C₂²×C₄₇	#397	1	1	yes	characteristic
81	C₂²×C₄₇	#488	1	1	yes	characteristic
82	C₂²×C₄₇	#490	1	1	yes	characteristic
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	—	16	1	no	—
86	C₄×C₄₇	#463	1	2	yes	—
87	C₄×C₄₇	#482	1	2	yes	—
88	#1961	C₂×C₄₇	1	1	yes	characteristic
89	#1838	C₂×C₄₇	1	1	yes	characteristic, complemented
90	#1545	C₂×C₄₇	1	2	yes	—
91	#1286	C₂×C₄₇	1	2	yes	complemented
92	#1281	C₂×C₄₇	1	1	yes	characteristic, complemented
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	—	16	2	no	—
96	C₂×C₄₇	#2073	1	1	yes	characteristic
97	C₂×C₄₇	#2081	1	2	yes	—
98	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
99	#431	C₂²×C₄₇	1	2	yes	—
100	#423	C₂²×C₄₇	1	2	yes	—
101	C₂×#102	—	2	2	no	—
102	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
103	#382	C₂²×C₄₇	1	1	yes	characteristic
104	#341	—	2	2	no	—
105	#331	—	2	4	no	—
106	C₄₇	#10634	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×D₄	—	2	1	no	—
108	C₂²×D₄	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	D₄×C₄₇	1	2	yes	—
111	#127	—	2	1	no	—
112	#127	—	2	2	no	—
113	#127	D₄×C₄₇	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	#102	—	4	4	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	#95	—	2	2	no	—
124	C₂⁴	—	4	1	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	4	1	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	—	8	4	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
134	C₂²×C₄	#41×C₄₇	1	2	yes	—
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
144	C₄²	#41×C₄₇	1	1	yes	characteristic
145	C₄²	#41×C₄₇	1	2	yes	—
146	C₂³	—	4	2	no	—
147	C₂³	—	8	4	no	—
148	C₂³	—	8	1	no	—
149	C₂³	—	8	1	no	—
150	C₂³	#124×C₄₇	1	1	yes	characteristic
151	D₄	—	4	4	no	—
152	D₄	—	8	4	no	—
153	D₄	—	8	4	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
159	C₂×C₄	#123×C₄₇	1	2	yes	—
160	C₂×C₄	#123×C₄₇	1	1	yes	characteristic
161	C₂×C₄	#126×C₄₇	1	2	yes	—
162	C₂×C₄	#135×C₄₇	1	2	yes	—
163	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	8	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	4	no	—
170	C₂²	—	16	2	no	—
171	C₂²	—	16	2	no	—
172	C₂²	C₄₇×#397	1	1	yes	characteristic
173	C₂²	C₄₇×#488	1	1	yes	characteristic
174	C₂²	C₄₇×#490	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	16	1	no	—
178	C₄	C₄₇×#463	1	2	yes	—
179	C₄	C₄₇×#482	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	16	2	no	—
183	C₂	C₄₇×#2073	1	1	yes	characteristic
184	C₂	C₄₇×#2081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#1961

C₂

yes

maximal, characteristic

C₄₇×#1838

C₂

yes

maximal, characteristic, complemented

C₄₇×#1545

C₂

yes

maximal

C₄₇×#1286

C₂

yes

maximal, complemented

C₄₇×#1281

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₇

C₂²

yes

characteristic

C₄₇×#431

C₂²

yes

—

C₄₇×#423

C₂²

yes

—

C₂×#102×C₄₇

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#382

C₂²

yes

characteristic

C₄₇×#341

—

C₄₇×#331

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

#127×C₄₇

—

#127×C₄₇

—

#127×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#102×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#124

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#10634

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#463

yes

—

C₄×C₄₇

#482

yes

—

#1961

C₂×C₄₇

yes

characteristic

#1838

C₂×C₄₇

yes

characteristic, complemented

#1545

C₂×C₄₇

yes

—

#1286

C₂×C₄₇

yes

complemented

#1281

C₂×C₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2073

yes

characteristic

C₂×C₄₇

#2081

yes

—

C₂×#127

C₂²×C₄₇

yes

characteristic

#431

C₂²×C₄₇

yes

—

100

#423

C₂²×C₄₇

yes

—

101

C₂×#102

—

102

C₂×#97

C₂²×C₄₇

yes

characteristic

103

#382

C₂²×C₄₇

yes

characteristic

104

#341

—

105

#331

—

106

C₄₇

#10634

yes

characteristic, direct product factor, Sylow

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×M₁₆

—

110

C₂²×C₈

D₄×C₄₇

yes

—

111

#127

—

112

#127

—

113

#127

D₄×C₄₇

yes

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂³×C₄₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#102

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

#95

—

124

C₂⁴

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂×D₄×C₄₇

yes

characteristic

134

C₂²×C₄

#41×C₄₇

yes

—

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄²

C₂×D₄×C₄₇

yes

characteristic

144

C₄²

#41×C₄₇

yes

characteristic

145

C₄²

#41×C₄₇

yes

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

#124×C₄₇

yes

characteristic

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#121×C₄₇

yes

characteristic

159

C₂×C₄

#123×C₄₇

yes

—

160

C₂×C₄

#123×C₄₇

yes

characteristic

161

C₂×C₄

#126×C₄₇

yes

—

162

C₂×C₄

#135×C₄₇

yes

—

163

C₂×C₄

AGL(1,ℤ/8)×C₄₇

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₄₇×#397

yes

characteristic

173

C₂²

C₄₇×#488

yes

characteristic

174

C₂²

C₄₇×#490

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

C₄₇×#463

yes

—

179

C₄

C₄₇×#482

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

C₄₇×#2073

yes

characteristic

184

C₂

C₄₇×#2081

yes

—

	subgroup	G/H
center, socle	C₂²×C₄₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₂×#127×C₄₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₇

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₄₇

prime radical

C₂×#127×C₄₇

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
2. C₄₇×#1838	C₂	semidirect
4. C₄₇×#1286	C₂	semidirect
5. C₄₇×#1281	C₂	semidirect
74. #10634	C₄₇	direct (N × H)
89. #1838	C₂×C₄₇	semidirect
91. #1286	C₂×C₄₇	semidirect
92. #1281	C₂×C₄₇	semidirect
106. C₄₇	#10634	direct (N × H)

H ≅ G/N

type

2. C₄₇×#1838

C₂

semidirect

4. C₄₇×#1286

C₂

semidirect

5. C₄₇×#1281

C₂

semidirect

74. #10634

C₄₇

direct (N × H)

89. #1838

C₂×C₄₇

semidirect

91. #1286

C₂×C₄₇

semidirect

92. #1281

C₂×C₄₇

semidirect

106. C₄₇

#10634

direct (N × H)

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

subgroup	index
C₄₇×#1961	2
C₄₇×#1838	2
C₄₇×#1545	2
C₄₇×#1286	2
C₄₇×#1281	2
#10634	47

subgroup

index

C₄₇×#1961

C₄₇×#1838

C₄₇×#1545

C₄₇×#1286

C₄₇×#1281

#10634

Subgroups — C₄₇×#10634 #26050332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#10634 #26050332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10634 #26050332