Group #26121056 — subgroups

Subgroups — #26121056

Order 12032.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20243508	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242037	C₂	1	2	yes	maximal
3	#20241283	C₂	1	2	yes	maximal, complemented
4	#20241217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1963	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15688878	—	2	2	no	—
9	#15688869	—	2	2	no	—
10	#15688867	—	2	2	no	—
11	C₄₇×#532	C₂²	1	1	yes	characteristic
12	C₄₇×#423	C₂²	1	1	yes	characteristic
13	C₂×#104×C₄₇	—	2	2	no	—
14	C₄₇×#376	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₂×#41×C₄₇	—	2	1	no	—
22	C₂×M₁₆×C₄₇	—	2	1	no	—
23	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
24	#127×C₄₇	D₄	1	1	yes	characteristic
25	#124×C₄₇	—	2	1	no	—
26	#121×C₄₇	—	2	2	no	—
27	C₄×D₄×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	#104×C₄₇	—	4	4	no	—
32	#95×C₄₇	—	2	2	no	—
33	#41×C₄₇	—	4	2	no	—
34	#41×C₄₇	—	4	2	no	—
35	C₂×Q₈×C₄₇	—	2	1	no	—
36	C₂×D₄×C₄₇	—	2	1	no	—
37	C₂×D₄×C₄₇	—	2	2	no	—
38	C₂×D₄×C₄₇	—	4	1	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	4	1	no	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	2	yes	—
43	M₁₆×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	—	4	1	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂²⋊C₄×C₄₇	—	4	2	no	—
47	C₄²×C₄₇	—	2	2	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂×C₄₇⋊C₈	—	4	2	no	—
56	C₂³×C₄₇	—	4	1	no	—
57	C₂³×C₄₇	#123	1	1	yes	characteristic
58	Q₈×C₄₇	—	4	2	no	—
59	D₄×C₄₇	—	4	2	no	—
60	D₄×C₄₇	—	8	2	no	—
61	D₄×C₄₇	—	8	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	4	1	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	8	1	no	—
67	C₂×C₄×C₄₇	—	8	1	no	—
68	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	2	yes	—
70	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#126	1	2	yes	—
72	C₂×C₄×C₄₇	#135	1	2	yes	—
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₄₇	—	8	1	no	—
75	C₄₇⋊C₈	—	8	2	no	—
76	C₄₇⋊C₈	—	8	2	no	—
77	#10629	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	—	8	1	no	—
80	C₂²×C₄₇	—	8	2	no	—
81	C₂²×C₄₇	#397	1	1	yes	characteristic
82	C₂²×C₄₇	#482	1	2	yes	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	—	8	2	no	—
86	C₄×C₄₇	—	8	1	no	—
87	C₄×C₄₇	#463	1	2	yes	—
88	C₄×C₄₇	#488	1	1	yes	characteristic
89	C₄×C₄₇	#490	1	1	yes	characteristic
90	#1963	D₄₇	1	1	yes	characteristic
91	#1832	—	47	1	no	—
92	#1540	—	47	2	no	—
93	#1301	—	47	2	no	—
94	#1274	—	47	1	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	—	16	1	no	—
97	C₂×C₄₇	#2073	1	1	yes	characteristic
98	C₂×C₄₇	#2080	1	2	yes	—
99	#532	C₂×D₄₇	1	1	yes	characteristic
100	#431	—	47	2	no	—
101	#423	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	47	2	no	—
103	C₂×#104	—	2	2	no	—
104	#376	C₂×D₄₇	1	1	yes	characteristic
105	#336	—	94	2	no	—
106	#332	—	94	2	no	—
107	#331	—	94	2	no	—
108	C₄₇	#10629	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	94	2	no	—
112	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
113	#127	D₁₈₈	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
118	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	94	2	no	—
121	C₄⋊C₈	—	94	2	no	—
122	C₄⋊C₈	—	94	2	no	—
123	#104	—	4	4	no	—
124	C₂²⋊C₈	—	94	2	no	—
125	C₄×C₈	—	94	2	no	—
126	#95	—	2	2	no	—
127	#41	—	4	2	no	—
128	#41	—	4	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	#10625352	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	188	2	no	—
140	C₂×C₈	—	188	2	no	—
141	C₂×C₈	—	188	2	no	—
142	C₂×C₈	—	188	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
148	C₄²	#10625350	1	1	yes	characteristic
149	C₄²	#10625352	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#13094005	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#13093998	1	1	yes	characteristic
163	C₂×C₄	#13094008	1	2	yes	—
164	C₂×C₄	#13094011	1	1	yes	characteristic
165	C₂×C₄	#13094017	1	2	yes	—
166	C₂×C₄	#13094044	1	2	yes	—
167	C₂×C₄	#13094050	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	376	2	no	—
170	C₈	—	376	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#15689048	1	1	yes	characteristic
175	C₂²	#15689444	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	#15689338	1	2	yes	—
181	C₄	#15689477	1	1	yes	characteristic
182	C₄	#15689487	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#20244801	1	1	yes	characteristic
186	C₂	#20244838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243508

C₂

yes

maximal, characteristic, complemented

#20242037

C₂

yes

maximal

#20241283

C₂

yes

maximal, complemented

#20241217

C₂

yes

maximal, characteristic, complemented

C₄₇×#1963

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688878

—

#15688869

—

#15688867

—

C₄₇×#532

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

characteristic

C₂×#104×C₄₇

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×#41×C₄₇

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

characteristic

#127×C₄₇

D₄

yes

characteristic

#124×C₄₇

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

#104×C₄₇

—

#95×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10629

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#397

yes

characteristic

C₂²×C₄₇

#482

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#463

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2073

yes

characteristic

C₂×C₄₇

#2080

yes

—

#532

C₂×D₄₇

yes

characteristic

100

#431

—

101

#423

C₂×D₄₇

yes

characteristic

102

C₂×C₄⋊C₈

—

103

C₂×#104

—

104

#376

C₂×D₄₇

yes

characteristic

105

#336

—

106

#332

—

107

#331

—

108

C₄₇

#10629

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₄⋊Q₈

D₁₈₈

yes

characteristic

113

#127

D₁₈₈

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₄₇

yes

—

118

C₂×C₄²

C₂²×D₄₇

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#104

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

#10625352

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

188

—

140

C₂×C₈

—

188

—

141

C₂×C₈

—

188

—

142

C₂×C₈

—

188

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

C₂×D₁₈₈

yes

characteristic

148

C₄²

#10625350

yes

characteristic

149

C₄²

#10625352

yes

—

150

C₂³

—

151

C₂³

#13094005

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#13093998

yes

characteristic

163

C₂×C₄

#13094008

yes

—

164

C₂×C₄

#13094011

yes

characteristic

165

C₂×C₄

#13094017

yes

—

166

C₂×C₄

#13094044

yes

—

167

C₂×C₄

#13094050

yes

—

168

C₈

—

169

C₈

—

376

—

170

C₈

—

376

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#15689048

yes

characteristic

175

C₂²

#15689444

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#15689338

yes

—

181

C₄

#15689477

yes

characteristic

182

C₄

#15689487

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#20244801

yes

characteristic

186

C₂

#20244838

yes

—

	subgroup	G/H
center	C₂×C₄	#13094011
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1963	C₂
socle	C₂²×C₄₇	#397
prime radical	C₄₇×#532	C₂²
hypercenter	#1963	D₄₇
hypocenter	C₄₇	#10629
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#13094011

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1963

C₂

socle

C₂²×C₄₇

#397

prime radical

C₄₇×#532

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #20243508	C₂	semidirect
3. #20241283	C₂	semidirect
4. #20241217	C₂	semidirect
108. C₄₇	#10629	semidirect

H ≅ G/N

type

1. #20243508

C₂

semidirect

3. #20241283

C₂

semidirect

4. #20241217

C₂

semidirect

108. C₄₇

#10629

semidirect

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

subgroup	index
#20243508	2
#20242037	2
#20241283	2
#20241217	2
C₄₇×#1963	2
#10629	47

subgroup

index

C₄₇×#1963

Subgroups — #26121056

Conjugacy classes of subgroups

Complements

Maximal subgroups