Group #26121084 — subgroups

Subgroups — #26121084

Order 12032.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20244212	C₂	1	1	yes	maximal, characteristic
2	#20242057	C₂	1	2	yes	maximal
3	#20241248	C₂	1	2	yes	maximal, complemented
4	#20241233	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1838	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094021	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	C₂×#13093946	—	2	2	no	—
9	#15688989	C₂²	1	1	yes	characteristic
10	#15688866	—	2	4	no	—
11	C₄₇×#423	C₂²	1	2	yes	—
12	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#341	—	2	2	no	—
14	C₂²×D₁₈₈	—	2	1	no	—
15	C₂²×D₁₈₈	—	2	2	no	—
16	#13094021	—	2	1	no	—
17	#13094021	—	2	2	no	—
18	#13094021	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	—	2	1	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	#13093946	—	4	4	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	#97×C₄₇	—	2	2	no	—
31	C₂³×D₄₇	—	4	1	no	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	#41	1	2	yes	—
34	C₂²×Dic₄₇	—	4	1	no	—
35	C₂×D₁₈₈	—	2	2	no	—
36	C₂×D₁₈₈	—	2	4	no	—
37	C₂×D₁₈₈	—	4	2	no	—
38	C₂×D₁₈₈	—	4	2	no	—
39	C₂×D₁₈₈	—	4	1	no	—
40	C₂×D₁₈₈	—	4	2	no	—
41	C₂×D₁₈₈	—	8	4	no	—
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	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	2	yes	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂³×C₄₇	#124	1	1	yes	characteristic
54	C₂²×D₄₇	—	4	2	no	—
55	C₂²×D₄₇	—	8	4	no	—
56	C₂²×D₄₇	—	8	1	no	—
57	C₂²×D₄₇	—	8	1	no	—
58	C₂×C₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#123	1	2	yes	—
62	C₂×C₄×C₄₇	#126	1	2	yes	—
63	C₂×C₄×C₄₇	#135	1	2	yes	—
64	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₄₇	—	8	2	no	—
66	C₂×Dic₄₇	—	8	1	no	—
67	D₁₈₈	—	4	4	no	—
68	D₁₈₈	—	8	4	no	—
69	D₁₈₈	—	8	4	no	—
70	D₁₈₈	—	8	2	no	—
71	C₈×C₄₇	—	4	2	no	—
72	C₈×C₄₇	—	8	2	no	—
73	C₄₇⋊C₈	—	8	2	no	—
74	#10634	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	#397	1	1	yes	characteristic
77	C₂²×C₄₇	#488	1	1	yes	characteristic
78	C₂²×C₄₇	#490	1	1	yes	characteristic
79	C₂×D₄₇	—	8	2	no	—
80	C₂×D₄₇	—	8	4	no	—
81	C₂×D₄₇	—	16	2	no	—
82	C₂×D₄₇	—	16	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	#463	1	2	yes	—
86	C₄×C₄₇	#482	1	2	yes	—
87	Dic₄₇	—	16	1	no	—
88	#1961	—	47	1	no	—
89	#1838	D₄₇	1	1	yes	characteristic, complemented
90	#1545	—	47	2	no	—
91	#1286	—	47	2	no	—
92	#1281	—	47	1	no	—
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	#2073	1	1	yes	characteristic
96	C₂×C₄₇	#2081	1	2	yes	—
97	D₄₇	—	16	2	no	—
98	C₂×#127	—	47	1	no	—
99	#431	—	47	2	no	—
100	#423	C₂×D₄₇	1	2	yes	—
101	C₂×#102	—	94	2	no	—
102	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
103	#382	—	47	1	no	—
104	#341	—	2	2	no	—
105	#331	—	94	4	no	—
106	C₄₇	#10634	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	—	94	2	no	—
108	C₂²×D₄	—	94	1	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
111	#127	—	47	2	no	—
112	#127	—	94	2	no	—
113	#127	—	94	1	no	—
114	C₂×C₄⋊C₄	—	94	1	no	—
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	94	2	no	—
119	#102	—	188	4	no	—
120	C₂²⋊C₈	—	94	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	94	2	no	—
123	#95	—	94	2	no	—
124	C₂⁴	—	188	1	no	—
125	C₂×D₄	—	94	2	no	—
126	C₂×D₄	—	94	4	no	—
127	C₂×D₄	—	376	4	no	—
128	C₂×D₄	—	188	1	no	—
129	C₂×D₄	—	188	2	no	—
130	C₂×D₄	—	188	2	no	—
131	C₂×D₄	—	188	2	no	—
132	C₂²×C₄	—	188	1	no	—
133	C₂²×C₄	#10625354	1	2	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
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	—	188	2	no	—
142	C₄⋊C₄	—	188	2	no	—
143	C₄²	#10625350	1	2	yes	—
144	C₄²	D₄×D₄₇	1	1	yes	characteristic
145	C₄²	#10625352	1	1	yes	characteristic
146	C₂³	—	376	4	no	—
147	C₂³	—	376	1	no	—
148	C₂³	—	376	1	no	—
149	C₂³	—	188	2	no	—
150	C₂³	#13094013	1	1	yes	characteristic
151	D₄	—	376	4	no	—
152	D₄	—	376	4	no	—
153	D₄	—	376	2	no	—
154	D₄	—	188	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	376	2	no	—
157	C₂×C₄	—	376	1	no	—
158	C₂×C₄	#13093995	1	1	yes	characteristic
159	C₂×C₄	#13094007	1	1	yes	characteristic
160	C₂×C₄	#13094009	1	2	yes	—
161	C₂×C₄	#13094019	1	2	yes	—
162	C₂×C₄	#13094044	1	2	yes	—
163	C₂×C₄	#13094049	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	376	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	376	2	no	—
169	C₂²	—	376	4	no	—
170	C₂²	—	752	2	no	—
171	C₂²	—	752	2	no	—
172	C₂²	#15689049	1	1	yes	characteristic
173	C₂²	#15689480	1	1	yes	characteristic
174	C₂²	#15689489	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	752	1	no	—
178	C₄	#15689335	1	2	yes	—
179	C₄	#15689447	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	752	2	no	—
183	C₂	#20244802	1	1	yes	characteristic
184	C₂	#20244846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244212

C₂

yes

maximal, characteristic

#20242057

C₂

yes

maximal

#20241248

C₂

yes

maximal, complemented

#20241233

C₂

yes

maximal, characteristic, complemented

C₄₇×#1838

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

characteristic

#15689176

C₂²

yes

—

C₂×#13093946

—

#15688989

C₂²

yes

characteristic

#15688866

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

#13094021

—

#13094021

—

#13094021

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

#13093946

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#10634

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#463

yes

—

C₄×C₄₇

#482

yes

—

Dic₄₇

—

#1961

—

#1838

D₄₇

yes

characteristic, complemented

#1545

—

#1286

—

#1281

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2073

yes

characteristic

C₂×C₄₇

#2081

yes

—

D₄₇

—

C₂×#127

—

#431

—

100

#423

C₂×D₄₇

yes

—

101

C₂×#102

—

102

C₂×#97

C₂×D₄₇

yes

characteristic

103

#382

—

104

#341

—

105

#331

—

106

C₄₇

#10634

yes

characteristic, complemented, Sylow

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×M₁₆

—

110

C₂²×C₈

C₂²⋊D₄₇

yes

—

111

#127

—

112

#127

—

113

#127

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#102

—

188

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

#95

—

124

C₂⁴

—

188

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

376

—

128

C₂×D₄

—

188

—

129

C₂×D₄

—

188

—

130

C₂×D₄

—

188

—

131

C₂×D₄

—

188

—

132

C₂²×C₄

—

188

—

133

C₂²×C₄

#10625354

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

188

—

142

C₄⋊C₄

—

188

—

143

C₄²

#10625350

yes

—

144

C₄²

D₄×D₄₇

yes

characteristic

145

C₄²

#10625352

yes

characteristic

146

C₂³

—

376

—

147

C₂³

—

376

—

148

C₂³

—

376

—

149

C₂³

—

188

—

150

C₂³

#13094013

yes

characteristic

151

D₄

—

376

—

152

D₄

—

376

—

153

D₄

—

376

—

154

D₄

—

188

—

155

C₂×C₄

—

156

C₂×C₄

—

376

—

157

C₂×C₄

—

376

—

158

C₂×C₄

#13093995

yes

characteristic

159

C₂×C₄

#13094007

yes

characteristic

160

C₂×C₄

#13094009

yes

—

161

C₂×C₄

#13094019

yes

—

162

C₂×C₄

#13094044

yes

—

163

C₂×C₄

#13094049

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

376

—

167

C₂²

—

168

C₂²

—

376

—

169

C₂²

—

376

—

170

C₂²

—

752

—

171

C₂²

—

752

—

172

C₂²

#15689049

yes

characteristic

173

C₂²

#15689480

yes

characteristic

174

C₂²

#15689489

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

752

—

178

C₄

#15689335

yes

—

179

C₄

#15689447

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

752

—

183

C₂

#20244802

yes

characteristic

184

C₂

#20244846

yes

—

	subgroup	G/H
center	C₂²	#15689049
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1838	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂×#13094021	C₂²
hypercenter	#1838	D₄₇
hypocenter	C₄₇	#10634
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#15689049

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1838

C₂

C₂²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
3. #20241248	C₂	semidirect
4. #20241233	C₂	semidirect
5. C₄₇×#1838	C₂	semidirect
89. #1838	D₄₇	semidirect
106. C₄₇	#10634	semidirect

H ≅ G/N

type

3. #20241248

C₂

semidirect

4. #20241233

C₂

semidirect

5. C₄₇×#1838

C₂

semidirect

89. #1838

D₄₇

semidirect

106. C₄₇

#10634

semidirect

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

subgroup	index
#20244212	2
#20242057	2
#20241248	2
#20241233	2
C₄₇×#1838	2
#10634	47

subgroup

index

C₄₇×#1838

Subgroups — #26121084

Conjugacy classes of subgroups

Complements

Maximal subgroups