Group #26121290 — subgroups

Subgroups — #26121290

Order 12032.

354 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20244220	C₂	1	1	yes	maximal, characteristic
2	#20242158	C₂	1	2	yes	maximal, complemented
3	#20241786	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689673	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	#15689147	C₂²	1	1	yes	characteristic
9	C₂×#13093952	—	2	2	no	—
10	#15688873	—	2	4	no	—
11	#15688850	—	2	4	no	—
12	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#376	C₂²	1	1	yes	characteristic
15	C₂×#10625350	—	2	1	no	—
16	C₂×#10625329	—	2	1	no	—
17	#13094024	Q₈	1	1	yes	characteristic
18	#13094021	Q₈	1	1	yes	characteristic
19	#13094012	—	2	1	no	—
20	#13093998	—	2	2	no	—
21	C₄×D₁₈₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	#13093952	—	4	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
27	C₂²×C₈×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	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄⋊C₈×C₄₇	—	4	4	no	—
33	#95×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	—	2	2	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	#41	1	2	yes	—
37	C₂×C₂²⋊D₄₇	—	4	1	no	—
38	#10625350	—	4	2	no	—
39	#10625350	—	4	2	no	—
40	C₂×D₁₈₈	—	2	2	no	—
41	C₂×D₁₈₈	—	2	1	no	—
42	C₂×C₄×D₄₇	—	4	1	no	—
43	C₂×Dic₉₄	—	2	1	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	4	4	no	—
46	C₄²×C₄₇	—	2	2	no	—
47	C₄²×C₄₇	SD₁₆	1	4	yes	—
48	C₄²×C₄₇	C₂×D₄	1	2	yes	—
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	1	yes	characteristic
51	(C₂×D₄₇)⋊C₄	—	4	2	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	#10625329	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	2	4	no	—
55	C₂×C₄₇⋊C₈	—	4	1	no	—
56	C₂³×C₄₇	#123	1	1	yes	characteristic
57	C₂²×D₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	#123	1	2	yes	—
64	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₄₇	—	8	2	no	—
68	C₂×Dic₄₇	—	4	1	no	—
69	D₁₈₈	—	4	4	no	—
70	D₁₈₈	—	4	2	no	—
71	C₄×D₄₇	—	8	1	no	—
72	C₄×D₄₇	—	8	1	no	—
73	Dic₉₄	—	4	2	no	—
74	C₈×C₄₇	—	8	2	no	—
75	C₄₇⋊C₈	—	4	4	no	—
76	C₄₇⋊C₈	—	8	1	no	—
77	#10666	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	#399	1	1	yes	characteristic
80	C₂²×C₄₇	#481	1	2	yes	—
81	C₂×D₄₇	—	8	1	no	—
82	C₂×D₄₇	—	8	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	—	4	4	no	—
86	C₄×C₄₇	#476	1	2	yes	—
87	C₄×C₄₇	#492	1	1	yes	characteristic
88	C₄×C₄₇	#496	1	1	yes	characteristic
89	Dic₄₇	—	8	1	no	—
90	#1963	—	47	1	no	—
91	#1843	D₄₇	1	1	yes	characteristic, complemented
92	#1570	—	47	2	no	—
93	#1480	—	47	2	no	—
94	#1274	—	47	1	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	#2094	1	1	yes	characteristic
97	C₂×C₄₇	#2098	1	2	yes	—
98	D₄₇	—	16	1	no	—
99	#532	—	47	1	no	—
100	#431	—	47	2	no	—
101	#423	—	47	1	no	—
102	C₂×C₄⋊C₈	—	2	2	no	—
103	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
104	C₂×#104	—	94	2	no	—
105	#376	C₂×D₄₇	1	1	yes	characteristic
106	#334	—	94	4	no	—
107	D₄⋊C₈	—	94	4	no	—
108	C₄₇	#10666	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	94	1	no	—
110	C₂×M₁₆	—	94	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	—	47	1	no	—
113	#127	—	47	1	no	—
114	#124	—	94	1	no	—
115	#121	—	94	2	no	—
116	C₄×D₄	—	94	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	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	94	2	no	—
123	C₄⋊C₈	—	94	2	no	—
124	#104	—	188	4	no	—
125	C₂²⋊C₈	—	94	2	no	—
126	C₄×C₈	—	47	4	no	—
127	#95	—	2	2	no	—
128	#41	—	188	2	no	—
129	#41	—	188	2	no	—
130	C₂×Q₈	—	94	1	no	—
131	C₂×D₄	—	94	1	no	—
132	C₂×D₄	—	94	2	no	—
133	C₂×D₄	—	188	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	188	1	no	—
136	C₂²×C₄	#10625350	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	M₁₆	—	188	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	94	4	no	—
142	C₂×C₈	—	188	1	no	—
143	C₄⋊C₄	—	94	2	no	—
144	C₂²⋊C₄	—	188	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#10625336	1	4	yes	—
147	C₄²	D₄×D₄₇	1	2	yes	—
148	C₄²	Q₈×D₄₇	1	1	yes	characteristic
149	C₄²	#10625354	1	1	yes	characteristic
150	C₂³	—	188	1	no	—
151	C₂³	#13094007	1	1	yes	characteristic
152	Q₈	—	188	2	no	—
153	D₄	—	376	2	no	—
154	D₄	—	188	4	no	—
155	D₄	—	188	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	376	1	no	—
160	C₂×C₄	—	376	1	no	—
161	C₂×C₄	—	188	1	no	—
162	C₂×C₄	#13094000	1	1	yes	characteristic
163	C₂×C₄	#13094002	1	2	yes	—
164	C₂×C₄	#13094009	1	2	yes	—
165	C₂×C₄	#13094020	1	1	yes	characteristic
166	C₂×C₄	C₂×#10625336	1	2	yes	—
167	C₂×C₄	#13094050	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	376	1	no	—
170	C₈	—	188	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	376	1	no	—
173	C₂²	—	376	2	no	—
174	C₂²	#15689058	1	1	yes	characteristic
175	C₂²	#15689441	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	376	1	no	—
180	C₄	#15689410	1	2	yes	—
181	C₄	#15689497	1	1	yes	characteristic
182	C₄	#15689510	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	752	1	no	—
185	C₂	#20244922	1	1	yes	characteristic
186	C₂	#20244944	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244220

C₂

yes

maximal, characteristic

#20242158

C₂

yes

maximal, complemented

#20241786

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic, complemented

C₄₇×#1843

C₂

yes

maximal, characteristic, complemented

#15689673

C₂²

yes

characteristic

#15689176

C₂²

yes

—

#15689147

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄₇×#376

C₂²

yes

—

—

Q₈

yes

characteristic

#13094021

Q₈

yes

characteristic

#13094012

—

#13093998

—

C₄×D₁₈₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093952

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10666

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#399

yes

characteristic

C₂²×C₄₇

#481

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1963

—

#1843

D₄₇

yes

characteristic, complemented

#1570

—

#1480

—

#1274

—

C₂×C₄₇

—

C₂×C₄₇

#2094

yes

characteristic

C₂×C₄₇

#2098

yes

—

D₄₇

—

#532

—

100

#431

—

101

#423

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

104

C₂×#104

—

105

#376

C₂×D₄₇

yes

characteristic

106

#334

—

107

D₄⋊C₈

—

108

C₄₇

#10666

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

#127

—

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

C₄⋊C₈

—

124

#104

—

188

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

#95

—

128

#41

—

188

—

129

#41

—

188

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

188

—

134

C₂²×C₄

—

135

C₂²×C₄

—

188

—

136

C₂²×C₄

#10625350

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

M₁₆

—

188

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

188

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

188

—

145

C₄²

—

146

C₄²

#10625336

yes

—

147

C₄²

D₄×D₄₇

yes

—

148

C₄²

Q₈×D₄₇

yes

characteristic

149

C₄²

#10625354

yes

characteristic

150

C₂³

—

188

—

151

C₂³

#13094007

yes

characteristic

152

Q₈

—

188

—

153

D₄

—

376

—

154

D₄

—

188

—

155

D₄

—

188

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

376

—

160

C₂×C₄

—

376

—

161

C₂×C₄

—

188

—

162

C₂×C₄

#13094000

yes

characteristic

163

C₂×C₄

#13094002

yes

—

164

C₂×C₄

#13094009

yes

—

165

C₂×C₄

#13094020

yes

characteristic

166

C₂×C₄

C₂×#10625336

yes

—

167

C₂×C₄

#13094050

yes

—

168

C₈

—

169

C₈

—

376

—

170

C₈

—

188

—

171

C₂²

—

172

C₂²

—

376

—

173

C₂²

—

376

—

174

C₂²

#15689058

yes

characteristic

175

C₂²

#15689441

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

376

—

180

C₄

#15689410

yes

—

181

C₄

#15689497

yes

characteristic

182

C₄

#15689510

yes

characteristic

183

C₂

—

184

C₂

—

752

—

185

C₂

#20244922

yes

characteristic

186

C₂

#20244944

yes

—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1843	C₂
socle	C₂²×C₄₇	#399
prime radical	#15689673	C₂²
hypercenter	#1843	D₄₇
hypocenter	C₄₇	#10666
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#13094020

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1843

C₂

C₂²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
2. #20242158	C₂	semidirect
4. #20241217	C₂	semidirect
5. C₄₇×#1843	C₂	semidirect
91. #1843	D₄₇	semidirect
108. C₄₇	#10666	semidirect

H ≅ G/N

type

2. #20242158

C₂

semidirect

4. #20241217

C₂

semidirect

5. C₄₇×#1843

C₂

semidirect

91. #1843

D₄₇

semidirect

108. C₄₇

#10666

semidirect

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

subgroup	index
#20244220	2
#20242158	2
#20241786	2
#20241217	2
C₄₇×#1843	2
#10666	47

subgroup

index

C₄₇×#1843

Subgroups — #26121290

Conjugacy classes of subgroups

Complements

Maximal subgroups