Group #26109307 — subgroups

Subgroups — #26109307

Order 12032.

336 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20252318	C₂	1	1	yes	maximal, characteristic
2	#20242146	C₂	1	2	yes	maximal
3	#20242125	C₂	1	2	yes	maximal
4	C₄₇⋊#1309	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	1	yes	characteristic
7	#15689250	—	2	2	no	—
8	#15689247	—	2	2	no	—
9	#15689151	C₄	1	2	yes	—
10	#15689147	C₂²	1	2	yes	—
11	#15689081	C₄	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688875	—	2	2	no	—
14	#15688875	—	2	2	no	—
15	#15688870	—	2	2	no	—
16	#15688868	—	2	2	no	—
17	C₄₇×#423	C₂²	1	2	yes	—
18	C₄₇×#423	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	#13094024	D₄	1	2	yes	—
23	#13094016	Q₈	1	2	yes	—
24	#13094003	—	2	2	no	—
25	C₄×Dic₉₄	—	2	2	no	—
26	#13093983	C₂×C₄	1	2	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
29	Dic₄₇⋊C₈	—	2	4	no	—
30	#13093934	—	2	2	no	—
31	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
32	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
33	C₈×Dic₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	—	2	2	no	—
35	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	—	2	2	no	—
38	C₄⋊C₈×C₄₇	—	2	2	no	—
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	#41	1	2	yes	—
42	C₂×Dic₉₄	—	2	2	no	—
43	M₁₆×C₄₇	—	4	2	no	—
44	M₁₆×C₄₇	—	8	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	4	2	no	—
47	C₄²×C₄₇	C₂×D₄	1	2	yes	—
48	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²⋊Dic₄₇	—	2	2	no	—
51	C₄⋊Dic₄₇	—	2	2	no	—
52	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
53	Dic₄₇⋊C₄	—	2	4	no	—
54	C₄×Dic₄₇	—	2	2	no	—
55	#10625329	—	8	2	no	—
56	C₂×C₄₇⋊C₈	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₄₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#121	1	2	yes	—
70	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₄₇	—	2	4	no	—
72	Dic₉₄	—	4	2	no	—
73	Dic₉₄	—	4	2	no	—
74	C₈×C₄₇	—	4	2	no	—
75	C₈×C₄₇	—	8	2	no	—
76	C₄₇⋊C₈	—	2	1	no	—
77	C₄₇⋊C₈	—	2	2	no	—
78	C₄₇⋊C₈	—	8	2	no	—
79	C₄₇⋊C₈	#101	1	2	yes	—
80	#8639	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	1	no	—
82	C₂²×C₄₇	—	2	2	no	—
83	C₂²×C₄₇	#380	1	1	yes	characteristic
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
87	C₄×C₄₇	#413	1	1	yes	characteristic
88	C₄×C₄₇	#472	1	2	yes	—
89	C₄×C₄₇	#473	1	2	yes	—
90	Dic₄₇	—	4	4	no	—
91	#2960	—	47	1	no	—
92	#1567	—	47	2	no	—
93	#1561	—	47	2	no	—
94	#1309	—	47	1	no	—
95	#1309	D₄₇	1	1	yes	characteristic
96	C₂×C₄₇	—	4	1	no	—
97	C₂×C₄₇	#1801	1	1	yes	characteristic
98	C₂×C₄₇	#1816	1	2	yes	—
99	#533	—	47	1	no	—
100	#446	—	94	2	no	—
101	C₈×Q₈	—	94	2	no	—
102	#424	—	47	2	no	—
103	#423	—	47	2	no	—
104	#423	C₂×D₄₇	1	2	yes	—
105	#423	C₂×D₄₇	1	1	yes	characteristic
106	#405	—	47	2	no	—
107	C₂×C₄×C₈	—	47	1	no	—
108	#335	—	94	2	no	—
109	#335	—	94	2	no	—
110	#332	—	94	2	no	—
111	#332	—	94	2	no	—
112	C₄₇	#8639	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	94	2	no	—
115	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
116	C₂²×C₈	—	47	1	no	—
117	C₂²×C₈	—	47	1	no	—
118	C₄⋊Q₈	—	47	2	no	—
119	#125	—	47	2	no	—
120	C₂²⋊Q₈	—	94	2	no	—
121	C₄×Q₈	—	94	2	no	—
122	#117	—	47	2	no	—
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	94	4	no	—
128	C₄⋊C₈	—	94	2	no	—
129	C₄⋊C₈	—	94	2	no	—
130	#97	—	94	2	no	—
131	C₄×C₈	—	47	2	no	—
132	C₄×C₈	—	47	2	no	—
133	C₄×C₈	—	94	2	no	—
134	C₂×Q₈	—	94	2	no	—
135	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	#10625350	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	8	2	no	—
139	M₁₆	—	376	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	47	1	no	—
143	C₂×C₈	—	47	1	no	—
144	C₂×C₈	—	47	2	no	—
145	C₂×C₈	—	94	2	no	—
146	C₂×C₈	—	188	1	no	—
147	C₂×C₈	—	188	2	no	—
148	C₄⋊C₄	—	47	4	no	—
149	C₄⋊C₄	—	94	2	no	—
150	C₄⋊C₄	—	94	4	no	—
151	C₂²⋊C₄	—	94	2	no	—
152	C₄²	—	94	2	no	—
153	C₄²	D₄×D₄₇	1	2	yes	—
154	C₄²	D₄×D₄₇	1	1	yes	characteristic
155	C₄²	Q₈×D₄₇	1	1	yes	characteristic
156	C₂³	#13093981	1	1	yes	characteristic
157	Q₈	—	188	2	no	—
158	Q₈	—	188	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
166	C₂×C₄	#13093997	1	2	yes	—
167	C₂×C₄	#13094002	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	47	2	no	—
171	C₈	—	94	1	no	—
172	C₈	—	94	2	no	—
173	C₈	—	376	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#15688982	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	188	4	no	—
180	C₄	#101×D₄₇	1	1	yes	characteristic
181	C₄	#15689104	1	1	yes	characteristic
182	C₄	#15689392	1	2	yes	—
183	C₄	#15689397	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#20243383	1	1	yes	characteristic
186	C₂	#20243451	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252318

C₂

yes

maximal, characteristic

#20242146

C₂

yes

maximal

#20242125

C₂

yes

maximal

C₄₇⋊#1309

C₂

yes

maximal, characteristic

C₄₇×#1309

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

—

C₄

yes

—

#15689147

C₂²

yes

—

#15689081

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

—

—

—

C₄₇×#423

C₂²

yes

—

C₄₇×#423

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

#13094024

D₄

yes

—

#13094016

Q₈

yes

—

#13094003

—

C₄×Dic₉₄

—

#13093983

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093934

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₈×Dic₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₄⋊C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#101

yes

—

#8639

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#380

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#473

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1801

yes

characteristic

C₂×C₄₇

#1816

yes

—

#533

—

100

#446

—

101

C₈×Q₈

—

102

#424

—

103

#423

—

104

#423

C₂×D₄₇

yes

—

105

#423

C₂×D₄₇

yes

characteristic

106

#405

—

107

C₂×C₄×C₈

—

108

#335

—

109

#335

—

110

#332

—

111

#332

—

112

C₄₇

#8639

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₄×D₄₇

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

121

C₄×Q₈

—

122

#117

—

123

C₂×C₄²

C₂²×D₄₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#97

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂²×C₄

#10625350

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

376

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

188

—

147

C₂×C₈

—

188

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

D₄×D₄₇

yes

—

154

C₄²

D₄×D₄₇

yes

characteristic

155

C₄²

Q₈×D₄₇

yes

characteristic

156

C₂³

#13093981

yes

characteristic

157

Q₈

—

188

—

158

Q₈

—

188

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂²⋊C₄×D₄₇

yes

characteristic

165

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

166

C₂×C₄

#13093997

yes

—

167

C₂×C₄

#13094002

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

376

—

174

C₂²

—

175

C₂²

—

176

C₂²

#15688982

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

188

—

180

C₄

#101×D₄₇

yes

characteristic

181

C₄

#15689104

yes

characteristic

182

C₄

#15689392

yes

—

183

C₄

#15689397

yes

—

184

C₂

—

185

C₂

#20243383

yes

characteristic

186

C₂

#20243451

yes

—

	subgroup	G/H
center	C₂²	#15688982
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1309	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1309	D₄₇
hypocenter	C₄₇	#8639
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#15688982

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1309

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
112. C₄₇	#8639	semidirect

H ≅ G/N

type

112. C₄₇

#8639

semidirect

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

subgroup	index
#20252318	2
#20242146	2
#20242125	2
C₄₇⋊#1309	2
C₄₇×#1309	2
#8639	47

subgroup

index

#20252318

#20242146

#20242125

C₄₇⋊#1309

C₄₇×#1309

#8639

Subgroups — #26109307

Conjugacy classes of subgroups

Complements

Maximal subgroups