Group #26102880 — subgroups

Subgroups — #26102880

Order 12032.

352 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20243368	C₂	1	2	yes	maximal
2	C₄₇×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
5	#15688878	C₄	1	4	yes	—
6	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
11	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	—	2	2	no	—
17	C₂×C₄²×C₄₇	—	4	2	no	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₄₇	D₄	1	2	yes	—
20	C₂²⋊C₈×C₄₇	—	4	2	no	—
21	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	—	2	4	no	—
24	C₂²×C₄×C₄₇	—	4	1	no	—
25	C₂²×C₄×C₄₇	—	4	1	no	—
26	C₂²×C₄×C₄₇	—	4	2	no	—
27	C₂²×C₄×C₄₇	—	4	1	no	—
28	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	4	4	no	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	—	2	4	no	—
38	C₄²×C₄₇	—	4	2	no	—
39	C₂×C₄₇⋊C₈	—	4	4	no	—
40	C₂×C₄₇⋊C₈	—	4	4	no	—
41	C₂×C₄₇⋊C₈	—	4	4	no	—
42	C₂³×C₄₇	—	2	1	no	—
43	C₂³×C₄₇	—	2	2	no	—
44	C₂³×C₄₇	—	2	1	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	#95	1	1	yes	characteristic
47	C₂³×C₄₇	#106	1	1	yes	characteristic
48	C₂³×C₄₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	4	1	no	—
59	C₂×C₄×C₄₇	—	4	4	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	#99	1	2	yes	—
62	C₂×C₄×C₄₇	#102	1	2	yes	—
63	C₂×C₄×C₄₇	#103	1	2	yes	—
64	C₂×C₄×C₄₇	#108	1	4	yes	—
65	C₈×C₄₇	—	8	2	no	—
66	C₄₇⋊C₈	—	8	4	no	—
67	#7044	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	1	no	—
74	C₂²×C₄₇	—	4	1	no	—
75	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
76	C₂²×C₄₇	#340	1	1	yes	characteristic
77	C₂²×C₄₇	#341	1	2	yes	—
78	C₂²×C₄₇	#342	1	1	yes	characteristic
79	C₂²×C₄₇	#344	1	2	yes	—
80	C₄×C₄₇	—	2	2	no	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	2	4	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	4	1	no	—
85	#1817	D₄₇	1	1	yes	characteristic
86	#1797	—	47	2	no	—
87	C₂×C₄₇	—	4	1	no	—
88	C₂×C₄₇	—	4	1	no	—
89	C₂×C₄₇	#1287	1	2	yes	—
90	C₂×C₄₇	#1289	1	1	yes	characteristic
91	C₂×C₄₇	#1298	1	1	yes	characteristic
92	C₂×C₄₇	#1301	1	2	yes	—
93	C₂×C₄₇	#1306	1	1	yes	characteristic
94	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	94	2	no	—
96	C₂×C₂²⋊C₈	—	94	2	no	—
97	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
98	#336	—	2	2	no	—
99	#336	—	47	4	no	—
100	C₄₇	#7044	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
102	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
103	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	94	4	no	—
106	C₂×C₄²	—	4	2	no	—
107	C₂×C₄²	C₄×D₄₇	1	2	yes	—
108	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
109	C₄⋊C₈	—	188	4	no	—
110	C₂²⋊C₈	—	4	2	no	—
111	C₂²⋊C₈	—	188	2	no	—
112	C₂²⋊C₈	—	188	2	no	—
113	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	#10625325	1	2	yes	—
121	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
122	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
123	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
124	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
125	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
126	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	188	4	no	—
130	C₂×C₈	—	188	4	no	—
131	C₂×C₈	—	188	4	no	—
132	C₄²	—	2	4	no	—
133	C₄²	—	4	2	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	1	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₄₇⋊#95	1	1	yes	characteristic
139	C₂³	#13093956	1	1	yes	characteristic
140	C₂³	#13093958	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	#13093939	1	2	yes	—
154	C₂×C₄	#13093946	1	2	yes	—
155	C₂×C₄	#13093949	1	2	yes	—
156	C₂×C₄	#13093959	1	4	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	376	4	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#15688880	1	1	yes	characteristic
167	C₂²	#15688886	1	1	yes	characteristic
168	C₂²	#15688887	1	2	yes	—
169	C₂²	#15688890	1	1	yes	characteristic
170	C₂²	#15688893	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	4	1	no	—
178	C₂	#20241251	1	2	yes	—
179	C₂	#20241256	1	1	yes	characteristic
180	C₂	#20241275	1	1	yes	characteristic
181	C₂	#20241284	1	2	yes	—
182	C₂	#20241296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243368

C₂

yes

maximal

C₄₇×#1817

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₂²⋊(C₄₇⋊C₈)

—

#15688878

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₄₇

C₄

yes

—

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂³×C₄×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

yes

C₂³×C₄₇

yes

C₂³×C₄₇

yes

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#99

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7044

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#342

yes

characteristic

C₂²×C₄₇

#344

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1817

D₄₇

yes

characteristic

#1797

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1287

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1306

yes

characteristic

C₂²×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

Dic₄₇

yes

—

#336

—

#336

—

100

C₄₇

#7044

yes

characteristic, complemented, Sylow

101

C₂³×C₄

Dic₉₄

yes

characteristic

102

C₂³×C₄

D₁₈₈

yes

characteristic

103

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₄²

C₄×D₄₇

yes

—

108

C₂×C₄²

C₂²⋊D₄₇

yes

—

109

C₄⋊C₈

—

188

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

—

188

—

112

C₂²⋊C₈

—

188

—

113

C₂⁴

C₄⋊Dic₄₇

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

#10625325

yes

—

121

C₂²×C₄

D₃₇₆

yes

characteristic

122

C₂²×C₄

Dic₁₈₈

yes

characteristic

123

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

124

C₂²×C₄

Dic₄₇⋊C₄

yes

—

125

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

126

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

188

—

130

C₂×C₈

—

188

—

131

C₂×C₈

—

188

—

132

C₄²

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₄₇⋊#95

yes

characteristic

139

C₂³

#13093956

yes

characteristic

140

C₂³

#13093958

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#13093939

yes

—

154

C₂×C₄

#13093946

yes

—

155

C₂×C₄

#13093949

yes

—

156

C₂×C₄

#13093959

yes

—

157

C₈

—

158

C₈

—

376

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#15688880

yes

characteristic

167

C₂²

#15688886

yes

characteristic

168

C₂²

#15688887

yes

—

169

C₂²

#15688890

yes

characteristic

170

C₂²

#15688893

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₂

—

177

C₂

—

178

C₂

#20241251

yes

—

179

C₂

#20241256

yes

characteristic

180

C₂

#20241275

yes

characteristic

181

C₂

#20241284

yes

—

182

C₂

#20241296

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄₇×#1817	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	#1817	D₄₇
hypocenter	C₄₇	#7044
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₂²×C₄²

C₂×D₄₇

Fitting

C₄₇×#1817

C₂

socle

C₂³×C₄₇

#95

prime radical

C₂⁴×C₄₇

C₄⋊C₄

cosocle

C₂²×C₄²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
100. C₄₇	#7044	semidirect

H ≅ G/N

type

100. C₄₇

#7044

semidirect

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

subgroup	index
#20243368	2
C₄₇×#1817	2
#7044	47

subgroup

index

#20243368

C₄₇×#1817

#7044

Subgroups — #26102880

Conjugacy classes of subgroups

Complements

Maximal subgroups