Group #26225159 — subgroups

Subgroups — #26225159

Order 12032.

662 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 255 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	#20253912	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	2	yes	maximal
3	#20244368	C₂	1	2	yes	maximal
4	#20244134	C₂	1	4	yes	maximal
5	#20243275	C₂	1	4	yes	maximal
6	#20243076	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	2	yes	—
9	#15689532	C₄	1	8	yes	complemented
10	#15689250	—	2	4	no	—
11	#15689247	—	2	4	no	—
12	#15689161	C₂²	1	4	yes	—
13	#15689151	C₂²	1	4	yes	—
14	#15689081	C₂²	1	4	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	#15688969	C₂²	1	1	yes	characteristic
19	#15688884	C₂²	1	8	yes	—
20	C₂×#108×C₄₇	C₂²	1	4	yes	—
21	C₄₇×#423	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
25	#13094024	C₂×C₄	1	4	yes	—
26	#13094016	C₂×C₄	1	4	yes	—
27	#13094003	—	2	4	no	—
28	C₄×Dic₉₄	—	2	4	no	—
29	#13093983	C₂³	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
31	#13093958	C₂×C₄	1	8	yes	—
32	#13093956	C₂×C₄	1	8	yes	—
33	Dic₄₇⋊C₈	—	2	8	no	—
34	#13093934	—	2	4	no	—
35	#13093934	Q₈	1	8	yes	complemented
36	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
37	C₈×Dic₄₇	—	2	4	no	—
38	C₈×Dic₄₇	—	2	4	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#108×C₄₇	—	2	8	no	—
44	C₄⋊C₈×C₄₇	—	2	4	no	—
45	#97×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	#41	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₇	—	2	2	no	—
49	C₂×Dic₉₄	—	2	4	no	—
50	M₁₆×C₄₇	—	2	8	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
54	C₂×C₈×C₄₇	#41	1	4	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²⋊Dic₄₇	—	2	4	no	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
60	Dic₄₇⋊C₄	—	2	8	no	—
61	C₄×Dic₄₇	—	2	4	no	—
62	C₄×Dic₄₇	—	2	2	no	—
63	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
68	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	4	2	no	—
70	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
75	C₂×Dic₄₇	—	2	8	no	—
76	C₂×Dic₄₇	—	2	4	no	—
77	C₂×Dic₄₇	—	4	4	no	—
78	Dic₉₄	—	4	8	no	—
79	C₈×C₄₇	—	4	4	no	—
80	C₈×C₄₇	C₄×Q₈	1	8	yes	—
81	C₄₇⋊C₈	—	2	2	no	—
82	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
83	#21737	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄₇	#533	1	2	yes	—
87	C₄×C₄₇	—	2	4	no	—
88	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
89	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₄×C₄₇	#532	1	1	yes	characteristic
91	Dic₄₇	—	4	4	no	—
92	Dic₄₇	—	4	8	no	—
93	#3162	—	47	1	no	—
94	#2960	—	47	2	no	—
95	#1995	—	47	2	no	—
96	#1945	—	47	4	no	—
97	#1849	D₄₇	1	1	yes	characteristic
98	#1773	—	47	4	no	—
99	#1726	—	47	1	no	—
100	C₂×C₄₇	—	2	2	no	—
101	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄₇	#2955	1	2	yes	—
103	#533	—	47	2	no	—
104	#501	—	47	8	no	—
105	#446	—	94	4	no	—
106	C₈×Q₈	—	94	4	no	—
107	C₂×#108	C₂×D₄₇	1	4	yes	—
108	#427	—	47	4	no	—
109	#424	—	47	4	no	—
110	#423	C₂×D₄₇	1	2	yes	—
111	#405	—	47	4	no	—
112	C₂×#97	—	47	2	no	—
113	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₄×C₈	—	47	1	no	—
115	C₂×C₄×C₈	—	47	2	no	—
116	#376	—	47	1	no	—
117	#339	—	47	8	no	—
118	C₄₇	#21737	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	—	47	1	no	—
122	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
123	C₄⋊Q₈	—	47	4	no	—
124	#125	—	47	4	no	—
125	C₂²⋊Q₈	—	94	4	no	—
126	C₄×Q₈	—	94	4	no	—
127	#117	—	47	4	no	—
128	C₂×C₄²	—	47	2	no	—
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	#108	—	2	8	no	—
131	#107	—	47	8	no	—
132	#106	—	47	8	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	94	8	no	—
135	#97	—	47	8	no	—
136	#97	—	94	4	no	—
137	#97	C₄×D₄₇	1	4	yes	—
138	C₄×C₈	—	47	4	no	—
139	C₄×C₈	—	94	4	no	—
140	C₄×C₈	—	94	4	no	—
141	#95	—	94	2	no	—
142	C₂×Q₈	—	94	4	no	—
143	C₂²×C₄	—	94	2	no	—
144	C₂²×C₄	#10625354	1	2	yes	—
145	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
146	M₁₆	—	2	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	47	2	no	—
149	C₂×C₈	—	47	4	no	—
150	C₂×C₈	—	47	2	no	—
151	C₂×C₈	—	94	2	no	—
152	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
153	C₂×C₈	#10625352	1	4	yes	—
154	C₂×C₈	Q₈×D₄₇	1	4	yes	—
155	C₄⋊C₄	—	47	8	no	—
156	C₄⋊C₄	—	94	8	no	—
157	C₄⋊C₄	—	94	4	no	—
158	C₂²⋊C₄	—	94	4	no	—
159	C₄²	—	47	4	no	—
160	C₄²	—	94	4	no	—
161	C₄²	—	94	2	no	—
162	C₄²	C₂×C₄×D₄₇	1	2	yes	—
163	C₄²	D₄×D₄₇	1	2	yes	—
164	C₂³	C₂×#10625354	1	1	yes	characteristic
165	Q₈	—	188	8	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	94	8	no	—
168	C₂×C₄	—	94	4	no	—
169	C₂×C₄	—	188	4	no	—
170	C₂×C₄	#13093986	1	4	yes	—
171	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
174	C₂×C₄	C₂×#10625352	1	2	yes	—
175	C₈	—	4	4	no	—
176	C₈	—	47	4	no	—
177	C₈	—	94	2	no	—
178	C₈	#13093989	1	8	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×#13093986	1	1	yes	characteristic
181	C₂²	#15689678	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	188	4	no	—
184	C₄	—	188	8	no	—
185	C₄	C₂×#13093989	1	2	yes	—
186	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
187	C₄	#15689671	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	#20245955	1	1	yes	characteristic
190	C₂	#20252290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253912

C₂

yes

maximal, characteristic

#20252318

C₂

yes

maximal

#20244368

C₂

yes

maximal

#20244134

C₂

yes

maximal

#20243275

C₂

yes

maximal

#20243076

C₂

yes

maximal, characteristic

C₄₇×#1849

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689532

C₄

yes

—

—

C₂²

yes

—

#15689151

C₂²

yes

—

#15689081

C₂²

yes

—

C₂×#13093934

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688969

C₂²

yes

characteristic

#15688884

C₂²

yes

—

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

#13094024

C₂×C₄

yes

—

#13094016

C₂×C₄

yes

—

#13094003

—

C₄×Dic₉₄

—

#13093983

C₂³

yes

—

C₂×C₄×Dic₄₇

C₂³

yes

—

#13093958

C₂×C₄

yes

—

#13093956

C₂×C₄

yes

—

Dic₄₇⋊C₈

—

#13093934

—

#13093934

Q₈

yes

complemented

C₄×C₄₇⋊C₈

D₄

yes

—

C₈×Dic₄₇

—

C₈×Dic₄₇

—

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂²×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₇

C₂×#41

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

#21737

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₇

#533

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄×Q₈

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₄₇

#532

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

#1773

—

#1726

—

100

C₂×C₄₇

—

101

C₂×C₄₇

C₄×C₄⋊Q₈

yes

characteristic

102

C₂×C₄₇

#2955

yes

—

103

#533

—

104

#501

—

105

#446

—

106

C₈×Q₈

—

107

C₂×#108

C₂×D₄₇

yes

—

108

#427

—

109

#424

—

110

#423

C₂×D₄₇

yes

—

111

#405

—

112

C₂×#97

—

113

C₂×#97

C₂×D₄₇

yes

characteristic

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

#376

—

117

#339

—

118

C₄₇

#21737

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

C₂²×D₄₇

yes

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²×D₄₇

yes

—

123

C₄⋊Q₈

—

124

#125

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

#108

—

131

#107

—

132

#106

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#97

—

136

#97

—

137

#97

C₄×D₄₇

yes

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

#95

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

#10625354

yes

—

145

C₂²×C₄

C₂³×D₄₇

yes

characteristic

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂×C₄×D₄₇

yes

—

153

C₂×C₈

#10625352

yes

—

154

C₂×C₈

Q₈×D₄₇

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂×C₄×D₄₇

yes

—

163

C₄²

D₄×D₄₇

yes

—

164

C₂³

C₂×#10625354

yes

characteristic

165

Q₈

—

188

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

188

—

170

C₂×C₄

#13093986

yes

—

171

C₂×C₄

C₂²×C₄×D₄₇

yes

characteristic

172

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

173

C₂×C₄

C₂×Q₈×D₄₇

yes

—

174

C₂×C₄

C₂×#10625352

yes

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

#13093989

yes

—

179

C₂²

—

180

C₂²

C₂×#13093986

yes

characteristic

181

C₂²

#15689678

yes

—

182

C₄

—

183

C₄

—

188

—

184

C₄

—

188

—

185

C₄

C₂×#13093989

yes

—

186

C₄

C₄⋊Q₈×D₄₇

yes

characteristic

187

C₄

#15689671

yes

characteristic

188

C₂

—

189

C₂

#20245955

yes

characteristic

190

C₂

#20252290

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#21737
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1849

C₂

socle

C₂²×C₄₇

C₂×C₄×D₄

prime radical

C₂³×C₄₇

C₂×#41

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
9. #15689532	C₄	semidirect
35. #13093934	Q₈	semidirect
118. C₄₇	#21737	semidirect

H ≅ G/N

type

9. #15689532

C₄

semidirect

35. #13093934

Q₈

semidirect

118. C₄₇

#21737

semidirect

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

subgroup	index
#20253912	2
#20252318	2
#20244368	2
#20244134	2
#20243275	2
#20243076	2
C₄₇×#1849	2
#21737	47

subgroup

index

C₄₇×#1849

Subgroups — #26225159

Conjugacy classes of subgroups

Complements

Maximal subgroups