Group #26115775 — subgroups

Subgroups — #26115775

Order 12032.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20243526	C₂	1	1	yes	maximal, characteristic
2	#20242073	C₂	1	2	yes	maximal
3	#20242069	C₂	1	2	yes	maximal
4	#20241680	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	1	yes	characteristic
8	#15689147	C₂²	1	2	yes	—
9	#15688888	—	2	2	no	—
10	#15688872	—	2	4	no	—
11	#15688869	—	2	4	no	—
12	#15688845	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#500	—	2	2	no	—
15	C₄₇×#500	C₄	1	4	yes	—
16	C₂×#107×C₄₇	—	2	2	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#10625329	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₇	—	2	2	no	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	—	2	2	no	—
32	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#107×C₄₇	—	2	8	no	—
37	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₇	D₄	1	2	yes	—
39	C₂×Q₈×C₄₇	—	2	4	no	—
40	C₂×Q₈×C₄₇	—	4	2	no	—
41	C₂×Q₈×C₄₇	—	8	4	no	—
42	C₂²×C₄×C₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
48	C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	4	2	no	—
52	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
56	#10625329	—	4	4	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₂×C₄₇⋊C₈	—	2	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂³×C₄₇	#124	1	1	yes	characteristic
61	Q₈×C₄₇	—	8	4	no	—
62	Q₈×C₄₇	—	8	4	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	4	4	no	—
66	C₂×C₄×C₄₇	—	8	4	no	—
67	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#137	1	4	yes	—
74	C₈×C₄₇	—	2	4	no	—
75	C₄₇⋊C₈	—	4	2	no	—
76	C₄₇⋊C₈	—	4	2	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	#9778	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	1	no	—
80	C₂²×C₄₇	—	2	1	no	—
81	C₂²×C₄₇	#390	1	1	yes	characteristic
82	C₂²×C₄₇	#491	1	2	yes	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	8	4	no	—
86	C₄×C₄₇	#419	1	2	yes	—
87	C₄×C₄₇	#472	1	2	yes	—
88	C₄×C₄₇	#473	1	2	yes	—
89	C₄×C₄₇	#497	1	2	yes	—
90	C₂×#500	D₄₇	1	1	yes	characteristic
91	#1837	—	47	1	no	—
92	#1549	—	47	2	no	—
93	#1548	—	47	2	no	—
94	#1452	—	47	1	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	#1960	1	1	yes	characteristic
97	C₂×C₄₇	#1978	1	1	yes	characteristic
98	C₂×C₄₇	#1979	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
100	#500	—	2	2	no	—
101	#500	Dic₄₇	1	4	yes	—
102	#431	—	47	1	no	—
103	#431	—	47	1	no	—
104	C₂×#107	—	2	2	no	—
105	#423	—	47	2	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	#341	—	94	2	no	—
108	#333	—	94	4	no	—
109	#332	—	94	4	no	—
110	#322	—	94	2	no	—
111	C₄₇	#9778	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂×M₁₆	—	94	2	no	—
114	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	#107	—	2	8	no	—
123	C₄⋊C₈	—	47	2	no	—
124	C₄⋊C₈	—	47	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄⋊C₈	—	94	2	no	—
127	C₂²⋊C₈	—	94	1	no	—
128	C₂²⋊C₈	—	94	1	no	—
129	C₄×C₈	—	94	1	no	—
130	C₄×C₈	—	94	1	no	—
131	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	#10625352	1	1	yes	characteristic
138	C₂²×C₄	#10625352	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	M₁₆	—	188	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	D₄×D₄₇	1	1	yes	characteristic
151	C₄²	D₄×D₄₇	1	1	yes	characteristic
152	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
154	C₂³	#13094011	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	#13093995	1	1	yes	characteristic
165	C₂×C₄	#13093995	1	1	yes	characteristic
166	C₂×C₄	#13094020	1	1	yes	characteristic
167	C₂×C₄	#13094055	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	376	2	no	—
170	C₈	—	188	2	no	—
171	C₈	—	188	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#15689026	1	1	yes	characteristic
175	C₂²	#15689491	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#15689132	1	2	yes	—
180	C₄	#15689390	1	2	yes	—
181	C₄	#15689395	1	2	yes	—
182	C₄	#15689515	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#20244209	1	1	yes	characteristic
185	C₂	#20244299	1	1	yes	characteristic
186	C₂	#20244304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243526

C₂

yes

maximal, characteristic

#20242073

C₂

yes

maximal

#20242069

C₂

yes

maximal

#20241680

C₂

yes

maximal, characteristic

C₂×C₄₇×#500

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

characteristic

#15689176

C₂²

yes

characteristic

#15689147

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#500

—

C₄₇×#500

C₄

yes

—

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9778

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#491

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#419

yes

—

C₄×C₄₇

#472

yes

—

C₄×C₄₇

#473

yes

—

C₄×C₄₇

#497

yes

—

C₂×#500

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

100

#500

—

101

#500

Dic₄₇

yes

—

102

#431

—

103

#431

—

104

C₂×#107

—

105

#423

—

106

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

107

#341

—

108

#333

—

109

#332

—

110

#322

—

111

C₄₇

#9778

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×M₁₆

—

114

C₂²×C₈

C₂²⋊D₄₇

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×Dic₄₇

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₄₇

yes

characteristic

122

#107

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

C₂×Dic₄₇

yes

—

132

C₄×C₈

C₂²⋊D₄₇

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

#10625352

yes

characteristic

138

C₂²×C₄

#10625352

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

140

M₁₆

—

188

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

188

—

145

C₂×C₈

C₂²⋊Dic₄₇

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

D₄×D₄₇

yes

characteristic

151

C₄²

D₄×D₄₇

yes

characteristic

152

C₄²

C₂²×Dic₄₇

yes

characteristic

153

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

154

C₂³

#13094011

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

162

C₂×C₄

D₄×Dic₄₇

yes

characteristic

163

C₂×C₄

D₄×Dic₄₇

yes

characteristic

164

C₂×C₄

#13093995

yes

characteristic

165

C₂×C₄

#13093995

yes

characteristic

166

C₂×C₄

#13094020

yes

characteristic

167

C₂×C₄

#13094055

yes

—

168

C₈

—

169

C₈

—

376

—

170

C₈

—

188

—

171

C₈

—

188

—

172

C₂²

—

173

C₂²

—

174

C₂²

#15689026

yes

characteristic

175

C₂²

#15689491

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#15689132

yes

—

180

C₄

#15689390

yes

—

181

C₄

#15689395

yes

—

182

C₄

#15689515

yes

—

183

C₂

—

184

C₂

#20244209

yes

characteristic

185

C₂

#20244299

yes

characteristic

186

C₂

#20244304

yes

characteristic

	subgroup	G/H
center	C₂²	#15689026
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#9778
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#15689026

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#500

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
111. C₄₇	#9778	semidirect

H ≅ G/N

type

111. C₄₇

#9778

semidirect

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

subgroup	index
#20243526	2
#20242073	2
#20242069	2
#20241680	2
C₂×C₄₇×#500	2
#9778	47

subgroup

index

C₂×C₄₇×#500

Subgroups — #26115775

Conjugacy classes of subgroups

Complements

Maximal subgroups