Group #26115042 — subgroups

Subgroups — #26115042

Order 12032.

260 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20242177	C₂	1	2	yes	maximal
2	#20242144	C₂	1	1	yes	maximal, characteristic
3	#20241677	C₂	1	1	yes	maximal, characteristic
4	#20241279	C₂	1	2	yes	maximal
5	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	#15689147	C₂²	1	2	yes	—
8	C₄×#10625329	C₂²	1	1	yes	characteristic
9	#15688876	—	2	2	no	—
10	#15688874	—	2	1	no	—
11	#15688874	—	2	1	no	—
12	#15688874	—	2	2	no	—
13	#15688845	—	2	1	no	—
14	#15688844	—	2	1	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#335	C₄	1	2	yes	—
18	C₄₇×#334	C₄	1	2	yes	—
19	C₂×#10625329	—	2	1	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
25	#13093935	Q₈	1	2	yes	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	4	1	no	—
44	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	#10625329	—	4	2	no	—
53	#10625329	—	8	2	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂³×C₄₇	#125	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	—	2	1	no	—
71	C₈×C₄₇	—	4	4	no	—
72	C₈×C₄₇	#108	1	2	yes	—
73	C₄₇⋊C₈	—	4	2	no	—
74	C₄₇⋊C₈	—	4	2	no	—
75	C₄₇⋊C₈	—	8	2	no	—
76	#9635	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	—	2	1	no	—
80	C₂²×C₄₇	#389	1	1	yes	characteristic
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
85	C₄×C₄₇	#430	1	1	yes	characteristic
86	C₄×C₄₇	#444	1	2	yes	—
87	C₄×C₄₇	#471	1	2	yes	—
88	#1574	—	47	2	no	—
89	#1566	—	47	1	no	—
90	#1564	D₄₇	1	1	yes	characteristic
91	#1451	—	47	1	no	—
92	#1299	—	47	2	no	—
93	C₂×C₄₇	—	4	1	no	—
94	C₂×C₄₇	#1946	1	1	yes	characteristic
95	C₂×C₄₇	#1950	1	1	yes	characteristic
96	C₂×C₄₇	#1951	1	1	yes	characteristic
97	#431	—	47	1	no	—
98	#431	C₂×D₄₇	1	2	yes	—
99	#423	—	47	2	no	—
100	C₄×M₁₆	—	47	1	no	—
101	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	#335	—	94	2	no	—
103	#335	Dic₄₇	1	2	yes	—
104	#334	—	94	2	no	—
105	#334	—	94	1	no	—
106	#334	—	94	1	no	—
107	#334	Dic₄₇	1	2	yes	—
108	#322	—	94	1	no	—
109	#322	—	94	1	no	—
110	C₄₇	#9635	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₁₆	—	94	1	no	—
112	C₂×M₁₆	—	94	2	no	—
113	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
114	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	C₄⋊C₈	—	47	2	no	—
117	C₄⋊C₈	—	94	2	no	—
118	C₄⋊C₈	—	94	2	no	—
119	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	94	1	no	—
122	#97	—	47	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	94	1	no	—
125	C₄×C₈	—	94	1	no	—
126	C₄×C₈	D₁₈₈	1	1	yes	characteristic
127	C₄×C₈	D₁₈₈	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
131	C₂²×C₄	#10625352	1	1	yes	characteristic
132	C₂²×C₄	#10625354	1	1	yes	characteristic
133	M₁₆	—	376	2	no	—
134	M₁₆	—	188	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	94	2	no	—
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	188	2	no	—
141	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
145	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
146	C₄²	D₄×D₄₇	1	1	yes	characteristic
147	C₄²	Q₈×D₄₇	1	1	yes	characteristic
148	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
149	C₂³	#13094015	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
156	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
157	C₂×C₄	#13093999	1	1	yes	characteristic
158	C₂×C₄	#13094004	1	1	yes	characteristic
159	C₂×C₄	#13094023	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	4	no	—
164	C₈	—	376	2	no	—
165	C₈	—	188	2	no	—
166	C₈	—	188	2	no	—
167	C₈	#13093960	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#15689022	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#13093960	1	1	yes	characteristic
176	C₄	#15689172	1	1	yes	characteristic
177	C₄	#15689245	1	2	yes	—
178	C₄	#15689387	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#20244142	1	1	yes	characteristic
181	C₂	#20244164	1	1	yes	characteristic
182	C₂	#20244169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242177

C₂

yes

maximal

#20242144

C₂

yes

maximal, characteristic

#20241677

C₂

yes

maximal, characteristic

#20241279

C₂

yes

maximal

C₄₇×#1564

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

characteristic

#15689147

C₂²

yes

—

C₄×#10625329

C₂²

yes

—

—

—

—

—

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

C₄

yes

—

C₄₇×#334

C₄

yes

—

C₂×#10625329

—

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

Q₈

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

C₂×C₄

yes

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#108

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9635

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#389

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#471

yes

—

#1574

—

#1566

—

#1564

D₄₇

yes

characteristic

#1451

—

#1299

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

#423

—

100

C₄×M₁₆

—

101

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

102

#335

—

103

#335

Dic₄₇

yes

—

104

#334

—

105

#334

—

106

#334

—

107

#334

Dic₄₇

yes

—

108

#322

—

109

#322

—

110

C₄₇

#9635

yes

characteristic, complemented, Sylow

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂²×C₈

Dic₉₄

yes

characteristic

114

C₂²×C₈

Dic₉₄

yes

characteristic

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

C₂×Dic₄₇

yes

—

120

C₂²⋊C₈

—

121

C₂²⋊C₈

—

122

#97

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

D₁₈₈

yes

characteristic

127

C₄×C₈

D₁₈₈

yes

characteristic

128

C₄×C₈

C₂×Dic₄₇

yes

characteristic

129

C₄×C₈

C₂×Dic₄₇

yes

characteristic

130

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

131

C₂²×C₄

#10625352

yes

characteristic

132

C₂²×C₄

#10625354

yes

characteristic

133

M₁₆

—

376

—

134

M₁₆

—

188

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

188

—

141

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

142

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

143

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

144

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

145

C₄²

C₂×D₁₈₈

yes

characteristic

146

C₄²

D₄×D₄₇

yes

characteristic

147

C₄²

Q₈×D₄₇

yes

characteristic

148

C₄²

C₂²×Dic₄₇

yes

characteristic

149

C₂³

#13094015

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

155

C₂×C₄

D₄×Dic₄₇

yes

characteristic

156

C₂×C₄

Q₈×Dic₄₇

yes

characteristic

157

C₂×C₄

#13093999

yes

characteristic

158

C₂×C₄

#13094004

yes

characteristic

159

C₂×C₄

#13094023

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

376

—

165

C₈

—

188

—

166

C₈

—

188

—

167

C₈

#13093960

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#15689022

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#13093960

yes

characteristic

176

C₄

#15689172

yes

characteristic

177

C₄

#15689245

yes

—

178

C₄

#15689387

yes

—

179

C₂

—

180

C₂

#20244142

yes

characteristic

181

C₂

#20244164

yes

characteristic

182

C₂

#20244169

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	#125
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#9635
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#13093999

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1564

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
110. C₄₇	#9635	semidirect

H ≅ G/N

type

110. C₄₇

#9635

semidirect

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

subgroup	index
#20242177	2
#20242144	2
#20241677	2
#20241279	2
C₄₇×#1564	2
#9635	47

subgroup

index

C₄₇×#1564

Subgroups — #26115042

Conjugacy classes of subgroups

Complements

Maximal subgroups