Group #26107437 — subgroups

Subgroups — #26107437

Order 12032.

414 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241671	C₂	1	1	yes	maximal, characteristic
4	#20241220	C₂	1	2	yes	maximal
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
10	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688875	C₄	1	4	yes	complemented
12	#15688873	C₄	1	4	yes	complemented
13	#15688844	—	2	1	no	—
14	C₈×C₄₇⋊C₈	—	2	1	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#322	—	2	1	no	—
18	C₈²×C₄₇	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	—	2	2	no	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₈×Dic₄₇	—	2	4	no	—
29	C₈×Dic₄₇	C₈	1	8	yes	complemented
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	2	yes	—
40	C₂²×Dic₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
46	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₄₇	—	2	8	no	—
50	C₄×Dic₄₇	—	2	2	no	—
51	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
55	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
56	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
57	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
58	C₂³×C₄₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₄₇	—	2	8	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₈×C₄₇	—	4	4	no	—
69	C₈×C₄₇	C₄⋊C₈	1	4	yes	—
70	C₈×C₄₇	C₄⋊C₈	1	4	yes	—
71	C₄₇⋊C₈	—	2	2	no	—
72	C₄₇⋊C₈	—	4	4	no	—
73	C₄₇⋊C₈	C₄×C₈	1	2	yes	—
74	C₄₇⋊C₈	C₄×C₈	1	2	yes	—
75	#8266	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	#431	1	2	yes	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
83	C₄×C₄₇	#443	1	4	yes	—
84	Dic₄₇	—	2	8	no	—
85	C₄²×C₈	—	47	1	no	—
86	#1564	—	47	2	no	—
87	#1450	—	47	1	no	—
88	#1450	D₄₇	1	1	yes	characteristic
89	#1276	—	47	2	no	—
90	C₂×C₄₇	—	2	2	no	—
91	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄₇	#1772	1	2	yes	—
93	#431	—	47	2	no	—
94	#431	C₂×D₄₇	1	2	yes	—
95	C₂×C₄×C₈	—	47	2	no	—
96	C₂×C₄×C₈	—	47	1	no	—
97	C₂×C₄×C₈	—	47	2	no	—
98	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₄³	—	47	1	no	—
100	#335	—	47	4	no	—
101	#334	—	47	4	no	—
102	#322	—	2	1	no	—
103	#322	—	94	1	no	—
104	C₈²	—	2	1	no	—
105	C₈²	—	94	1	no	—
106	C₄₇	#8266	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	47	1	no	—
108	C₂²×C₈	—	47	1	no	—
109	C₂²×C₈	C₄×D₄₇	1	2	yes	—
110	C₂×C₄²	—	47	2	no	—
111	C₂×C₄²	—	94	2	no	—
112	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
113	C₄⋊C₈	—	47	4	no	—
114	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	94	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	47	8	no	—
119	C₄×C₈	—	47	2	no	—
120	C₄×C₈	—	47	2	no	—
121	C₄×C₈	—	94	4	no	—
122	C₄×C₈	—	94	2	no	—
123	C₄×C₈	Dic₉₄	1	2	yes	—
124	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂²×C₄	—	94	2	no	—
127	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
128	C₂²×C₄	#10625350	1	2	yes	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	47	1	no	—
131	C₂×C₈	—	47	1	no	—
132	C₂×C₈	—	47	4	no	—
133	C₂×C₈	—	47	2	no	—
134	C₂×C₈	—	94	2	no	—
135	C₂×C₈	—	94	4	no	—
136	C₂×C₈	C₈×D₄₇	1	4	yes	—
137	C₂×C₈	#10625324	1	4	yes	—
138	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
139	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
140	C₄²	—	47	4	no	—
141	C₄²	—	94	8	no	—
142	C₄²	—	94	2	no	—
143	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
144	C₄²	C₂×C₄×D₄₇	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₂³	#13093981	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	94	8	no	—
149	C₂×C₄	—	94	4	no	—
150	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
153	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
155	C₂×C₄	C₂×#10625324	1	2	yes	—
156	C₈	—	4	4	no	—
157	C₈	—	47	2	no	—
158	C₈	—	47	2	no	—
159	C₈	—	94	2	no	—
160	C₈	—	188	4	no	—
161	C₈	Dic₄₇⋊C₈	1	4	yes	—
162	C₈	Dic₄₇⋊C₈	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
165	C₂²	#15689174	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	94	8	no	—
168	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
169	C₄	C₄×#10625324	1	1	yes	characteristic
170	C₄	C₂×Dic₄₇⋊C₈	1	2	yes	—
171	C₄	#15689238	1	4	yes	—
172	C₂	—	2	2	no	—
173	C₂	C₄×Dic₄₇⋊C₈	1	1	yes	characteristic
174	C₂	#20243268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×Dic₄₇

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241671

C₂

yes

maximal, characteristic

#20241220

C₂

yes

maximal

C₄₇×#1450

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₇

C₄

yes

—

C₂×C₈×Dic₄₇

C₄

yes

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688875

C₄

yes

complemented

#15688873

C₄

yes

complemented

#15688844

—

C₈×C₄₇⋊C₈

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₈²×C₄₇

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₈×Dic₄₇

—

C₈×Dic₄₇

C₈

yes

complemented

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

C₂×C₄

yes

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄²

yes

—

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

C₄⋊C₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄×C₈

yes

—

C₄₇⋊C₈

C₄×C₈

yes

—

#8266

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄×C₈

yes

characteristic

C₄×C₄₇

C₄×M₁₆

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₈

yes

—

C₄×C₄₇

#443

yes

—

Dic₄₇

—

C₄²×C₈

—

#1564

—

#1450

—

#1450

D₄₇

yes

characteristic

#1276

—

C₂×C₄₇

—

C₂×C₄₇

C₄×C₄⋊C₈

yes

characteristic

C₂×C₄₇

#1772

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄³

—

100

#335

—

101

#334

—

102

#322

—

103

#322

—

104

C₈²

—

105

C₈²

—

106

C₄₇

#8266

yes

characteristic, complemented, Sylow

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₄×D₄₇

yes

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂²×D₄₇

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

C₄×D₄₇

yes

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

Dic₉₄

yes

—

124

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

125

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

126

C₂²×C₄

—

127

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

128

C₂²×C₄

#10625350

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₈×D₄₇

yes

—

137

C₂×C₈

#10625324

yes

—

138

C₂×C₈

Dic₄₇⋊C₄

yes

—

139

C₂×C₈

Dic₄₇⋊C₄

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₂×Dic₉₄

yes

characteristic

144

C₄²

C₂×C₄×D₄₇

yes

—

145

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

146

C₂³

#13093981

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

C₄²×D₄₇

yes

characteristic

151

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

152

C₂×C₄

C₄×C₂²⋊D₄₇

yes

—

153

C₂×C₄

C₄×Dic₉₄

yes

—

154

C₂×C₄

C₂×C₈×D₄₇

yes

—

155

C₂×C₄

C₂×#10625324

yes

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

188

—

161

C₈

Dic₄₇⋊C₈

yes

—

162

C₈

Dic₄₇⋊C₈

yes

—

163

C₂²

—

164

C₂²

C₄×Dic₄₇⋊C₄

yes

characteristic

165

C₂²

#15689174

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₄×C₈×D₄₇

yes

characteristic

169

C₄

C₄×#10625324

yes

characteristic

170

C₄

C₂×Dic₄₇⋊C₈

yes

—

171

C₄

#15689238

yes

—

172

C₂

—

173

C₂

C₄×Dic₄₇⋊C₈

yes

characteristic

174

C₂

#20243268

yes

—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	C₄×C₄⋊C₄
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#8266
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ #1450

subgroup

G/H

center

C₄×C₈

C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1450

C₂

socle

C₂²×C₄₇

C₄×C₄⋊C₄

prime radical

C₂³×C₄₇

#117

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ #1450

N	H ≅ G/N	type
11. #15688875	C₄	semidirect
12. #15688873	C₄	semidirect
29. C₈×Dic₄₇	C₈	semidirect
106. C₄₇	#8266	semidirect

H ≅ G/N

type

11. #15688875

C₄

semidirect

12. #15688873

C₄

semidirect

29. C₈×Dic₄₇

C₈

semidirect

106. C₄₇

#8266

semidirect

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

subgroup	index
C₄×C₈×Dic₄₇	2
#20242134	2
#20241671	2
#20241220	2
C₄₇×#1450	2
#8266	47

subgroup

index

C₄×C₈×Dic₄₇

C₄₇×#1450

Subgroups — #26107437

Conjugacy classes of subgroups

Complements

Maximal subgroups