Group #26141562 — subgroups

Subgroups — #26141562

Order 12032.

1494 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 427 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	C₂×#15688969	C₂	1	2	yes	maximal
2	#20241656	C₂	1	4	yes	maximal
3	#20241620	C₂	1	8	yes	maximal
4	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
6	#15688969	—	2	8	no	—
7	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
8	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
9	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
10	C₂×#125×C₄₇	—	2	8	no	—
11	C₂×#125×C₄₇	C₄	1	8	yes	complemented
12	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×Dic₄₇	C₂³	1	8	yes	—
16	C₂×C₄×Dic₄₇	—	2	8	no	—
17	C₂×C₄×Dic₄₇	—	2	16	no	—
18	C₄₇⋊#95	—	2	8	no	—
19	C₄₇⋊#95	—	2	8	no	—
20	C₄₇⋊#95	—	2	32	no	—
21	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
24	#125×C₄₇	—	2	32	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	8	yes	—
33	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	—	2	4	no	—
38	C₂²×C₄×C₄₇	—	2	16	no	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
45	C₂²×Dic₄₇	—	2	8	no	—
46	C₂²×Dic₄₇	—	2	16	no	—
47	C₂²×Dic₄₇	—	2	8	no	—
48	C₂²×Dic₄₇	—	2	16	no	—
49	C₂²×Dic₄₇	#41	1	16	yes	—
50	C₄⋊C₄×C₄₇	—	2	16	no	—
51	C₄⋊C₄×C₄₇	—	2	32	no	—
52	C₄²×C₄₇	C₂²⋊C₄	1	16	yes	—
53	C₄×Dic₄₇	—	2	32	no	—
54	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
55	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
56	C₂³×C₄₇	C₂×#41	1	8	yes	—
57	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
58	C₂×C₄×C₄₇	—	2	8	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	—	2	8	no	—
61	C₂×C₄×C₄₇	—	2	8	no	—
62	C₂×C₄×C₄₇	—	2	8	no	—
63	C₂×C₄×C₄₇	—	2	8	no	—
64	C₂×C₄×C₄₇	—	2	16	no	—
65	C₂×C₄×C₄₇	—	4	8	no	—
66	C₂×C₄×C₄₇	—	4	8	no	—
67	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
68	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
69	C₂×Dic₄₇	—	2	16	no	—
70	C₂×Dic₄₇	—	2	32	no	—
71	C₂×Dic₄₇	—	4	8	no	—
72	C₂×Dic₄₇	—	4	8	no	—
73	C₂×Dic₄₇	—	4	16	no	—
74	#14170	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄₇	C₄×#41	1	4	yes	—
77	C₂²×C₄₇	#519	1	2	yes	—
78	C₂²×C₄₇	#525	1	4	yes	—
79	C₂²×C₄₇	#532	1	4	yes	—
80	C₂²×C₄₇	#536	1	4	yes	—
81	C₂²×C₄₇	#552	1	16	yes	—
82	C₄×C₄₇	—	2	4	no	—
83	C₄×C₄₇	—	2	8	no	—
84	C₄×C₄₇	—	4	8	no	—
85	Dic₄₇	—	4	16	no	—
86	C₂²×#125	D₄₇	1	1	yes	characteristic
87	C₂×#376	—	47	2	no	—
88	#1445	—	47	4	no	—
89	#1436	—	47	8	no	—
90	C₂×C₄₇	#2297	1	2	yes	—
91	C₂×C₄₇	#2310	1	1	yes	characteristic
92	C₂×C₄₇	#2371	1	4	yes	—
93	C₂×C₄₇	#2611	1	8	yes	—
94	C₂×#125	—	2	8	no	—
95	C₂×#125	Dic₄₇	1	8	yes	complemented
96	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
97	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
98	C₂²×C₄²	—	47	4	no	—
99	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
100	#376	—	94	8	no	—
101	C₂×#95	—	47	16	no	—
102	C₂×#95	—	47	4	no	—
103	C₂×#95	—	47	4	no	—
104	C₄₇	#14170	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	—	47	8	no	—
106	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
107	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
109	#125	—	2	32	no	—
110	C₂×C₄⋊C₄	—	2	8	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	8	no	—
114	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
116	C₂×C₄²	—	94	16	no	—
117	C₂×C₄²	—	94	8	no	—
118	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
119	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
120	#95	—	94	32	no	—
121	#95	—	94	8	no	—
122	#95	—	94	8	no	—
123	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	47	16	no	—
130	C₂²×C₄	—	94	8	no	—
131	C₂²×C₄	—	94	16	no	—
132	C₂²×C₄	—	94	8	no	—
133	C₂²×C₄	—	94	16	no	—
134	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
135	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
140	C₄⋊C₄	—	2	32	no	—
141	C₄⋊C₄	—	2	16	no	—
142	C₄²	—	94	32	no	—
143	C₄²	C₂²⋊Dic₄₇	1	16	yes	—
144	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
145	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
146	C₂³	#41×D₄₇	1	8	yes	—
147	C₂³	#13094072	1	4	yes	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	2	8	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	—	94	32	no	—
158	C₂×C₄	—	94	16	no	—
159	C₂×C₄	—	188	16	no	—
160	C₂×C₄	—	188	8	no	—
161	C₂×C₄	—	188	8	no	—
162	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
163	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
164	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
165	C₂²	#41×Dic₄₇	1	4	yes	—
166	C₂²	#15689601	1	2	yes	—
167	C₂²	#15689630	1	4	yes	—
168	C₂²	#15689669	1	4	yes	—
169	C₂²	#15689697	1	4	yes	—
170	C₂²	#15689805	1	16	yes	—
171	C₄	—	2	8	no	—
172	C₄	—	2	4	no	—
173	C₄	—	4	8	no	—
174	C₄	—	188	16	no	—
175	C₂	#20245879	1	2	yes	—
176	C₂	#20245974	1	1	yes	characteristic
177	C₂	#20246494	1	4	yes	—
178	C₂	#20249088	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688969

C₂

yes

maximal

#20241656

C₂

yes

maximal

#20241620

C₂

yes

maximal

C₂²×#125×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₄₇

C₂²

yes

—

#15688969

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×#125×C₄₇

—

C₂×#125×C₄₇

C₄

yes

complemented

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂³×C₄×C₄₇

C₂³

yes

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

C₂×#41

yes

—

C₂³×C₄₇

2₋¹⁺⁴

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×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#14170

—

maximal, Sylow

C₂²×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×#41

yes

—

C₂²×C₄₇

#519

yes

—

C₂²×C₄₇

#525

yes

—

C₂²×C₄₇

#532

yes

—

C₂²×C₄₇

#536

yes

—

C₂²×C₄₇

#552

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

C₂²×#125

D₄₇

yes

—

—

—

C₂×C₄₇

#2297

yes

—

C₂×C₄₇

#2310

yes

characteristic

C₂×C₄₇

#2371

yes

—

C₂×C₄₇

#2611

yes

—

C₂×#125

—

C₂×#125

Dic₄₇

yes

complemented

C₂²×C₄⋊C₄

C₂×D₄₇

yes

—

C₂²×C₄⋊C₄

C₂×D₄₇

yes

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₄₇

yes

characteristic

100

#376

—

101

C₂×#95

—

102

C₂×#95

—

103

C₂×#95

—

104

C₄₇

#14170

yes

characteristic, complemented, Sylow

105

C₂³×C₄

—

106

C₂³×C₄

C₂²×D₄₇

yes

—

107

C₂³×C₄

C₂²×D₄₇

yes

characteristic

108

C₂³×C₄

C₂²×D₄₇

yes

—

109

#125

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

115

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂×Dic₄₇

yes

—

119

C₂×C₄²

C₂²⋊D₄₇

yes

—

120

#95

—

121

#95

—

122

#95

—

123

C₂⁴

C₂³×D₄₇

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂²×Dic₄₇

yes

—

135

C₂²×C₄

C₂²×Dic₄₇

yes

—

136

C₂²×C₄

C₂²×Dic₄₇

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

C₂²⋊Dic₄₇

yes

—

144

C₂³

C₂³×Dic₄₇

yes

characteristic

145

C₂³

C₂²×C₂²⋊D₄₇

yes

—

146

C₂³

#41×D₄₇

yes

—

147

C₂³

#13094072

yes

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

188

—

160

C₂×C₄

—

188

—

161

C₂×C₄

—

188

—

162

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

163

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

164

C₂²

C₂²×C₂²⋊Dic₄₇

yes

characteristic

165

C₂²

#41×Dic₄₇

yes

—

166

C₂²

#15689601

yes

—

167

C₂²

#15689630

yes

—

168

C₂²

#15689669

yes

—

169

C₂²

#15689697

yes

—

170

C₂²

#15689805

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

188

—

175

C₂

#20245879

yes

—

176

C₂

#20245974

yes

characteristic

177

C₂

#20246494

yes

—

178

C₂

#20249088

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#14170
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#125

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂²×#125×C₄₇

C₂

socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

hypercenter

C₂²×#125

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂²×#125

N	H ≅ G/N	type
11. C₂×#125×C₄₇	C₄	semidirect
95. C₂×#125	Dic₄₇	semidirect
104. C₄₇	#14170	semidirect

H ≅ G/N

type

11. C₂×#125×C₄₇

C₄

semidirect

95. C₂×#125

Dic₄₇

semidirect

104. C₄₇

#14170

semidirect

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

subgroup	index
C₂×#15688969	2
#20241656	2
#20241620	2
C₂²×#125×C₄₇	2
#14170	47

subgroup

index

C₂²×#125×C₄₇

Subgroups — #26141562

Conjugacy classes of subgroups

Complements

Maximal subgroups