Group #26147472 — subgroups

Subgroups — #26147472

Order 12032.

1422 conjugacy classes of subgroups; only the 443 normal subgroups are shown.

normal subgroups only

This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 443 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×#15688991	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	Dic₄₇⋊#95	C₂	1	6	yes	maximal
4	#20241614	C₂	1	6	yes	maximal
5	C₂×C₄₇×#383	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
9	#15688991	—	2	4	no	—
10	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
12	C₄²×Dic₄₇	C₄	1	8	yes	complemented
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#383	—	2	4	no	—
15	C₂×#95×C₄₇	C₂²	1	6	yes	—
16	C₂³×Dic₄₇	C₂³	1	4	yes	—
17	C₂³×Dic₄₇	C₂³	1	3	yes	—
18	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₇⋊C₄	—	2	24	no	—
20	C₂×C₄×Dic₄₇	—	2	12	no	—
21	C₂×C₄×Dic₄₇	—	2	12	no	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	12	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	12	yes	—
24	C₄₇⋊#95	—	2	12	no	—
25	C₄₇⋊#95	—	2	24	no	—
26	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	4	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
31	#95×C₄₇	—	2	12	no	—
32	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	—	2	6	no	—
34	C₂²×C₄×C₄₇	—	2	6	no	—
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	—	2	12	no	—
37	C₂²×C₄×C₄₇	—	2	12	no	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
40	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
41	C₂²×Dic₄₇	—	2	6	no	—
42	C₂²×Dic₄₇	—	2	6	no	—
43	C₂²×Dic₄₇	—	2	4	no	—
44	C₂²×Dic₄₇	—	2	12	no	—
45	C₂²×Dic₄₇	—	2	12	no	—
46	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
47	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
48	C₂²×Dic₄₇	#41	1	12	yes	—
49	C₂²×Dic₄₇	#41	1	12	yes	—
50	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
51	Dic₄₇⋊C₄	—	4	16	no	—
52	C₄×Dic₄₇	—	2	24	no	—
53	C₄×Dic₄₇	—	2	24	no	—
54	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
55	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
56	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
57	C₂³×C₄₇	C₂×#41	1	6	yes	—
58	C₂³×C₄₇	C₂×#41	1	6	yes	—
59	C₂×C₄×C₄₇	—	2	6	no	—
60	C₂×C₄×C₄₇	—	2	6	no	—
61	C₂×C₄×C₄₇	—	2	12	no	—
62	C₂×C₄×C₄₇	—	2	12	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	4	12	no	—
65	C₂×C₄×C₄₇	—	4	12	no	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
67	C₂×Dic₄₇	—	2	6	no	—
68	C₂×Dic₄₇	—	2	6	no	—
69	C₂×Dic₄₇	—	2	24	no	—
70	C₂×Dic₄₇	—	4	4	no	—
71	C₂×Dic₄₇	—	4	12	no	—
72	C₂×Dic₄₇	—	4	12	no	—
73	C₂×Dic₄₇	#117	1	12	yes	—
74	C₂×Dic₄₇	#117	1	12	yes	—
75	C₂×Dic₄₇	#126	1	16	yes	—
76	#14688	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×#117	1	3	yes	—
79	C₂²×C₄₇	C₂×#117	1	3	yes	—
80	C₂²×C₄₇	C₂×#126	1	4	yes	—
81	C₂²×C₄₇	#529	1	12	yes	—
82	C₂²×C₄₇	#532	1	6	yes	—
83	C₂²×C₄₇	#533	1	6	yes	—
84	C₄×C₄₇	—	2	12	no	—
85	C₄×C₄₇	—	8	4	no	—
86	Dic₄₇	—	2	12	no	—
87	Dic₄₇	—	8	4	no	—
88	Dic₄₇	#383	1	8	yes	complemented
89	C₂×#383	—	47	1	no	—
90	C₂×#383	D₄₇	1	1	yes	characteristic
91	C₂×C₄³	—	47	1	no	—
92	C₄⋊#95	—	47	6	no	—
93	#1434	—	47	6	no	—
94	C₂×C₄₇	C₂×#383	1	1	yes	characteristic
95	C₂×C₄₇	#2283	1	3	yes	—
96	C₂×C₄₇	#2285	1	3	yes	—
97	C₂×C₄₇	#2860	1	4	yes	—
98	C₂×C₄₇	#2863	1	4	yes	—
99	C₂²×C₄⋊C₄	—	47	4	no	—
100	C₂²×C₄²	—	47	3	no	—
101	C₂²×C₄²	—	47	3	no	—
102	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	#383	—	2	4	no	—
104	#383	—	94	4	no	—
105	C₂×#95	—	47	12	no	—
106	C₂×#95	—	47	6	no	—
107	C₂×#95	C₂×D₄₇	1	6	yes	—
108	C₄³	—	47	8	no	—
109	C₄₇	#14688	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	47	1	no	—
111	C₂³×C₄	—	47	3	no	—
112	C₂³×C₄	—	47	4	no	—
113	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
114	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
115	C₂×C₄⋊C₄	—	94	24	no	—
116	C₂×C₄²	—	47	12	no	—
117	C₂×C₄²	—	47	12	no	—
118	C₂×C₄²	—	94	12	no	—
119	C₂×C₄²	—	94	12	no	—
120	C₂×C₄²	Dic₉₄	1	4	yes	—
121	C₂×C₄²	C₄×D₄₇	1	4	yes	—
122	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
123	#95	—	2	12	no	—
124	#95	—	94	24	no	—
125	#95	—	94	12	no	—
126	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
127	C₂²×C₄	—	2	6	no	—
128	C₂²×C₄	—	2	6	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	—	2	12	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	—	47	12	no	—
134	C₂²×C₄	—	47	6	no	—
135	C₂²×C₄	—	47	12	no	—
136	C₂²×C₄	—	94	6	no	—
137	C₂²×C₄	—	94	6	no	—
138	C₂²×C₄	—	94	4	no	—
139	C₂²×C₄	—	94	12	no	—
140	C₂²×C₄	—	94	12	no	—
141	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
142	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
144	C₄⋊C₄	—	188	16	no	—
145	C₄²	—	94	24	no	—
146	C₄²	—	94	24	no	—
147	C₄²	Dic₄₇⋊C₄	1	16	yes	complemented
148	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
149	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
151	C₂³	#41×D₄₇	1	6	yes	—
152	C₂³	#41×D₄₇	1	6	yes	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	2	12	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	12	no	—
159	C₂×C₄	—	4	12	no	—
160	C₂×C₄	—	47	12	no	—
161	C₂×C₄	—	47	16	no	—
162	C₂×C₄	—	47	12	no	—
163	C₂×C₄	—	94	6	no	—
164	C₂×C₄	—	94	6	no	—
165	C₂×C₄	—	94	24	no	—
166	C₂×C₄	—	188	4	no	—
167	C₂×C₄	—	188	12	no	—
168	C₂×C₄	—	188	12	no	—
169	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
170	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
171	C₂²	#117×D₄₇	1	3	yes	—
172	C₂²	#117×D₄₇	1	3	yes	—
173	C₂²	#126×D₄₇	1	4	yes	—
174	C₂²	#15689656	1	12	yes	—
175	C₂²	#15689669	1	6	yes	—
176	C₂²	#15689676	1	6	yes	—
177	C₄	—	2	12	no	—
178	C₄	—	8	4	no	—
179	C₄	—	47	8	no	—
180	C₄	—	94	12	no	—
181	C₄	—	376	4	no	—
182	C₂	#383×D₄₇	1	1	yes	characteristic
183	C₂	#20245785	1	3	yes	—
184	C₂	#20245791	1	3	yes	—
185	C₂	#20251753	1	4	yes	—
186	C₂	#20251768	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688991

C₂

yes

maximal, characteristic

C₂×C₄²×Dic₄₇

C₂

yes

maximal, characteristic

Dic₄₇⋊#95

C₂

yes

maximal

#20241614

C₂

yes

maximal

C₂×C₄₇×#383

C₂

yes

maximal, characteristic

C₂²×Dic₄₇⋊C₄

C₂²

yes

—

C₂²×C₄×Dic₄₇

C₂²

yes

—

C₂²×C₄×Dic₄₇

C₂²

yes

—

#15688991

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₄²×Dic₄₇

C₄

yes

complemented

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#383

—

C₂×#95×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂³×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

#95×C₄₇

—

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

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

#41

yes

—

C₂²×Dic₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂³×C₄₇

C₂²×Q₈

yes

characteristic

C₂³×C₄₇

C₂×#41

yes

—

C₂³×C₄₇

C₂×#41

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⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#117

yes

—

C₂×Dic₄₇

#117

yes

—

C₂×Dic₄₇

#126

yes

—

#14688

—

maximal, Sylow

C₂²×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×#117

yes

—

C₂²×C₄₇

C₂×#117

yes

—

C₂²×C₄₇

C₂×#126

yes

—

C₂²×C₄₇

#529

yes

—

C₂²×C₄₇

#532

yes

—

C₂²×C₄₇

#533

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

yes

—

D₄₇

yes

characteristic

C₂×C₄³

—

C₄⋊#95

—

#1434

—

C₂×C₄₇

C₂×#383

yes

characteristic

C₂×C₄₇

#2283

yes

—

C₂×C₄₇

#2285

yes

—

C₂×C₄₇

#2860

yes

—

C₂×C₄₇

#2863

yes

—

C₂²×C₄⋊C₄

—

100

C₂²×C₄²

—

101

C₂²×C₄²

—

102

C₂²×C₄²

C₂×D₄₇

yes

characteristic

103

#383

—

104

#383

—

105

C₂×#95

—

106

C₂×#95

—

107

C₂×#95

C₂×D₄₇

yes

—

108

C₄³

—

109

C₄₇

#14688

yes

characteristic, complemented, Sylow

110

C₂³×C₄

—

111

C₂³×C₄

—

112

C₂³×C₄

—

113

C₂³×C₄

C₂²×D₄₇

yes

—

114

C₂³×C₄

C₂²×D₄₇

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

Dic₉₄

yes

—

121

C₂×C₄²

C₄×D₄₇

yes

—

122

C₂×C₄²

C₂²⋊D₄₇

yes

—

123

#95

—

124

#95

—

125

#95

—

126

C₂⁴

C₂³×D₄₇

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×Dic₉₄

yes

—

142

C₂²×C₄

C₂×C₄×D₄₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

144

C₄⋊C₄

—

188

—

145

C₄²

—

146

C₄²

—

147

C₄²

Dic₄₇⋊C₄

yes

complemented

148

C₂³

C₂²×C₄×D₄₇

yes

characteristic

149

C₂³

C₂²×C₂²⋊D₄₇

yes

characteristic

150

C₂³

C₂²×Dic₉₄

yes

characteristic

151

C₂³

#41×D₄₇

yes

—

152

C₂³

#41×D₄₇

yes

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

188

—

167

C₂×C₄

—

188

—

168

C₂×C₄

—

188

—

169

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

—

170

C₂²

C₂²×Dic₄₇⋊C₄

yes

characteristic

171

C₂²

#117×D₄₇

yes

—

172

C₂²

#117×D₄₇

yes

—

173

C₂²

#126×D₄₇

yes

—

174

C₂²

#15689656

yes

—

175

C₂²

#15689669

yes

—

176

C₂²

#15689676

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

376

—

182

C₂

#383×D₄₇

yes

characteristic

183

C₂

#20245785

yes

—

184

C₂

#20245791

yes

—

185

C₂

#20251753

yes

—

186

C₂

#20251768

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#383	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#383	D₄₇
hypocenter	C₄₇	#14688
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂×C₄₇×#383

C₂

socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂×#383

N	H ≅ G/N	type
12. C₄²×Dic₄₇	C₄	semidirect
50. C₄²×C₄₇	C₄⋊C₄	semidirect
88. Dic₄₇	#383	semidirect
109. C₄₇	#14688	semidirect
147. C₄²	Dic₄₇⋊C₄	semidirect

H ≅ G/N

type

12. C₄²×Dic₄₇

C₄

semidirect

50. C₄²×C₄₇

C₄⋊C₄

semidirect

88. Dic₄₇

#383

semidirect

109. C₄₇

#14688

semidirect

147. C₄²

Dic₄₇⋊C₄

semidirect

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

subgroup	index
C₂×#15688991	2
C₂×C₄²×Dic₄₇	2
Dic₄₇⋊#95	2
#20241614	2
C₂×C₄₇×#383	2
#14688	47

subgroup

index

C₂×#15688991

C₂×C₄²×Dic₄₇

Dic₄₇⋊#95

#20241614

C₂×C₄₇×#383

#14688

Subgroups — #26147472

Conjugacy classes of subgroups

Complements

Maximal subgroups