Group #26110040 — subgroups

Subgroups — #26110040

Order 12032.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20243505	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688875	C₂	1	2	yes	maximal
4	C₂×#15688873	C₂	1	2	yes	maximal
5	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	2	no	—
10	#15688878	C₄	1	2	yes	—
11	#15688878	C₄	1	2	yes	—
12	#15688875	—	2	4	no	—
13	#15688873	—	2	4	no	—
14	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	1	no	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
23	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄⋊C₈×C₄₇	—	4	4	no	—
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	4	no	—
40	C₄²×C₄₇	—	2	4	no	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	C₂×D₄	1	2	yes	—
46	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄₇⋊C₈	—	2	4	no	—
48	C₂×C₄₇⋊C₈	—	2	4	no	—
49	C₂×C₄₇⋊C₈	—	4	1	no	—
50	C₂×C₄₇⋊C₈	—	4	1	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
54	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	#108	1	4	yes	—
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₄₇	#131	1	2	yes	—
67	C₂×C₄×C₄₇	#131	1	2	yes	—
68	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₄₇	#137	1	2	yes	—
70	C₈×C₄₇	—	8	2	no	—
71	C₄₇⋊C₈	—	2	8	no	—
72	C₄₇⋊C₈	—	8	1	no	—
73	C₄₇⋊C₈	—	8	1	no	—
74	#8812	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	#384	1	1	yes	characteristic
76	C₂²×C₄₇	#424	1	1	yes	characteristic
77	C₂²×C₄₇	#424	1	1	yes	characteristic
78	C₂²×C₄₇	#428	1	2	yes	—
79	C₂²×C₄₇	C₂×#108	1	2	yes	—
80	C₄×C₄₇	—	2	4	no	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	4	2	no	—
83	C₄×C₄₇	#424	1	4	yes	—
84	C₄×C₄₇	#496	1	1	yes	characteristic
85	C₄×C₄₇	#497	1	1	yes	characteristic
86	C₄×C₄₇	#501	1	2	yes	—
87	#1842	D₄₇	1	1	yes	characteristic
88	#1831	—	47	1	no	—
89	#1744	—	47	1	no	—
90	C₂×#335	—	47	2	no	—
91	C₂×#334	—	47	2	no	—
92	C₂×C₄₇	#1573	1	2	yes	—
93	C₂×C₄₇	#1574	1	2	yes	—
94	C₂×C₄₇	#1843	1	1	yes	characteristic
95	C₂×C₄₇	#1847	1	1	yes	characteristic
96	C₂×C₄₇	#1848	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	2	2	no	—
98	C₂×C₄⋊C₈	—	47	1	no	—
99	C₂×C₄⋊C₈	—	47	1	no	—
100	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
101	C₂×C₄×C₈	—	47	2	no	—
102	C₄³	C₂×D₄₇	1	1	yes	characteristic
103	#336	—	47	2	no	—
104	#336	—	47	2	no	—
105	#336	—	94	2	no	—
106	#335	—	94	4	no	—
107	#334	—	94	4	no	—
108	C₄₇	#8812	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	47	4	no	—
111	C₂²×C₈	—	94	1	no	—
112	C₂²×C₈	—	94	1	no	—
113	C₂×C₄²	Dic₉₄	1	2	yes	—
114	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	—	94	2	no	—
122	C₄⋊C₈	—	94	2	no	—
123	C₄×C₈	—	47	4	no	—
124	C₄×C₈	—	47	4	no	—
125	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
126	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	47	8	no	—
133	C₂×C₈	—	94	4	no	—
134	C₂×C₈	—	94	4	no	—
135	C₂×C₈	—	188	1	no	—
136	C₂×C₈	—	188	1	no	—
137	C₂×C₈	—	188	2	no	—
138	C₂×C₈	—	188	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	D₄×D₄₇	1	2	yes	—
145	C₄²	Q₈×D₄₇	1	2	yes	—
146	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#13093959	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
155	C₂×C₄	#13094020	1	1	yes	characteristic
156	C₂×C₄	#13094022	1	1	yes	characteristic
157	C₂×C₄	#13094023	1	2	yes	—
158	C₂×C₄	#13094030	1	2	yes	—
159	C₂×C₄	#13094032	1	2	yes	—
160	C₂×C₄	#13094050	1	2	yes	—
161	C₂×C₄	#13094055	1	2	yes	—
162	C₈	—	8	2	no	—
163	C₈	—	94	8	no	—
164	C₈	—	376	1	no	—
165	C₈	—	376	1	no	—
166	C₂²	#15688992	1	1	yes	characteristic
167	C₂²	#15689149	1	1	yes	characteristic
168	C₂²	#15689151	1	1	yes	characteristic
169	C₂²	#15689163	1	2	yes	—
170	C₂²	C₂×#13093959	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	#15689148	1	4	yes	—
175	C₄	#15689510	1	1	yes	characteristic
176	C₄	#15689515	1	1	yes	characteristic
177	C₄	#15689529	1	2	yes	—
178	C₂	#20242171	1	2	yes	—
179	C₂	#20242179	1	2	yes	—
180	C₂	#20243541	1	1	yes	characteristic
181	C₂	#20243554	1	1	yes	characteristic
182	C₂	#20243559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal, characteristic

#20243134

C₂

yes

maximal, characteristic

C₂×#15688875

C₂

yes

maximal

C₂×#15688873

C₂

yes

maximal

C₄₇×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

—

#15688878

C₄

yes

—

#15688878

C₄

yes

—

#15688875

—

#15688873

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#108

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#108

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#424

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#1842

D₄₇

yes

—

—

—

—

C₂×C₄₇

#1573

yes

—

C₂×C₄₇

#1574

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

100

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

101

C₂×C₄×C₈

—

102

C₄³

C₂×D₄₇

yes

characteristic

103

#336

—

104

#336

—

105

#336

—

106

#335

—

107

#334

—

108

C₄₇

#8812

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂×C₄²

Dic₉₄

yes

—

114

C₂×C₄²

C₄×D₄₇

yes

characteristic

115

C₂×C₄²

C₄×D₄₇

yes

characteristic

116

C₂×C₄²

C₂²⋊D₄₇

yes

—

117

C₂×C₄²

C₂²×D₄₇

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

Dic₄₇⋊C₄

yes

—

126

C₂²×C₄

Dic₄₇⋊C₄

yes

—

127

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

128

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

188

—

136

C₂×C₈

—

188

—

137

C₂×C₈

—

188

—

138

C₂×C₈

—

188

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

D₄×D₄₇

yes

—

145

C₄²

Q₈×D₄₇

yes

—

146

C₂³

C₂×Dic₄₇⋊C₄

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#13093959

yes

—

154

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

155

C₂×C₄

#13094020

yes

characteristic

156

C₂×C₄

#13094022

yes

characteristic

157

C₂×C₄

#13094023

yes

—

158

C₂×C₄

#13094030

yes

—

159

C₂×C₄

#13094032

yes

—

160

C₂×C₄

#13094050

yes

—

161

C₂×C₄

#13094055

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

376

—

165

C₈

—

376

—

166

C₂²

#15688992

yes

characteristic

167

C₂²

#15689149

yes

characteristic

168

C₂²

#15689151

yes

characteristic

169

C₂²

#15689163

yes

—

170

C₂²

C₂×#13093959

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#15689148

yes

—

175

C₄

#15689510

yes

characteristic

176

C₄

#15689515

yes

characteristic

177

C₄

#15689529

yes

—

178

C₂

#20242171

yes

—

179

C₂

#20242179

yes

—

180

C₂

#20243541

yes

characteristic

181

C₂

#20243554

yes

characteristic

182

C₂

#20243559

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8812
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1842

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
108. C₄₇	#8812	semidirect

H ≅ G/N

type

108. C₄₇

#8812

semidirect

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

subgroup	index
#20243505	2
#20243134	2
C₂×#15688875	2
C₂×#15688873	2
C₄₇×#1842	2
#8812	47

subgroup

index

C₄₇×#1842

Subgroups — #26110040

Conjugacy classes of subgroups

Complements

Maximal subgroups