Group #26132168 — subgroups

Subgroups — #26132168

Order 12032.

784 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15688922	C₂	1	3	yes	maximal
2	#20243782	C₂	1	3	yes	maximal
3	C₂×Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689601	C₂²	1	1	yes	characteristic
5	#15689132	—	2	3	no	—
6	C₂×#13093943	C₂²	1	3	yes	—
7	#15688922	—	2	6	no	—
8	#15688883	—	2	3	no	—
9	Q₈²×C₄₇	—	2	6	no	—
10	Q₈²×C₄₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊Q₈×C₄₇	—	2	3	no	—
12	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
13	C₂×C₄×Q₈×C₄₇	—	2	3	no	—
14	C₂×#10625329	—	2	3	no	—
15	Q₈×Dic₄₇	—	4	2	no	—
16	#13093983	—	2	3	no	—
17	#13093948	—	4	6	no	—
18	#13093943	—	2	6	no	—
19	C₂²×Q₈×C₄₇	—	2	1	no	—
20	C₂²×Q₈×C₄₇	—	2	1	no	—
21	C₂²×Q₈×C₄₇	D₄	1	3	yes	—
22	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₄₇	—	2	12	no	—
24	C₄⋊Q₈×C₄₇	—	2	12	no	—
25	C₄⋊Q₈×C₄₇	—	2	6	no	—
26	C₄⋊Q₈×C₄₇	C₂×C₄	1	6	yes	—
27	C₄⋊Q₈×C₄₇	D₄	1	6	yes	—
28	C₄×Q₈×C₄₇	—	2	12	no	—
29	C₄×Q₈×C₄₇	—	2	12	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
32	C₂×C₄²×C₄₇	—	2	3	no	—
33	C₂×C₄²×C₄₇	D₄	1	3	yes	—
34	C₂×Q₈×C₄₇	—	2	4	no	—
35	C₂×Q₈×C₄₇	—	2	8	no	—
36	C₂×Q₈×C₄₇	—	2	4	no	—
37	C₂×Q₈×C₄₇	—	2	3	no	—
38	C₂×Q₈×C₄₇	—	4	1	no	—
39	C₂×Q₈×C₄₇	—	4	1	no	—
40	C₂×Q₈×C₄₇	—	4	3	no	—
41	C₂×Q₈×C₄₇	—	4	3	no	—
42	C₂×Q₈×C₄₇	—	8	1	no	—
43	C₂×Q₈×C₄₇	—	8	1	no	—
44	C₂×Q₈×C₄₇	C₂²⋊C₄	1	6	yes	—
45	C₂×Q₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Q₈×C₄₇	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₄₇	—	2	3	no	—
48	C₂²×C₄×C₄₇	—	2	3	no	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
50	C₄⋊C₄×C₄₇	—	2	12	no	—
51	C₄⋊C₄×C₄₇	—	2	12	no	—
52	C₄⋊C₄×C₄₇	—	2	12	no	—
53	C₄²×C₄₇	—	2	12	no	—
54	C₄²×C₄₇	—	2	3	no	—
55	C₄²×C₄₇	C₂²⋊C₄	1	6	yes	—
56	C₂²⋊Dic₄₇	—	4	1	no	—
57	C₄⋊Dic₄₇	—	2	6	no	—
58	C₄×Dic₄₇	—	4	3	no	—
59	#10625329	—	4	3	no	—
60	#10625329	—	4	3	no	—
61	C₂×C₄₇⋊C₈	—	4	3	no	—
62	C₂³×C₄₇	#120	1	1	yes	characteristic
63	Q₈×C₄₇	—	2	16	no	—
64	Q₈×C₄₇	—	4	1	no	—
65	Q₈×C₄₇	—	4	3	no	—
66	Q₈×C₄₇	—	4	6	no	—
67	Q₈×C₄₇	—	8	3	no	—
68	Q₈×C₄₇	—	8	4	no	—
69	C₂×C₄×C₄₇	—	2	12	no	—
70	C₂×C₄×C₄₇	—	2	6	no	—
71	C₂×C₄×C₄₇	—	2	3	no	—
72	C₂×C₄×C₄₇	—	2	3	no	—
73	C₂×C₄×C₄₇	—	2	6	no	—
74	C₂×C₄×C₄₇	—	4	3	no	—
75	C₂×C₄×C₄₇	—	4	3	no	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
77	C₂×C₄×C₄₇	#120	1	3	yes	—
78	C₂×Dic₄₇	—	4	2	no	—
79	C₄₇⋊C₈	—	8	3	no	—
80	#12777	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	#379	1	1	yes	characteristic
83	C₂²×C₄₇	B(4,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₄₇	#458	1	1	yes	characteristic
85	C₄×C₄₇	—	2	12	no	—
86	C₄×C₄₇	—	2	3	no	—
87	C₄×C₄₇	—	2	3	no	—
88	C₄×C₄₇	—	4	6	no	—
89	Dic₄₇	—	8	2	no	—
90	C₂×Q₈²	D₄₇	1	1	yes	characteristic
91	C₂×#356	—	47	3	no	—
92	#1886	—	47	3	no	—
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	#1790	1	1	yes	characteristic
96	C₂×C₄₇	#2205	1	2	yes	—
97	Q₈²	—	2	6	no	—
98	Q₈²	Dic₄₇	1	4	yes	complemented
99	C₂×C₄⋊Q₈	—	2	3	no	—
100	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
101	#519	—	47	1	no	—
102	C₂×C₄×Q₈	—	2	3	no	—
103	#419	—	94	3	no	—
104	C₂×#101	—	47	3	no	—
105	#356	—	94	6	no	—
106	#339	—	94	3	no	—
107	C₄₇	#12777	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	2	1	no	—
109	C₂²×Q₈	—	2	1	no	—
110	C₂²×Q₈	C₂²⋊D₄₇	1	3	yes	—
111	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
112	C₂×M₁₆	—	94	3	no	—
113	C₄⋊Q₈	—	2	12	no	—
114	C₄⋊Q₈	—	2	12	no	—
115	C₄⋊Q₈	—	2	6	no	—
116	C₄⋊Q₈	C₂×Dic₄₇	1	6	yes	—
117	C₄⋊Q₈	C₂²⋊D₄₇	1	6	yes	—
118	C₄×Q₈	—	2	12	no	—
119	C₄×Q₈	—	2	12	no	—
120	C₄×Q₈	—	188	2	no	—
121	#117	—	94	3	no	—
122	C₂×C₄⋊C₄	—	2	3	no	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄²	—	2	3	no	—
125	C₂×C₄²	C₂²⋊D₄₇	1	3	yes	—
126	#103	—	188	6	no	—
127	#101	—	94	6	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	8	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	3	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	3	no	—
135	C₂×Q₈	—	4	3	no	—
136	C₂×Q₈	—	8	1	no	—
137	C₂×Q₈	—	8	1	no	—
138	C₂×Q₈	C₂²⋊Dic₄₇	1	6	yes	—
139	C₂×Q₈	C₂²×Dic₄₇	1	1	yes	characteristic
140	C₂×Q₈	C₂×C₂²⋊D₄₇	1	3	yes	—
141	C₂²×C₄	—	2	3	no	—
142	C₂²×C₄	—	2	3	no	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
144	M₁₆	—	188	3	no	—
145	M₁₆	—	188	3	no	—
146	C₂×C₈	—	188	3	no	—
147	C₄⋊C₄	—	2	12	no	—
148	C₄⋊C₄	—	2	12	no	—
149	C₄⋊C₄	—	2	12	no	—
150	C₄⋊C₄	—	94	6	no	—
151	C₂²⋊C₄	—	188	1	no	—
152	C₄²	—	2	12	no	—
153	C₄²	—	2	3	no	—
154	C₄²	—	188	3	no	—
155	C₄²	C₂²⋊Dic₄₇	1	6	yes	—
156	C₂³	#13093993	1	1	yes	characteristic
157	Q₈	—	2	16	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	4	3	no	—
160	Q₈	—	4	6	no	—
161	Q₈	—	8	3	no	—
162	Q₈	—	8	4	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	6	no	—
165	C₂×C₄	—	2	3	no	—
166	C₂×C₄	—	2	3	no	—
167	C₂×C₄	—	2	6	no	—
168	C₂×C₄	—	4	3	no	—
169	C₂×C₄	—	4	3	no	—
170	C₂×C₄	—	188	2	no	—
171	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
172	C₂×C₄	#13093993	1	3	yes	—
173	C₈	—	376	3	no	—
174	C₂²	—	2	2	no	—
175	C₂²	#15688979	1	1	yes	characteristic
176	C₂²	#15689310	1	1	yes	characteristic
177	C₂²	#15689313	1	1	yes	characteristic
178	C₄	—	2	12	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	—	4	6	no	—
182	C₄	—	376	2	no	—
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#20243336	1	1	yes	characteristic
186	C₂	#20245418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688922

C₂

yes

maximal

#20243782

C₂

yes

maximal

C₂×Q₈²×C₄₇

C₂

yes

maximal, characteristic

#15689601

C₂²

yes

characteristic

#15689132

—

C₂×#13093943

C₂²

yes

—

#15688922

—

#15688883

—

Q₈²×C₄₇

—

Q₈²×C₄₇

C₄

yes

complemented

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂×C₄×Q₈×C₄₇

—

C₂×#10625329

—

Q₈×Dic₄₇

—

#13093983

—

#13093948

—

#13093943

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

D₄

yes

—

C₂²×Q₈×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

C₄⋊Q₈×C₄₇

D₄

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂²⋊C₄

yes

—

C₂×Q₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×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

#120

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

#12777

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

C₂×Q₈²

D₄₇

yes

characteristic

C₂×#356

—

#1886

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1790

yes

characteristic

C₂×C₄₇

#2205

yes

—

Q₈²

—

Q₈²

Dic₄₇

yes

complemented

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

101

#519

—

102

C₂×C₄×Q₈

—

103

#419

—

104

C₂×#101

—

105

#356

—

106

#339

—

107

C₄₇

#12777

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

—

109

C₂²×Q₈

—

110

C₂²×Q₈

C₂²⋊D₄₇

yes

—

111

C₂²×Q₈

C₂²×D₄₇

yes

characteristic

112

C₂×M₁₆

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

C₂×Dic₄₇

yes

—

117

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

118

C₄×Q₈

—

119

C₄×Q₈

—

120

C₄×Q₈

—

188

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂²⋊D₄₇

yes

—

126

#103

—

188

—

127

#101

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

C₂²⋊Dic₄₇

yes

—

139

C₂×Q₈

C₂²×Dic₄₇

yes

characteristic

140

C₂×Q₈

C₂×C₂²⋊D₄₇

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

144

M₁₆

—

188

—

145

M₁₆

—

188

—

146

C₂×C₈

—

188

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

188

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

188

—

155

C₄²

C₂²⋊Dic₄₇

yes

—

156

C₂³

#13093993

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

188

—

171

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

172

C₂×C₄

#13093993

yes

—

173

C₈

—

376

—

174

C₂²

—

175

C₂²

#15688979

yes

characteristic

176

C₂²

#15689310

yes

characteristic

177

C₂²

#15689313

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

376

—

183

C₂

—

184

C₂

—

185

C₂

#20243336

yes

characteristic

186

C₂

#20245418

yes

—

	subgroup	G/H
center	C₂²	#15689310
commutator	C₂×Q₈×C₄₇	C₂²×C₄
Frattini	C₂²×Q₈	C₂²×D₄₇
Fitting	C₂×Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	B(4,𝔽₂)
prime radical	C₂³×C₄₇	#120
cosocle	C₂²×Q₈×C₄₇	C₂³
hypercenter	C₂×Q₈²	D₄₇
hypocenter	C₄₇	#12777
LCS	G ▷ C₂×Q₈×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×Q₈²

subgroup

G/H

center

C₂²

#15689310

commutator

C₂×Q₈×C₄₇

C₂²×C₄

Frattini

C₂²×Q₈

C₂²×D₄₇

Fitting

C₂×Q₈²×C₄₇

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂²×Q₈×C₄₇

C₂³

hypercenter

C₂×Q₈²

D₄₇

hypocenter

C₄₇

#12777

LCS

G ▷ C₂×Q₈×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×Q₈²

N	H ≅ G/N	type
10. Q₈²×C₄₇	C₄	semidirect
98. Q₈²	Dic₄₇	semidirect
107. C₄₇	#12777	semidirect

H ≅ G/N

type

10. Q₈²×C₄₇

C₄

semidirect

98. Q₈²

Dic₄₇

semidirect

107. C₄₇

#12777

semidirect

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

subgroup	index
C₂×#15688922	2
#20243782	2
C₂×Q₈²×C₄₇	2
#12777	47

subgroup

index

C₂×#15688922

#20243782

C₂×Q₈²×C₄₇

#12777

Subgroups — #26132168

Conjugacy classes of subgroups

Complements

Maximal subgroups