Group #26135931 — subgroups

Subgroups — #26135931

Order 12032.

324 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 103 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×#15689525	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688947	C₂	1	2	yes	maximal
3	#20241493	C₂	1	2	yes	maximal
4	#20241470	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	—	2	1	no	—
7	C₂×#13093964	C₂²	1	2	yes	—
8	#15689525	—	2	4	no	—
9	C₂×#13093958	—	2	1	no	—
10	C₂×#13093956	C₂²	1	2	yes	—
11	#15688947	—	2	4	no	—
12	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#427	C₂²	1	1	yes	characteristic
14	C₄₇×#424	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#339	—	2	1	no	—
17	#13094016	—	2	4	no	—
18	#13094016	—	4	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×Dic₄₇⋊C₄	—	4	2	no	—
21	#13093964	—	2	2	no	—
22	#13093964	Q₈	1	4	yes	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	#13093958	—	2	4	no	—
25	#13093956	—	2	2	no	—
26	#13093956	D₄	1	4	yes	—
27	C₂×M₁₆×C₄₇	—	2	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	#117×C₄₇	—	2	1	no	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	#108×C₄₇	—	2	2	no	—
33	#107×C₄₇	Q₈	1	2	yes	—
34	#106×C₄₇	—	2	1	no	—
35	C₄⋊C₈×C₄₇	—	4	2	no	—
36	C₄×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	4	2	no	—
40	M₁₆×C₄₇	—	4	1	no	—
41	M₁₆×C₄₇	—	4	1	no	—
42	C₂×C₈×C₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₂×C₈×C₄₇	—	4	1	no	—
45	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂²⋊C₄×C₄₇	—	4	1	no	—
52	C₄²×C₄₇	—	2	1	no	—
53	C₄²×C₄₇	—	4	1	no	—
54	C₄²×C₄₇	SD₁₆	1	2	yes	—
55	C₄⋊Dic₄₇	—	2	4	no	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	Dic₄₇⋊C₄	—	4	8	no	—
58	C₄₇⋊C₁₆	—	2	4	no	—
59	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
69	C₂×Dic₄₇	—	4	4	no	—
70	C₂×Dic₄₇	—	8	4	no	—
71	C₈×C₄₇	—	2	2	no	—
72	C₈×C₄₇	—	8	1	no	—
73	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
75	#13458	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	#450	1	1	yes	characteristic
78	C₂²×C₄₇	#476	1	1	yes	characteristic
79	C₂²×C₄₇	#501	1	1	yes	characteristic
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	—	8	2	no	—
82	C₄×C₄₇	#395	1	1	yes	characteristic
83	C₄×C₄₇	#448	1	1	yes	characteristic
84	C₄×C₄₇	#475	1	1	yes	characteristic
85	C₄×C₄₇	#498	1	1	yes	characteristic
86	Dic₄₇	—	8	4	no	—
87	C₂×#499	—	47	1	no	—
88	C₂×#365	—	47	2	no	—
89	#1945	D₄₇	1	1	yes	characteristic
90	#1386	—	47	2	no	—
91	#1376	—	47	1	no	—
92	C₂×C₄₇	—	2	1	no	—
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	#2058	1	1	yes	characteristic
95	C₂×C₄₇	#2231	1	2	yes	—
96	C₂×#125	—	94	1	no	—
97	C₂×M₃₂	—	47	2	no	—
98	#499	—	94	4	no	—
99	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
100	#427	C₂×D₄₇	1	1	yes	characteristic
101	C₂×#107	—	94	1	no	—
102	C₂×#106	—	47	2	no	—
103	#424	—	2	1	no	—
104	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
105	#365	—	94	4	no	—
106	#339	—	2	1	no	—
107	C₄₇	#13458	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	2	1	no	—
109	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
111	#125	—	94	4	no	—
112	#125	—	188	2	no	—
113	#117	—	2	1	no	—
114	C₂×C₄⋊C₄	—	94	2	no	—
115	C₂×C₄⋊C₄	—	188	2	no	—
116	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
117	M₃₂	—	47	4	no	—
118	M₃₂	—	94	2	no	—
119	C₂×C₁₆	—	94	2	no	—
120	#108	—	2	2	no	—
121	#107	—	94	4	no	—
122	#107	Dic₉₄	1	2	yes	—
123	#106	—	2	1	no	—
124	#106	—	47	4	no	—
125	#106	—	94	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	188	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	M₁₆	—	4	1	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	#10625335	1	2	yes	—
137	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
138	C₂×C₈	#10625350	1	1	yes	characteristic
139	C₂×C₈	D₄×D₄₇	1	2	yes	—
140	C₂×C₈	Q₈×D₄₇	1	2	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	94	4	no	—
143	C₄⋊C₄	—	188	8	no	—
144	C₄⋊C₄	—	188	2	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	#10625336	1	2	yes	—
149	C₁₆	—	94	4	no	—
150	C₂³	#13094049	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	376	4	no	—
155	C₂×C₄	—	188	4	no	—
156	C₂×C₄	#13093992	1	1	yes	characteristic
157	C₂×C₄	#13094003	1	1	yes	characteristic
158	C₂×C₄	#13094022	1	1	yes	characteristic
159	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
160	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
161	C₂×C₄	#13094054	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	8	1	no	—
164	C₈	#13094002	1	2	yes	—
165	C₈	#13094023	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#15689273	1	1	yes	characteristic
168	C₂²	#15689413	1	1	yes	characteristic
169	C₂²	#15689533	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	376	4	no	—
173	C₄	#15689043	1	1	yes	characteristic
174	C₄	#15689261	1	1	yes	characteristic
175	C₄	#15689408	1	1	yes	characteristic
176	C₄	#15689519	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#20244721	1	1	yes	characteristic
180	C₂	#20245564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689525

C₂

yes

maximal, characteristic

C₂×#15688947

C₂

yes

maximal

#20241493

C₂

yes

maximal

#20241470

C₂

yes

maximal, characteristic

C₄₇×#1945

C₂

yes

maximal, characteristic

C₂×#13094016

—

C₂×#13093964

C₂²

yes

—

#15689525

—

C₂×#13093958

—

C₂×#13093956

C₂²

yes

—

#15688947

—

C₂×#108×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

characteristic

C₄₇×#424

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#339

—

#13094016

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

#13093964

—

#13093964

Q₈

yes

—

C₂×C₄₇⋊C₁₆

—

#13093958

—

#13093956

—

#13093956

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#108×C₄₇

—

#107×C₄₇

Q₈

yes

—

#106×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

#137

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

C₄⋊Q₈

yes

—

#13458

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×#499

—

C₂×#365

—

#1945

D₄₇

yes

characteristic

#1386

—

#1376

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2058

yes

characteristic

C₂×C₄₇

#2231

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

C₂×#108

C₂×D₄₇

yes

characteristic

100

#427

C₂×D₄₇

yes

characteristic

101

C₂×#107

—

102

C₂×#106

—

103

#424

—

104

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

105

#365

—

106

#339

—

107

C₄₇

#13458

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

109

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

110

C₂²×C₈

C₂²×D₄₇

yes

characteristic

111

#125

—

112

#125

—

188

—

113

#117

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

188

—

116

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

117

M₃₂

—

118

M₃₂

—

119

C₂×C₁₆

—

120

#108

—

121

#107

—

122

#107

Dic₉₄

yes

—

123

#106

—

124

#106

—

125

#106

—

126

C₄⋊C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

188

—

130

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

#10625335

yes

—

137

C₂×C₈

C₂×Dic₉₄

yes

characteristic

138

C₂×C₈

#10625350

yes

characteristic

139

C₂×C₈

D₄×D₄₇

yes

—

140

C₂×C₈

Q₈×D₄₇

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

188

—

144

C₄⋊C₄

—

188

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

#10625336

yes

—

149

C₁₆

—

150

C₂³

#13094049

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

376

—

155

C₂×C₄

—

188

—

156

C₂×C₄

#13093992

yes

characteristic

157

C₂×C₄

#13094003

yes

characteristic

158

C₂×C₄

#13094022

yes

characteristic

159

C₂×C₄

C₂×#10625336

yes

characteristic

160

C₂×C₄

C₂×#10625335

yes

characteristic

161

C₂×C₄

#13094054

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

#13094002

yes

—

165

C₈

#13094023

yes

—

166

C₂²

—

167

C₂²

#15689273

yes

characteristic

168

C₂²

#15689413

yes

characteristic

169

C₂²

#15689533

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

376

—

173

C₄

#15689043

yes

characteristic

174

C₄

#15689261

yes

characteristic

175

C₄

#15689408

yes

characteristic

176

C₄

#15689519

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#20244721

yes

characteristic

180

C₂

#20245564

yes

—

	subgroup	G/H
center	C₂²	#15689413
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1945	C₂
socle	C₂²×C₄₇	#476
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
hypercenter	#1945	D₄₇
hypocenter	C₄₇	#13458
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#15689413

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1945

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
107. C₄₇	#13458	semidirect

H ≅ G/N

type

107. C₄₇

#13458

semidirect

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

subgroup	index
C₂×#15689525	2
C₂×#15688947	2
#20241493	2
#20241470	2
C₄₇×#1945	2
#13458	47

subgroup

index

C₄₇×#1945

Subgroups — #26135931

Conjugacy classes of subgroups

Complements

Maximal subgroups