Group #26132728 — subgroups

Subgroups — #26132728

Order 12032.

306 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15688953	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688952	C₂	1	2	yes	maximal
3	C₂×#15688898	C₂	1	1	yes	maximal, characteristic
4	#20243558	C₂	1	2	yes	maximal
5	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	C₂×#13093959	—	2	2	no	—
9	#15689146	—	2	2	no	—
10	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688953	—	2	2	no	—
12	#15688952	—	2	4	no	—
13	#15688898	C₄	1	4	yes	—
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#368	—	2	2	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
22	C₂×C₄₇⋊C₁₆	Q₈	1	2	yes	—
23	#13093960	—	2	2	no	—
24	#13093960	—	2	2	no	—
25	#13093959	—	4	4	no	—
26	Dic₄₇⋊C₈	—	4	4	no	—
27	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₈×Dic₄₇	Q₈	1	2	yes	—
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
33	C₂×C₁₆×C₄₇	Q₈	1	2	yes	—
34	#108×C₄₇	—	2	2	no	—
35	#108×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
38	M₁₆×C₄₇	—	4	2	no	—
39	M₁₆×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	4	2	no	—
41	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
48	C₄×Dic₄₇	D₈	1	2	yes	—
49	#10625329	—	4	2	no	—
50	#10625329	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
54	C₂×C₄₇⋊C₈	Q₁₆	1	2	yes	—
55	C₁₆×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₄₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
57	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₄₇	—	4	2	no	—
65	C₂×Dic₄₇	#107	1	4	yes	—
66	C₈×C₄₇	—	8	2	no	—
67	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₄₇⋊C₈	—	8	2	no	—
72	C₄₇⋊C₈	#107	1	4	yes	—
73	#12910	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
76	C₂²×C₄₇	#494	1	1	yes	characteristic
77	C₂²×C₄₇	#500	1	1	yes	characteristic
78	C₄×C₄₇	#384	1	1	yes	characteristic
79	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₄₇	#493	1	1	yes	characteristic
81	C₄×C₄₇	#500	1	1	yes	characteristic
82	Dic₄₇	—	2	4	no	—
83	C₂×#368	—	47	1	no	—
84	C₂×#368	—	47	2	no	—
85	C₂×#368	D₄₇	1	1	yes	characteristic
86	C₂×#346	—	47	1	no	—
87	#1848	—	47	2	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	—	2	1	no	—
90	C₂×C₄₇	#1845	1	1	yes	characteristic
91	C₂×C₄₇	#2161	1	2	yes	—
92	C₂²×C₁₆	—	47	1	no	—
93	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
94	C₂×#108	—	47	2	no	—
95	C₂×#108	—	94	2	no	—
96	C₂×#108	C₂×D₄₇	1	2	yes	—
97	#423	—	94	2	no	—
98	C₂×C₄×C₈	—	47	1	no	—
99	#368	—	2	2	no	—
100	#368	—	94	4	no	—
101	#368	—	94	2	no	—
102	#346	—	47	4	no	—
103	C₄₇	#12910	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	94	2	no	—
106	C₂²×C₈	—	47	1	no	—
107	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
108	C₂×C₄²	—	47	1	no	—
109	C₂×C₁₆	—	47	2	no	—
110	C₂×C₁₆	—	47	2	no	—
111	C₂×C₁₆	—	47	2	no	—
112	C₂×C₁₆	Dic₉₄	1	2	yes	—
113	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
114	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
115	#108	—	2	2	no	—
116	#108	—	2	2	no	—
117	#108	—	94	2	no	—
118	#108	—	94	2	no	—
119	#108	—	188	4	no	—
120	C₄⋊C₈	—	188	4	no	—
121	C₄×C₈	—	47	2	no	—
122	C₄×C₈	—	47	2	no	—
123	C₂²×C₄	—	47	2	no	—
124	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	M₁₆	—	188	2	no	—
128	M₁₆	—	188	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	47	2	no	—
131	C₂×C₈	—	47	2	no	—
132	C₂×C₈	—	47	2	no	—
133	C₂×C₈	—	188	2	no	—
134	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
137	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₄²	—	47	2	no	—
141	C₄²	—	47	2	no	—
142	C₁₆	—	47	4	no	—
143	C₁₆	Dic₄₇⋊C₄	1	4	yes	—
144	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
145	C₂×C₄	—	47	4	no	—
146	C₂×C₄	—	188	2	no	—
147	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄	#13094020	1	1	yes	characteristic
149	C₂×C₄	#13094023	1	1	yes	characteristic
150	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
151	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
152	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	—	47	4	no	—
155	C₈	—	376	2	no	—
156	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
157	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
158	C₈	#13094022	1	1	yes	characteristic
159	C₈	#13094023	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	#107×D₄₇	1	1	yes	characteristic
162	C₂²	#15689501	1	1	yes	characteristic
163	C₂²	#15689527	1	1	yes	characteristic
164	C₄	—	94	4	no	—
165	C₄	#15688992	1	1	yes	characteristic
166	C₄	#107×D₄₇	1	1	yes	characteristic
167	C₄	#15689498	1	1	yes	characteristic
168	C₄	#15689527	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#20243546	1	1	yes	characteristic
172	C₂	#20245181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688953

C₂

yes

maximal, characteristic

C₂×#15688952

C₂

yes

maximal

C₂×#15688898

C₂

yes

maximal, characteristic

#20243558

C₂

yes

maximal

C₂×C₄₇×#368

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

C₂×#13093959

—

#15689146

—

C₂×C₈×Dic₄₇

C₂²

yes

—

—

C₄

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#368

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×C₄₇⋊C₁₆

Q₈

yes

—

#13093960

—

#13093960

—

#13093959

—

Dic₄₇⋊C₈

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₈×Dic₄₇

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₂×C₁₆×C₄₇

Q₈

yes

—

#108×C₄₇

—

#108×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

Q₁₆

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₄×Dic₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂×C₄₇⋊C₈

Q₁₆

yes

—

C₁₆×C₄₇

C₄⋊C₄

yes

—

C₄₇⋊C₁₆

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

#107

yes

—

#12910

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×#368

—

C₂×#368

—

C₂×#368

D₄₇

yes

characteristic

C₂×#346

—

#1848

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

#423

—

C₂×C₄×C₈

—

#368

—

100

#368

—

101

#368

—

102

#346

—

103

C₄₇

#12910

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂²×D₄₇

yes

characteristic

108

C₂×C₄²

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₉₄

yes

—

113

C₂×C₁₆

C₄×D₄₇

yes

—

114

C₂×C₁₆

C₂²⋊D₄₇

yes

—

115

#108

—

116

#108

—

117

#108

—

118

#108

—

119

#108

—

188

—

120

C₄⋊C₈

—

188

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₂²×C₄

—

124

C₂²×C₄

D₄×D₄₇

yes

characteristic

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

188

—

128

M₁₆

—

188

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

188

—

134

C₂×C₈

C₂×Dic₉₄

yes

characteristic

135

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

136

C₂×C₈

D₄×D₄₇

yes

characteristic

137

C₂×C₈

Q₈×D₄₇

yes

characteristic

138

C₂×C₈

Q₈×D₄₇

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₄²

—

141

C₄²

—

142

C₁₆

—

143

C₁₆

Dic₄₇⋊C₄

yes

—

144

C₂³

Q₁₆×D₄₇

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

188

—

147

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

148

C₂×C₄

#13094020

yes

characteristic

149

C₂×C₄

#13094023

yes

characteristic

150

C₂×C₄

D₈×D₄₇

yes

characteristic

151

C₂×C₄

D₈×D₄₇

yes

characteristic

152

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

376

—

156

C₈

C₄⋊C₄×D₄₇

yes

characteristic

157

C₈

C₂×Dic₄₇⋊C₄

yes

characteristic

158

C₈

#13094022

yes

characteristic

159

C₈

#13094023

yes

characteristic

160

C₂²

—

161

C₂²

#107×D₄₇

yes

characteristic

162

C₂²

#15689501

yes

characteristic

163

C₂²

#15689527

yes

characteristic

164

C₄

—

165

C₄

#15688992

yes

characteristic

166

C₄

#107×D₄₇

yes

characteristic

167

C₄

#15689498

yes

characteristic

168

C₄

#15689527

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

#20243546

yes

characteristic

172

C₂

#20245181

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#368	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#368	D₄₇
hypocenter	C₄₇	#12910
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#368

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
103. C₄₇	#12910	semidirect

H ≅ G/N

type

103. C₄₇

#12910

semidirect

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

subgroup	index
C₂×#15688953	2
C₂×#15688952	2
C₂×#15688898	2
#20243558	2
C₂×C₄₇×#368	2
#12910	47

subgroup

index

C₂×C₄₇×#368

Subgroups — #26132728

Conjugacy classes of subgroups

Complements

Maximal subgroups