Group #26134872 — subgroups

Subgroups — #26134872

Order 12032.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20245144	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
3	#20241496	C₂	1	4	yes	maximal
4	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×#13093960	C₂²	1	2	yes	—
7	C₄×#10625329	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
9	#15688888	—	2	1	no	—
10	C₂×#125×C₄₇	—	2	1	no	—
11	C₄₇×#499	—	2	2	no	—
12	C₄₇×#499	C₄	1	4	yes	—
13	C₂×#107×C₄₇	C₂²	1	2	yes	—
14	C₂×#106×C₄₇	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#10625329	—	2	2	no	—
17	#13093964	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	#13093960	—	2	2	no	—
20	#13093960	Q₈	1	4	yes	—
21	#13093935	—	4	1	no	—
22	C₄×C₄₇⋊C₈	—	4	1	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	#125×C₄₇	—	2	4	no	—
26	#125×C₄₇	—	4	1	no	—
27	#125×C₄₇	—	4	1	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	#107×C₄₇	—	2	2	no	—
32	#107×C₄₇	C₂×C₄	1	4	yes	—
33	#106×C₄₇	—	2	4	no	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₂²×C₄×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
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×Q₈	1	2	yes	—
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₄⋊C₄×C₄₇	—	4	4	no	—
47	C₄⋊C₄×C₄₇	—	4	4	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₄₇	D₈	1	1	yes	characteristic
51	C₄²×C₄₇	D₈	1	1	yes	characteristic
52	#10625329	—	2	4	no	—
53	#10625329	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₄₇⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	4	4	no	—
59	C₂×C₄×C₄₇	—	8	4	no	—
60	C₂×C₄×C₄₇	#102	1	2	yes	—
61	C₂×C₄×C₄₇	#102	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#137	1	2	yes	—
66	C₈×C₄₇	—	2	2	no	—
67	C₈×C₄₇	C₄×Q₈	1	2	yes	—
68	C₈×C₄₇	#124	1	1	yes	characteristic
69	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₄₇⋊C₈	—	4	4	no	—
71	#13280	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
74	C₂²×C₄₇	#475	1	2	yes	—
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	8	4	no	—
78	C₄×C₄₇	#391	1	1	yes	characteristic
79	C₄×C₄₇	#419	1	1	yes	characteristic
80	C₄×C₄₇	#474	1	2	yes	—
81	C₂×#499	D₄₇	1	1	yes	characteristic
82	#2150	—	47	1	no	—
83	C₄×#108	—	47	1	no	—
84	#1387	—	47	4	no	—
85	C₂×C₄₇	—	2	2	no	—
86	C₂×C₄₇	#1988	1	1	yes	characteristic
87	C₂×C₄₇	#2148	1	2	yes	—
88	C₂×#125	—	2	1	no	—
89	C₂×M₃₂	—	47	2	no	—
90	#499	—	2	2	no	—
91	#499	Dic₄₇	1	4	yes	—
92	C₂×#108	—	47	2	no	—
93	C₂×#107	C₂×D₄₇	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₄×M₁₆	—	94	1	no	—
96	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
97	C₄⋊C₁₆	—	94	2	no	—
98	#341	—	94	1	no	—
99	C₄₇	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	94	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
109	M₃₂	—	94	4	no	—
110	C₂×C₁₆	—	94	2	no	—
111	#108	—	47	4	no	—
112	#108	—	94	2	no	—
113	#107	—	2	2	no	—
114	#107	C₂×Dic₄₇	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	188	1	no	—
117	C₄×C₈	—	188	1	no	—
118	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
119	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#10625335	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	M₁₆	—	94	4	no	—
124	M₁₆	—	188	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	188	2	no	—
127	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
128	C₂×C₈	Q₈×D₄₇	1	2	yes	—
129	C₂×C₈	#10625354	1	2	yes	—
130	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	#10625334	1	1	yes	characteristic
137	C₄²	#10625334	1	1	yes	characteristic
138	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
139	C₁₆	—	188	2	no	—
140	C₂³	C₂×#10625335	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#13093945	1	2	yes	—
145	C₂×C₄	#13093945	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
147	C₂×C₄	#13094000	1	2	yes	—
148	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
149	C₂×C₄	#13094055	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	188	4	no	—
152	C₈	Q₈×Dic₄₇	1	2	yes	—
153	C₈	#13094013	1	1	yes	characteristic
154	C₈	#13094022	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#13093945	1	1	yes	characteristic
157	C₂²	#15689405	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#15689030	1	1	yes	characteristic
162	C₄	#15689132	1	1	yes	characteristic
163	C₄	#15689400	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#20244341	1	1	yes	characteristic
166	C₂	#20245137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245144

C₂

yes

maximal, characteristic

C₄×#13093960

C₂

yes

maximal, characteristic

#20241496

C₂

yes

maximal

C₂×C₄₇×#499

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093960

C₂²

yes

—

C₄×#10625329

—

C₄⋊(C₄₇⋊C₁₆)

—

#15688888

—

C₂×#125×C₄₇

—

C₄₇×#499

—

C₄₇×#499

C₄

yes

—

C₂×#107×C₄₇

C₂²

yes

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

Q₈

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#107×C₄₇

C₂×C₄

yes

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#137

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₄×Q₈

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₄₇⋊C₈

—

#13280

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#102

yes

characteristic

C₂²×C₄₇

#475

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×#499

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1988

yes

characteristic

C₂×C₄₇

#2148

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₄₇

yes

—

C₂×#108

—

C₂×#107

C₂×D₄₇

yes

—

C₂×#106

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

#341

—

C₄₇

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

102

C₂²×C₈

C₂²×D₄₇

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#108

—

113

#107

—

114

#107

C₂×Dic₄₇

yes

—

115

#106

—

116

#97

—

188

—

117

C₄×C₈

—

188

—

118

C₄×C₈

C₂×Dic₄₇

yes

—

119

C₄×C₈

C₂²⋊D₄₇

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

#10625335

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

188

—

125

C₂×C₈

—

126

C₂×C₈

—

188

—

127

C₂×C₈

C₂²⋊Dic₄₇

yes

—

128

C₂×C₈

Q₈×D₄₇

yes

—

129

C₂×C₈

#10625354

yes

—

130

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

#10625334

yes

characteristic

137

C₄²

#10625334

yes

characteristic

138

C₄²

C₂²⋊Dic₄₇

yes

—

139

C₁₆

—

188

—

140

C₂³

C₂×#10625335

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#13093945

yes

—

145

C₂×C₄

#13093945

yes

—

146

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

147

C₂×C₄

#13094000

yes

—

148

C₂×C₄

C₂×#10625334

yes

characteristic

149

C₂×C₄

#13094055

yes

—

150

C₈

—

151

C₈

—

188

—

152

C₈

Q₈×Dic₄₇

yes

—

153

C₈

#13094013

yes

characteristic

154

C₈

#13094022

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#13093945

yes

characteristic

157

C₂²

#15689405

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#15689030

yes

characteristic

162

C₄

#15689132

yes

characteristic

163

C₄

#15689400

yes

—

164

C₂

—

165

C₂

#20244341

yes

characteristic

166

C₂

#20245137

yes

—

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#499	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#499	D₄₇
hypocenter	C₄₇	#13280
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#13093945

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#499

C₂

socle

C₂²×C₄₇

C₂×#102

prime radical

C₂³×C₄₇

C₂×SD₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
99. C₄₇	#13280	semidirect

H ≅ G/N

type

99. C₄₇

#13280

semidirect

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

subgroup	index
#20245144	2
C₄×#13093960	2
#20241496	2
C₂×C₄₇×#499	2
#13280	47

subgroup

index

C₂×C₄₇×#499

Subgroups — #26134872

Conjugacy classes of subgroups

Complements

Maximal subgroups