Group #26132729 — subgroups

Subgroups — #26132729

Order 12032.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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×#15688952	C₂	1	4	yes	maximal
2	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
3	#20243560	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093960	—	2	1	no	—
7	C₂×#13093960	C₂²	1	2	yes	—
8	#15689147	—	2	1	no	—
9	#15688952	—	2	8	no	—
10	#15688899	C₄	1	4	yes	—
11	C₂×#108×C₄₇	—	2	1	no	—
12	C₂×#108×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#10625329	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
18	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	—
19	#13093960	—	2	4	no	—
20	#13093960	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	#108×C₄₇	—	2	4	no	—
27	#108×C₄₇	—	4	2	no	—
28	C₄⋊C₈×C₄₇	—	4	2	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₄₇	Q₈	1	2	yes	—
31	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	M₁₆×C₄₇	—	4	2	no	—
34	M₁₆×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₄₇	D₈	1	2	yes	—
38	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₄₇	D₈	1	2	yes	—
45	#10625329	—	4	2	no	—
46	#10625329	—	4	2	no	—
47	C₂×C₄₇⋊C₈	—	4	2	no	—
48	C₄₇⋊C₁₆	C₄⋊C₄	1	8	yes	—
49	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	#107	1	4	yes	—
52	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₄₇	—	8	2	no	—
59	C₈×C₄₇	#107	1	4	yes	—
60	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
61	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
62	C₄₇⋊C₈	—	8	2	no	—
63	#12910	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	2	2	no	—
65	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
66	C₂²×C₄₇	#494	1	1	yes	characteristic
67	C₂²×C₄₇	#500	1	1	yes	characteristic
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	#384	1	1	yes	characteristic
70	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
71	C₄×C₄₇	#493	1	1	yes	characteristic
72	C₄×C₄₇	#500	1	1	yes	characteristic
73	C₂×#368	—	47	4	no	—
74	C₂×#346	—	47	1	no	—
75	#1848	—	47	1	no	—
76	#1848	D₄₇	1	1	yes	characteristic
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	#1845	1	1	yes	characteristic
80	C₂×C₄₇	#2161	1	2	yes	—
81	C₂²×C₁₆	—	47	2	no	—
82	C₂×#108	—	2	1	no	—
83	C₂×#108	—	47	2	no	—
84	C₂×#108	—	94	1	no	—
85	C₂×#108	C₂×D₄₇	1	2	yes	—
86	#423	—	2	1	no	—
87	#423	—	94	1	no	—
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	#368	—	94	8	no	—
90	#346	—	47	4	no	—
91	C₄₇	#12910	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	94	2	no	—
94	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂×C₁₆	—	47	4	no	—
98	C₂×C₁₆	—	47	4	no	—
99	C₂×C₁₆	—	47	4	no	—
100	#108	—	2	4	no	—
101	#108	—	4	2	no	—
102	#108	—	94	4	no	—
103	#108	—	188	2	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	188	2	no	—
106	C₄×C₈	Dic₉₄	1	2	yes	—
107	C₄×C₈	C₄×D₄₇	1	2	yes	—
108	C₂²×C₄	#10625337	1	2	yes	—
109	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	188	2	no	—
113	M₁₆	—	188	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	188	2	no	—
116	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
117	C₂×C₈	#10625334	1	2	yes	—
118	C₂×C₈	#10625337	1	2	yes	—
119	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₈	D₄×D₄₇	1	2	yes	—
122	C₂×C₈	Q₈×D₄₇	1	2	yes	—
123	C₄²	Dic₄₇⋊C₄	1	2	yes	—
124	C₄²	#10625334	1	2	yes	—
125	C₁₆	—	47	8	no	—
126	C₂³	C₂×#10625337	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	#13093957	1	4	yes	—
129	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#13094020	1	1	yes	characteristic
131	C₂×C₄	#13094022	1	1	yes	characteristic
132	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
133	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
134	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
135	C₈	—	8	2	no	—
136	C₈	—	376	2	no	—
137	C₈	#13093957	1	4	yes	—
138	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
139	C₈	#13094023	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	C₂×#13093957	1	1	yes	characteristic
142	C₂²	#15689502	1	1	yes	characteristic
143	C₂²	#15689526	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	#15688992	1	1	yes	characteristic
146	C₄	C₂×#13093957	1	1	yes	characteristic
147	C₄	#15689499	1	1	yes	characteristic
148	C₄	#15689526	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#20243547	1	1	yes	characteristic
152	C₂	#20245182	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688952

C₂

yes

maximal

C₂×#15688899

C₂

yes

maximal, characteristic

#20243560

C₂

yes

maximal, characteristic

C₄₇×#1848

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₂×#13093960

—

C₂×#13093960

C₂²

yes

—

#15689147

—

#15688952

—

#15688899

C₄

yes

—

C₂×#108×C₄₇

—

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#423

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×C₄₇⋊C₁₆

Q₈

yes

—

#13093960

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

#108×C₄₇

—

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

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

C₈×C₄₇

—

C₈×C₄₇

#107

yes

—

C₈×C₄₇

C₂×C₄⋊C₄

yes

—

C₈×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

—

#12910

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

—

—

—

—

C₂×D₄₇

yes

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#368

—

#346

—

C₄₇

#12910

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

101

#108

—

102

#108

—

103

#108

—

188

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

188

—

106

C₄×C₈

Dic₉₄

yes

—

107

C₄×C₈

C₄×D₄₇

yes

—

108

C₂²×C₄

#10625337

yes

—

109

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

188

—

113

M₁₆

—

188

—

114

C₂×C₈

—

115

C₂×C₈

—

188

—

116

C₂×C₈

Dic₄₇⋊C₄

yes

—

117

C₂×C₈

#10625334

yes

—

118

C₂×C₈

#10625337

yes

—

119

C₂×C₈

C₂×Dic₉₄

yes

characteristic

120

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

121

C₂×C₈

D₄×D₄₇

yes

—

122

C₂×C₈

Q₈×D₄₇

yes

—

123

C₄²

Dic₄₇⋊C₄

yes

—

124

C₄²

#10625334

yes

—

125

C₁₆

—

126

C₂³

C₂×#10625337

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

#13093957

yes

—

129

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

130

C₂×C₄

#13094020

yes

characteristic

131

C₂×C₄

#13094022

yes

characteristic

132

C₂×C₄

C₂×#10625334

yes

characteristic

133

C₂×C₄

C₂×#10625334

yes

characteristic

134

C₂×C₄

C₂×#10625337

yes

characteristic

135

C₈

—

136

C₈

—

376

—

137

C₈

#13093957

yes

—

138

C₈

C₄⋊C₄×D₄₇

yes

—

139

C₈

#13094023

yes

—

140

C₂²

—

141

C₂²

C₂×#13093957

yes

characteristic

142

C₂²

#15689502

yes

characteristic

143

C₂²

#15689526

yes

characteristic

144

C₄

—

145

C₄

#15688992

yes

characteristic

146

C₄

C₂×#13093957

yes

characteristic

147

C₄

#15689499

yes

characteristic

148

C₄

#15689526

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#20243547

yes

characteristic

152

C₂

#20245182

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#12910
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₂×C₄

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1848

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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

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

H ≅ G/N

type

91. C₄₇

#12910

semidirect

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

subgroup	index
C₂×#15688952	2
C₂×#15688899	2
#20243560	2
C₄₇×#1848	2
#12910	47

subgroup

index

C₄₇×#1848

Subgroups — #26132729

Conjugacy classes of subgroups

Complements

Maximal subgroups