Group #26132669 — subgroups

Subgroups — #26132669

Order 12032.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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×#15688898	C₂	1	1	yes	maximal, characteristic
2	#20243558	C₂	1	2	yes	maximal
3	#20241482	C₂	1	1	yes	maximal, characteristic
4	#20241481	C₂	1	2	yes	maximal
5	C₄₇×#1381	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	#15688898	C₄	1	4	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₄₇	C₂²	1	2	yes	—
14	C₂×#10625329	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	#13093960	—	2	2	no	—
20	#13093960	—	2	2	no	—
21	#13093959	—	4	4	no	—
22	Dic₄₇⋊C₈	—	4	4	no	—
23	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
24	C₈×Dic₄₇	Q₈	1	2	yes	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	—	2	2	no	—
28	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
29	#108×C₄₇	—	2	2	no	—
30	#108×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
33	M₁₆×C₄₇	—	4	2	no	—
34	M₁₆×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	#41	1	2	yes	—
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₄₇	D₈	1	2	yes	—
42	#10625329	—	4	2	no	—
43	#10625329	—	4	2	no	—
44	C₂×C₄₇⋊C₈	—	4	2	no	—
45	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
47	C₁₆×C₄₇	—	2	2	no	—
48	C₄₇⋊C₁₆	—	2	2	no	—
49	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#124	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
54	C₂×Dic₄₇	—	4	2	no	—
55	C₂×Dic₄₇	#107	1	4	yes	—
56	C₈×C₄₇	—	8	2	no	—
57	C₈×C₄₇	#117	1	1	yes	characteristic
58	C₈×C₄₇	#117	1	1	yes	characteristic
59	C₈×C₄₇	#125	1	2	yes	—
60	C₄₇⋊C₈	—	8	2	no	—
61	C₄₇⋊C₈	#106	1	4	yes	—
62	#12892	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
65	C₂²×C₄₇	#487	1	2	yes	—
66	C₄×C₄₇	#382	1	1	yes	characteristic
67	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
68	C₄×C₄₇	#486	1	2	yes	—
69	Dic₄₇	—	2	4	no	—
70	C₂×#346	—	47	1	no	—
71	#1848	—	47	2	no	—
72	#1381	—	47	1	no	—
73	#1381	—	47	2	no	—
74	#1381	D₄₇	1	1	yes	characteristic
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	#1834	1	1	yes	characteristic
77	C₂×C₄₇	#2161	1	2	yes	—
78	C₂²×C₁₆	—	47	1	no	—
79	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
80	C₂×#108	—	47	2	no	—
81	C₂×#108	—	94	2	no	—
82	C₂×#108	C₂×D₄₇	1	2	yes	—
83	#423	—	94	2	no	—
84	C₂×C₄×C₈	—	47	1	no	—
85	#346	—	47	4	no	—
86	C₄₇	#12892	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	94	2	no	—
89	C₂²×C₈	—	47	1	no	—
90	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×C₄²	—	47	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	47	2	no	—
94	C₂×C₁₆	—	94	2	no	—
95	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	94	2	no	—
99	#108	—	94	2	no	—
100	#108	—	188	4	no	—
101	C₄⋊C₈	—	188	4	no	—
102	C₄×C₈	—	47	2	no	—
103	C₄×C₈	—	47	2	no	—
104	C₂²×C₄	—	47	2	no	—
105	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	188	2	no	—
109	M₁₆	—	188	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	47	2	no	—
112	C₂×C₈	—	47	4	no	—
113	C₂×C₈	—	188	2	no	—
114	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₈	#10625350	1	2	yes	—
116	C₂×C₈	#10625352	1	2	yes	—
117	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
118	C₄²	—	47	2	no	—
119	C₄²	—	47	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	94	2	no	—
122	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
123	C₂×C₄	—	47	4	no	—
124	C₂×C₄	—	188	2	no	—
125	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄	#13094010	1	2	yes	—
127	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
128	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
129	C₈	—	8	2	no	—
130	C₈	—	47	4	no	—
131	C₈	—	376	2	no	—
132	C₈	#13093980	1	1	yes	characteristic
133	C₈	#13093981	1	1	yes	characteristic
134	C₈	#13094014	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#107×D₄₇	1	1	yes	characteristic
137	C₂²	#15689472	1	2	yes	—
138	C₄	—	94	4	no	—
139	C₄	#15688987	1	1	yes	characteristic
140	C₄	#106×D₄₇	1	1	yes	characteristic
141	C₄	#15689468	1	2	yes	—
142	C₂	—	2	2	no	—
143	C₂	#20243514	1	1	yes	characteristic
144	C₂	#20245181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688898

C₂

yes

maximal, characteristic

#20243558

C₂

yes

maximal

#20241482

C₂

yes

maximal, characteristic

#20241481

C₂

yes

maximal

C₄₇×#1381

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

C₂×#13093959

—

#15689146

—

C₂×C₈×Dic₄₇

C₂²

yes

characteristic

#15688898

C₄

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

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×C₁₆×C₄₇

C₂×C₄

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×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₄×Dic₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

SD₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#106

yes

—

#12892

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#487

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

#423

—

C₂×C₄×C₈

—

#346

—

C₄₇

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

—

—

—

—

100

#108

—

188

—

101

C₄⋊C₈

—

188

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₂²×C₄

—

105

C₂²×C₄

D₄×D₄₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

188

—

109

M₁₆

—

188

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

188

—

114

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

115

C₂×C₈

#10625350

yes

—

116

C₂×C₈

#10625352

yes

—

117

C₂×C₈

Q₈×D₄₇

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

122

C₂³

Q₁₆×D₄₇

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

188

—

125

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

126

C₂×C₄

#13094010

yes

—

127

C₂×C₄

D₈×D₄₇

yes

characteristic

128

C₂×C₄

SD₁₆×D₄₇

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

376

—

132

C₈

#13093980

yes

characteristic

133

C₈

#13093981

yes

characteristic

134

C₈

#13094014

yes

—

135

C₂²

—

136

C₂²

#107×D₄₇

yes

characteristic

137

C₂²

#15689472

yes

—

138

C₄

—

139

C₄

#15688987

yes

characteristic

140

C₄

#106×D₄₇

yes

characteristic

141

C₄

#15689468

yes

—

142

C₂

—

143

C₂

#20243514

yes

characteristic

144

C₂

#20245181

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1381	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1381	D₄₇
hypocenter	C₄₇	#12892
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1381

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 ◁ #1381

N	H ≅ G/N	type
86. C₄₇	#12892	semidirect

H ≅ G/N

type

86. C₄₇

#12892

semidirect

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

subgroup	index
C₂×#15688898	2
#20243558	2
#20241482	2
#20241481	2
C₄₇×#1381	2
#12892	47

subgroup

index

C₄₇×#1381

Subgroups — #26132669

Conjugacy classes of subgroups

Complements

Maximal subgroups