Group #13143262 — subgroups

Subgroups — #13143262

Order 1728.

283 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 162 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	3₋¹⁺²×#131	C₂	1	1	yes	maximal, characteristic
2	#63371⋊C₃	C₂	1	3	yes	maximal, complemented
3	C₂×3₋¹⁺²⋊C₁₆	C₂	1	3	yes	maximal, complemented
4	C₃×#5140	—	3	1	no	maximal
5	#10588820	C₃	1	1	yes	maximal, characteristic, complemented
6	#41×3₋¹⁺²	C₄	1	1	yes	characteristic
7	M₁₆×3₋¹⁺²	C₄	1	3	yes	—
8	C₂×C₈×3₋¹⁺²	C₂²	1	3	yes	—
9	3₋¹⁺²⋊C₁₆	C₂²	1	1	yes	characteristic
10	3₋¹⁺²⋊C₁₆	C₂²	1	3	yes	—
11	C₃²×#131	D₃	1	1	yes	characteristic
12	C₃×#812	—	3	3	no	—
13	C₂×C₃×C₃⋊C₁₆	—	3	3	no	—
14	C₉×#131	—	2	1	no	—
15	C₉×#131	C₂×C₃	1	1	yes	characteristic
16	#63371	C₂×C₃	1	3	yes	complemented
17	C₂×C₉⋊C₁₆	C₂×C₃	1	3	yes	complemented
18	Q₈×3₋¹⁺²	C₈	1	1	yes	characteristic
19	D₄×3₋¹⁺²	C₈	1	3	yes	—
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	3	yes	—
21	C₈×3₋¹⁺²	C₂×C₄	1	3	yes	—
22	C₈×3₋¹⁺²	C₂³	1	1	yes	characteristic
23	C₃×#504	—	9	1	no	—
24	#5140	—	3	1	no	—
25	C₃²×#41	Dic₃	1	1	yes	characteristic
26	C₃²×M₁₆	Dic₃	1	3	yes	—
27	C₂×C₃²×C₈	C₂×D₃	1	3	yes	—
28	C₉×#41	—	2	1	no	—
29	C₉×#41	C₃×C₄	1	1	yes	characteristic
30	C₃×C₃⋊C₁₆	—	3	1	no	—
31	C₃×C₃⋊C₁₆	—	3	3	no	—
32	C₉×M₁₆	—	2	3	no	—
33	C₉×M₁₆	C₃×C₄	1	3	yes	—
34	C₂×C₈×C₉	—	2	3	no	—
35	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
36	C₉⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
37	C₉⋊C₁₆	C₂²×C₃	1	3	yes	—
38	C₂²×3₋¹⁺²	C₂×C₈	1	3	yes	—
39	C₄×3₋¹⁺²	C₂×C₈	1	3	yes	—
40	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
41	C₃×#131	—	3	1	no	—
42	C₃×#131	C₃×D₃	1	1	yes	characteristic
43	C₃×M₃₂	—	9	3	no	—
44	C₂×C₃×C₁₆	—	9	3	no	—
45	#812	—	3	3	no	—
46	C₂×C₃⋊C₁₆	—	3	3	no	—
47	C₃²×Q₈	C₃⋊C₈	1	1	yes	characteristic
48	C₃²×D₄	C₃⋊C₈	1	3	yes	—
49	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
50	C₃²×C₈	C₂×Dic₃	1	3	yes	—
51	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
52	Q₈×C₉	—	2	1	no	—
53	Q₈×C₉	C₃×C₈	1	1	yes	characteristic
54	D₄×C₉	—	2	3	no	—
55	D₄×C₉	C₃×C₈	1	3	yes	—
56	C₂×C₄×C₉	—	2	3	no	—
57	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
58	C₈×C₉	—	2	3	no	—
59	C₈×C₉	—	2	1	no	—
60	C₈×C₉	C₂×C₃×C₄	1	3	yes	—
61	C₈×C₉	C₂³×C₃	1	1	yes	characteristic
62	#504	—	9	1	no	Sylow
63	C₂×3₋¹⁺²	—	2	3	no	—
64	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
65	C₃×#41	—	3	1	no	—
66	C₃×#41	C₃×Dic₃	1	1	yes	characteristic
67	C₃×M₁₆	—	3	3	no	—
68	C₃×M₁₆	C₃×Dic₃	1	3	yes	—
69	C₂×C₃×C₈	—	3	3	no	—
70	C₂×C₃×C₈	C₂×C₃×D₃	1	3	yes	—
71	C₃×C₁₆	—	9	1	no	—
72	C₃×C₁₆	—	9	3	no	—
73	C₃⋊C₁₆	—	3	1	no	—
74	C₃⋊C₁₆	—	3	3	no	—
75	C₂²×C₃²	C₂×C₃⋊C₈	1	3	yes	—
76	C₃²×C₄	C₂×C₃⋊C₈	1	3	yes	—
77	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
78	C₂²×C₉	—	2	3	no	—
79	C₂²×C₉	C₂×C₃×C₈	1	3	yes	—
80	C₄×C₉	—	2	3	no	—
81	C₄×C₉	—	2	1	no	—
82	C₄×C₉	C₂×C₃×C₈	1	3	yes	—
83	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
84	#131	AGL(1,ℤ/9)	1	1	yes	characteristic
85	M₃₂	—	9	3	no	—
86	C₂×C₁₆	—	9	3	no	—
87	3₋¹⁺²	#504	1	1	yes	characteristic, complemented, Sylow
88	C₃×Q₈	—	3	1	no	—
89	C₃×Q₈	C₃×C₃⋊C₈	1	1	yes	characteristic
90	C₃×D₄	—	3	3	no	—
91	C₃×D₄	C₃×C₃⋊C₈	1	3	yes	—
92	C₂×C₃×C₄	—	3	3	no	—
93	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
94	C₃×C₈	—	3	3	no	—
95	C₃×C₈	—	3	1	no	—
96	C₃×C₈	C₂×C₃×Dic₃	1	3	yes	—
97	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
98	C₂×C₃²	—	2	3	no	—
99	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
100	C₂×C₉	—	2	3	no	—
101	C₂×C₉	—	2	1	no	—
102	C₂×C₉	—	4	3	no	—
103	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
104	#41	3₋¹⁺²⋊C₄	1	1	yes	characteristic
105	M₁₆	3₋¹⁺²⋊C₄	1	3	yes	—
106	C₂×C₈	C₂×AGL(1,ℤ/9)	1	3	yes	—
107	C₁₆	—	9	1	no	—
108	C₁₆	—	9	3	no	—
109	C₂²×C₃	—	3	3	no	—
110	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	3	yes	—
111	C₃×C₄	—	3	1	no	—
112	C₃×C₄	—	3	3	no	—
113	C₃×C₄	C₂×C₃×C₃⋊C₈	1	3	yes	—
114	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
115	C₃²	#5140	1	1	yes	characteristic
116	C₉	—	2	1	no	—
117	C₉	C₃×#504	1	1	yes	characteristic, complemented
118	Q₈	3₋¹⁺²⋊C₈	1	1	yes	characteristic
119	D₄	3₋¹⁺²⋊C₈	1	3	yes	—
120	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	3	yes	—
121	C₈	C₂×3₋¹⁺²⋊C₄	1	3	yes	—
122	C₈	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
123	C₂×C₃	—	2	3	no	—
124	C₂×C₃	—	3	1	no	—
125	C₂×C₃	—	6	3	no	—
126	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×3₋¹⁺²⋊C₈	1	3	yes	—
128	C₄	C₂×3₋¹⁺²⋊C₈	1	3	yes	—
129	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
130	C₃	—	3	1	no	—
131	C₃	C₃×#5140	1	1	yes	characteristic
132	C₂	—	2	3	no	—
133	C₂	C₂²×3₋¹⁺²⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#131

C₂

yes

maximal, characteristic

#63371⋊C₃

C₂

yes

maximal, complemented

C₂×3₋¹⁺²⋊C₁₆

C₂

yes

maximal, complemented

C₃×#5140

—

maximal

#10588820

C₃

yes

maximal, characteristic, complemented

#41×3₋¹⁺²

C₄

yes

characteristic

M₁₆×3₋¹⁺²

C₄

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

3₋¹⁺²⋊C₁₆

C₂²

yes

characteristic

3₋¹⁺²⋊C₁₆

C₂²

yes

—

C₃²×#131

D₃

yes

characteristic

C₃×#812

—

C₂×C₃×C₃⋊C₁₆

—

C₉×#131

—

C₉×#131

C₂×C₃

yes

characteristic

#63371

C₂×C₃

yes

complemented

C₂×C₉⋊C₁₆

C₂×C₃

yes

complemented

Q₈×3₋¹⁺²

C₈

yes

characteristic

D₄×3₋¹⁺²

C₈

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₈×3₋¹⁺²

C₂×C₄

yes

—

C₈×3₋¹⁺²

C₂³

yes

characteristic

C₃×#504

—

#5140

—

C₃²×#41

Dic₃

yes

characteristic

C₃²×M₁₆

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₉×#41

—

C₉×#41

C₃×C₄

yes

characteristic

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₉×M₁₆

—

C₉×M₁₆

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉⋊C₁₆

C₂²×C₃

yes

characteristic

C₉⋊C₁₆

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

C₂×C₈

yes

—

C₄×3₋¹⁺²

C₂×C₈

yes

—

C₄×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₃×#131

—

C₃×#131

C₃×D₃

yes

characteristic

C₃×M₃₂

—

C₂×C₃×C₁₆

—

#812

—

C₂×C₃⋊C₁₆

—

C₃²×Q₈

C₃⋊C₈

yes

characteristic

C₃²×D₄

C₃⋊C₈

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₃²×C₈

C₂×Dic₃

yes

—

C₃²×C₈

C₂²×D₃

yes

characteristic

Q₈×C₉

—

Q₈×C₉

C₃×C₈

yes

characteristic

D₄×C₉

—

D₄×C₉

C₃×C₈

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

C₂³×C₃

yes

characteristic

#504

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²×C₈

yes

characteristic

C₃×#41

—

C₃×#41

C₃×Dic₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

C₃×Dic₃

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

—

C₃²×C₄

C₂×C₃⋊C₈

yes

—

C₃²×C₄

C₂²×Dic₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

C₂²×C₃×C₄

yes

yes

M₃₂

—

C₂×C₁₆

—

3₋¹⁺²

#504

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

C₃×C₃⋊C₈

yes

characteristic

C₃×D₄

—

C₃×D₄

C₃×C₃⋊C₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃×Dic₃

yes

—

C₃×C₈

C₂²×C₃×D₃

yes

characteristic

C₂×C₃²

—

C₂×C₃²

C₂²×C₃⋊C₈

yes

characteristic

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

—

103

C₂×C₉

C₂²×C₃×C₈

yes

characteristic

104

#41

3₋¹⁺²⋊C₄

yes

characteristic

105

M₁₆

3₋¹⁺²⋊C₄

yes

—

106

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

107

C₁₆

—

108

C₁₆

—

109

C₂²×C₃

—

110

C₂²×C₃

C₂×C₃×C₃⋊C₈

yes

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

C₂×C₃×C₃⋊C₈

yes

—

114

C₃×C₄

C₂²×C₃×Dic₃

yes

characteristic

115

C₃²

#5140

yes

characteristic

116

C₉

—

117

C₉

C₃×#504

yes

characteristic, complemented

118

Q₈

3₋¹⁺²⋊C₈

yes

characteristic

119

D₄

3₋¹⁺²⋊C₈

yes

—

120

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

121

C₈

C₂×3₋¹⁺²⋊C₄

yes

—

122

C₈

C₂²×AGL(1,ℤ/9)

yes

characteristic

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

C₂²×C₃×C₃⋊C₈

yes

characteristic

127

C₂²

C₂×3₋¹⁺²⋊C₈

yes

—

128

C₄

C₂×3₋¹⁺²⋊C₈

yes

—

129

C₄

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

130

C₃

—

131

C₃

C₃×#5140

yes

characteristic

132

C₂

—

133

C₂

C₂²×3₋¹⁺²⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₃×C₈	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#131	C₂
socle	C₂×C₃	C₂²×C₃×C₃⋊C₈
prime radical	C₃²×#41	Dic₃
cosocle	C₈×C₉	C₂³×C₃
hypercenter	#131	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#504
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₈ ◁ #131

subgroup

G/H

center

C₈

C₂²×AGL(1,ℤ/9)

commutator

C₂×C₉

C₂²×C₃×C₈

Frattini

C₃×C₈

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#131

C₂

socle

C₂×C₃

C₂²×C₃×C₃⋊C₈

prime radical

C₃²×#41

Dic₃

cosocle

C₈×C₉

C₂³×C₃

C₉

G ▷ C₂×C₉ ▷ C₉

C₁ ◁ C₈ ◁ #131

N	H ≅ G/N	type
2. #63371⋊C₃	C₂	semidirect
3. C₂×3₋¹⁺²⋊C₁₆	C₂	semidirect
5. #10588820	C₃	semidirect
16. #63371	C₂×C₃	semidirect
17. C₂×C₉⋊C₁₆	C₂×C₃	semidirect
87. 3₋¹⁺²	#504	semidirect
117. C₉	C₃×#504	semidirect

H ≅ G/N

type

2. #63371⋊C₃

C₂

semidirect

3. C₂×3₋¹⁺²⋊C₁₆

C₂

semidirect

5. #10588820

C₃

semidirect

16. #63371

C₂×C₃

semidirect

17. C₂×C₉⋊C₁₆

C₂×C₃

semidirect

87. 3₋¹⁺²

#504

semidirect

117. C₉

C₃×#504

semidirect

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

subgroup	index
3₋¹⁺²×#131	2
#63371⋊C₃	2
C₂×3₋¹⁺²⋊C₁₆	2
C₃×#5140	3
#10588820	3

subgroup

index

3₋¹⁺²×#131

#63371⋊C₃

C₂×3₋¹⁺²⋊C₁₆

C₃×#5140

#10588820

Subgroups — #13143262

Conjugacy classes of subgroups

Complements

Maximal subgroups