Group #13141472 — subgroups

Subgroups — #10588365⋊C₃ #13141472

Order 1728.

388 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 132 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×Dic₃₆⋊C₃	C₂	1	4	yes	maximal
2	#63431⋊C₃	C₂	1	2	yes	maximal
3	C₄×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
4	C₃×#4699	—	3	1	no	maximal
5	#10588365	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×3₋¹⁺²⋊Q₈	C₂²	1	4	yes	—
7	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
8	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄⋊(3₋¹⁺²⋊C₄)	—	2	4	no	—
10	Dic₃₆⋊C₃	—	2	8	no	—
11	C₂×C₃×Dic₁₂	—	3	4	no	—
12	C₃×#869	—	3	2	no	—
13	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
14	C₂×Dic₃₆	C₂×C₃	1	4	yes	—
15	#63431	C₂×C₃	1	2	yes	—
16	C₄×C₈×C₉	—	2	1	no	—
17	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
19	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
20	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
21	3₋¹⁺²⋊Q₈	—	2	8	no	—
22	C₈×3₋¹⁺²	D₄	1	4	yes	—
23	C₃×#494	—	9	1	no	—
24	#4699	—	3	1	no	—
25	C₂×C₃×Dic₆	—	3	4	no	—
26	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
27	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄⋊Dic₃	—	6	4	no	—
29	C₃×Dic₁₂	—	6	8	no	—
30	C₂×Dic₁₈	C₂²×C₃	1	4	yes	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
33	C₄²×C₉	—	2	1	no	—
34	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
35	C₄⋊Dic₉	—	2	4	no	—
36	Dic₃₆	—	2	8	no	—
37	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
38	C₄×3₋¹⁺²	Q₁₆	1	4	yes	complemented
39	C₄×3₋¹⁺²	C₂×D₄	1	2	yes	—
40	3₋¹⁺²⋊C₄	—	4	4	no	—
41	C₂×C₃×Q₁₆	—	9	4	no	—
42	C₃×C₄⋊Q₈	—	9	2	no	—
43	C₂×Dic₁₂	—	3	4	no	—
44	#869	—	3	2	no	—
45	C₃×C₄×C₈	—	3	1	no	—
46	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
47	C₂×C₃²×C₄	D₁₂	1	2	yes	—
48	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
49	C₂×C₃×Dic₃	—	6	4	no	—
50	C₃×Dic₆	—	6	8	no	—
51	C₃²×C₈	D₁₂	1	4	yes	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	1	no	—
54	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
55	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
56	C₂×Dic₉	—	2	4	no	—
57	Dic₁₈	—	2	8	no	—
58	C₈×C₉	—	2	4	no	—
59	C₈×C₉	C₃×D₄	1	4	yes	—
60	#494	—	9	1	no	Sylow
61	C₂×3₋¹⁺²	#127	1	1	yes	characteristic
62	C₂×3₋¹⁺²	C₂×Q₁₆	1	2	yes	—
63	C₂×C₃×Q₈	—	9	4	no	—
64	C₂×Dic₆	—	3	4	no	—
65	C₃×Q₁₆	—	18	8	no	—
66	C₂×C₃×C₈	—	3	2	no	—
67	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
68	C₃×C₄⋊C₄	—	18	4	no	—
69	C₃×C₄²	—	3	1	no	—
70	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
71	C₄⋊Dic₃	—	6	4	no	—
72	Dic₁₂	—	6	8	no	—
73	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
74	C₃²×C₄	Dic₁₂	1	4	yes	—
75	C₃²×C₄	C₂×D₁₂	1	2	yes	—
76	C₃×Dic₃	—	12	4	no	—
77	C₂²×C₉	—	2	1	no	—
78	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
79	C₄×C₉	—	2	4	no	—
80	C₄×C₉	—	2	2	no	—
81	C₄×C₉	C₃×Q₁₆	1	4	yes	complemented
82	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
83	Dic₉	—	4	4	no	—
84	C₂×Q₁₆	—	9	4	no	—
85	C₄⋊Q₈	—	9	2	no	—
86	C₄×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
87	3₋¹⁺²	#494	1	1	yes	characteristic, complemented, Sylow
88	C₃×Q₈	—	18	8	no	—
89	C₂×C₃×C₄	—	3	2	no	—
90	C₂×C₃×C₄	—	3	1	no	—
91	C₂×C₃×C₄	—	18	4	no	—
92	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
93	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
94	C₂×Dic₃	—	6	4	no	—
95	Dic₆	—	6	8	no	—
96	C₃×C₈	—	3	4	no	—
97	C₃×C₈	C₃×D₁₂	1	4	yes	—
98	C₂×C₃²	#874	1	1	yes	characteristic
99	C₂×C₃²	C₂×Dic₁₂	1	2	yes	—
100	C₂×C₉	—	2	2	no	—
101	C₂×C₉	—	2	1	no	—
102	C₂×C₉	C₃×#127	1	1	yes	characteristic
103	C₂×C₉	C₂×C₃×Q₁₆	1	2	yes	—
104	C₂×Q₈	—	9	4	no	—
105	Q₁₆	—	18	8	no	—
106	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
107	C₄⋊C₄	—	18	4	no	—
108	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
109	C₂²×C₃	—	3	1	no	—
110	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
111	C₃×C₄	—	3	2	no	—
112	C₃×C₄	—	3	4	no	—
113	C₃×C₄	—	36	4	no	—
114	C₃×C₄	C₃×Dic₁₂	1	4	yes	—
115	C₃×C₄	C₂×C₃×D₁₂	1	2	yes	—
116	Dic₃	—	12	4	no	—
117	C₃²	#4699	1	1	yes	characteristic
118	C₉	—	2	1	no	—
119	C₉	C₃×#494	1	1	yes	characteristic, complemented
120	Q₈	—	18	8	no	—
121	C₂×C₄	—	18	4	no	—
122	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
123	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
124	C₈	C₄⋊AGL(1,ℤ/9)	1	4	yes	—
125	C₂×C₃	—	3	2	no	—
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	C₃×#874	1	1	yes	characteristic
128	C₂×C₃	C₂×C₃×Dic₁₂	1	2	yes	—
129	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
130	C₄	—	36	4	no	—
131	C₄	Dic₃₆⋊C₃	1	4	yes	complemented
132	C₄	C₂×C₄⋊AGL(1,ℤ/9)	1	2	yes	—
133	C₃	—	3	1	no	—
134	C₃	C₃×#4699	1	1	yes	characteristic
135	C₂	#11721394	1	1	yes	characteristic
136	C₂	C₂×Dic₃₆⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₃₆⋊C₃

C₂

yes

maximal

#63431⋊C₃

C₂

yes

maximal

C₄×C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

C₃×#4699

—

maximal

#10588365

C₃

yes

maximal, characteristic, complemented

C₂×3₋¹⁺²⋊Q₈

C₂²

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

—

Dic₃₆⋊C₃

—

C₂×C₃×Dic₁₂

—

C₃×#869

—

C₃²×C₄×C₈

D₃

yes

characteristic

C₂×Dic₃₆

C₂×C₃

yes

—

#63431

C₂×C₃

yes

—

C₄×C₈×C₉

—

C₄×C₈×C₉

C₂×C₃

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊Q₈

—

C₈×3₋¹⁺²

D₄

yes

—

C₃×#494

—

#4699

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×Dic₁₂

—

C₂×Dic₁₈

C₂²×C₃

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

—

Dic₃₆

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

Q₁₆

yes

complemented

C₄×3₋¹⁺²

C₂×D₄

yes

—

3₋¹⁺²⋊C₄

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

#869

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃×D₃

yes

characteristic

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

D₁₂

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

C₂×Q₁₆

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₁₂

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₃²×C₄

Dic₁₂

yes

—

C₃²×C₄

C₂×D₁₂

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×Q₁₆

yes

complemented

C₄×C₉

C₂×C₃×D₄

yes

—

Dic₉

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×C₈

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

#494

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×D₁₂

yes

—

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

C₃×D₁₂

yes

—

C₂×C₃²

#874

yes

characteristic

C₂×C₃²

C₂×Dic₁₂

yes

—

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

C₃×#127

yes

characteristic

103

C₂×C₉

C₂×C₃×Q₁₆

yes

—

104

C₂×Q₈

—

105

Q₁₆

—

106

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

107

C₄⋊C₄

—

108

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

109

C₂²×C₃

—

110

C₂²×C₃

C₂×C₃×D₁₂

yes

characteristic

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

C₃×Dic₁₂

yes

—

115

C₃×C₄

C₂×C₃×D₁₂

yes

—

116

Dic₃

—

117

C₃²

#4699

yes

characteristic

118

C₉

—

119

C₉

C₃×#494

yes

characteristic, complemented

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

123

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

124

C₈

C₄⋊AGL(1,ℤ/9)

yes

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

C₃×#874

yes

characteristic

128

C₂×C₃

C₂×C₃×Dic₁₂

yes

—

129

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

characteristic

130

C₄

—

131

C₄

Dic₃₆⋊C₃

yes

complemented

132

C₄

C₂×C₄⋊AGL(1,ℤ/9)

yes

—

133

C₃

—

134

C₃

C₃×#4699

yes

characteristic

135

C₂

#11721394

yes

characteristic

136

C₂

C₂×Dic₃₆⋊C₃

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×C₈×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₁₂
prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	C₄×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#494
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₄×C₈×3₋¹⁺²

C₂

socle

C₂²×C₃

C₂×C₃×D₁₂

prime radical

C₂²×C₃²

C₂×D₁₂

C₄×C₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
5. #10588365	C₃	semidirect
38. C₄×3₋¹⁺²	Q₁₆	semidirect
81. C₄×C₉	C₃×Q₁₆	semidirect
87. 3₋¹⁺²	#494	semidirect
119. C₉	C₃×#494	semidirect
131. C₄	Dic₃₆⋊C₃	semidirect

H ≅ G/N

type

5. #10588365

C₃

semidirect

38. C₄×3₋¹⁺²

Q₁₆

semidirect

81. C₄×C₉

C₃×Q₁₆

semidirect

87. 3₋¹⁺²

#494

semidirect

119. C₉

C₃×#494

semidirect

131. C₄

Dic₃₆⋊C₃

semidirect

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

subgroup	index
C₂×Dic₃₆⋊C₃	2
#63431⋊C₃	2
C₄×C₈×3₋¹⁺²	2
C₃×#4699	3
#10588365	3

subgroup

index

C₂×Dic₃₆⋊C₃

#63431⋊C₃

C₄×C₈×3₋¹⁺²

C₃×#4699

#10588365

Subgroups — #10588365⋊C₃ #13141472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588365⋊C3 #13141472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588365⋊C₃ #13141472