Group #13143742 — subgroups

Subgroups — #13143742

Order 1728.

490 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 231 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	#11721706	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	#11720540	C₂	1	4	yes	maximal
4	#10592379	—	3	1	no	maximal
5	#10588360	—	3	3	no	maximal
6	C₂×#6226	—	2	2	no	—
7	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
8	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
10	C₄⋊(C₃⋊Dic₉)	C₂²	1	4	yes	—
11	#64078	—	3	2	no	—
12	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
13	#63644	—	3	4	no	—
14	#63431	—	3	6	no	—
15	C₄×C₈×C₉	D₃	1	3	yes	—
16	#63377	—	3	12	no	—
17	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
18	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
19	C₂×C₃⋊Dic₉	—	2	4	no	—
20	#6226	—	4	4	no	—
21	C₃×C₈×C₉	Q₈	1	4	yes	complemented
22	#4680	—	9	3	no	—
23	#4680	—	9	1	no	—
24	C₂×#634	—	6	2	no	—
25	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
26	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
27	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
28	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
29	C₂×Dic₁₈	—	6	6	no	—
30	C₂×C₈×C₉	C₂×D₃	1	6	yes	—
31	C₄²×C₉	C₂×D₃	1	3	yes	—
32	C₄⋊Dic₉	—	3	12	no	—
33	C₄⋊Dic₉	—	6	6	no	—
34	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
35	C₃×C₄×C₉	SD₁₆	1	4	yes	—
36	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
37	C₃⋊Dic₉	—	4	4	no	—
38	#869	—	9	6	no	—
39	#869	—	9	2	no	—
40	C₃×C₄×C₈	D₉	1	3	yes	—
41	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
42	#817	—	9	12	no	—
43	#817	—	9	4	no	—
44	C₂×C₃²×C₄	D₁₂	1	2	yes	—
45	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
46	C₂×C₃⋊Dic₃	—	6	4	no	—
47	#634	—	12	4	no	—
48	C₃²×C₈	Dic₆	1	4	yes	—
49	C₂×C₄×C₉	D₁₂	1	6	yes	—
50	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
51	C₂×Dic₉	—	6	12	no	—
52	Dic₁₈	—	12	12	no	—
53	C₈×C₉	Dic₆	1	12	yes	complemented
54	#498	—	27	1	no	Sylow
55	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
57	C₂×Dic₆	—	18	6	no	—
58	C₂×Dic₆	—	18	2	no	—
59	C₂×C₃×C₈	C₂×D₉	1	6	yes	—
60	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
61	C₃×C₄²	C₂×D₉	1	3	yes	—
62	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₄⋊Dic₃	—	9	12	no	—
64	C₄⋊Dic₃	—	9	4	no	—
65	C₄⋊Dic₃	—	18	6	no	—
66	C₄⋊Dic₃	—	18	2	no	—
67	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
68	C₃²×C₄	#204	1	4	yes	—
69	C₃²×C₄	C₂×Dic₆	1	2	yes	—
70	C₃⋊Dic₃	—	12	4	no	—
71	C₂²×C₉	C₂×D₁₂	1	3	yes	—
72	C₄×C₉	#204	1	12	yes	—
73	C₄×C₉	C₂×Dic₆	1	6	yes	—
74	Dic₉	—	12	12	no	—
75	C₄⋊Q₈	—	27	2	no	—
76	#106	—	27	4	no	—
77	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic
78	C₃×C₉	#498	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃×C₄	D₃₆	1	6	yes	—
80	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
81	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
82	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
83	C₂×Dic₃	—	18	12	no	—
84	C₂×Dic₃	—	18	4	no	—
85	Dic₆	—	36	12	no	—
86	Dic₆	—	36	4	no	—
87	C₃×C₈	Dic₁₈	1	12	yes	complemented
88	C₃×C₈	#634	1	4	yes	—
89	C₂×C₃²	#869	1	1	yes	characteristic
90	C₂×C₃²	C₂×#204	1	2	yes	—
91	C₂×C₉	#869	1	3	yes	—
92	C₂×C₉	C₂×#204	1	6	yes	—
93	C₂×Q₈	—	54	2	no	—
94	C₂×C₈	C₂×C₃⋊D₉	1	2	yes	—
95	C₄⋊C₄	—	27	4	no	—
96	C₄⋊C₄	—	54	2	no	—
97	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
98	C₂²×C₃	C₂×D₃₆	1	3	yes	—
99	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
100	C₃×C₄	#3664	1	12	yes	—
101	C₃×C₄	C₂×Dic₁₈	1	6	yes	—
102	C₃×C₄	#3744	1	4	yes	—
103	C₃×C₄	C₂×#634	1	2	yes	—
104	Dic₃	—	36	4	no	—
105	Dic₃	—	36	12	no	—
106	C₃²	#4680	1	1	yes	characteristic
107	C₉	#4680	1	3	yes	complemented
108	Q₈	—	108	4	no	—
109	C₂×C₄	—	54	4	no	—
110	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
111	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
112	C₈	#6226	1	4	yes	complemented
113	C₂×C₃	#63431	1	3	yes	—
114	C₂×C₃	C₂×#3664	1	6	yes	—
115	C₂×C₃	#64078	1	1	yes	characteristic
116	C₂×C₃	C₂×#3744	1	2	yes	—
117	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
118	C₄	—	108	4	no	—
119	C₄	#88524	1	4	yes	—
120	C₄	C₂×#6226	1	2	yes	—
121	C₃	#10588360	1	3	yes	complemented
122	C₃	#10592379	1	1	yes	characteristic
123	C₂	#11721706	1	1	yes	characteristic
124	C₂	C₂×#88524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11721706

C₂

yes

maximal

C₃×C₄×C₈×C₉

C₂

yes

maximal, characteristic

#11720540

C₂

yes

—

—

—

C₂×C₃×C₈×C₉

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

#64078

—

C₃²×C₄×C₈

D₃

yes

characteristic

#63644

—

#63431

—

C₄×C₈×C₉

D₃

yes

—

#63377

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₃×C₈×C₉

Q₈

yes

—

—

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×Dic₁₈

—

C₂×C₈×C₉

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

SD₁₆

yes

—

C₃×C₄×C₉

C₂×Q₈

yes

—

C₃⋊Dic₉

—

#869

—

#869

—

C₃×C₄×C₈

D₉

yes

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

#817

—

#817

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

Dic₆

yes

—

C₂×C₄×C₉

D₁₂

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

Dic₁₈

—

C₈×C₉

Dic₆

yes

complemented

#498

—

Sylow

C₂×C₃×C₉

C₄⋊Q₈

yes

characteristic

C₂×C₃×C₉

C₂×SD₁₆

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×D₉

yes

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×D₉

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₃²×C₄

#204

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃⋊Dic₃

—

C₂²×C₉

C₂×D₁₂

yes

—

C₄×C₉

#204

yes

—

C₄×C₉

C₂×Dic₆

yes

—

Dic₉

—

C₄⋊Q₈

—

#106

—

C₄×C₈

C₃⋊D₉

yes

characteristic

C₃×C₉

#498

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

D₃₆

yes

—

C₂×C₃×C₄

C₂²×D₉

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

Dic₁₈

yes

complemented

C₃×C₈

#634

yes

—

C₂×C₃²

#869

yes

characteristic

C₂×C₃²

C₂×#204

yes

—

C₂×C₉

#869

yes

—

C₂×C₉

C₂×#204

yes

—

C₂×Q₈

—

C₂×C₈

C₂×C₃⋊D₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₃⋊D₉

yes

characteristic

C₂²×C₃

C₂×D₃₆

yes

—

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

100

C₃×C₄

#3664

yes

—

101

C₃×C₄

C₂×Dic₁₈

yes

—

102

C₃×C₄

#3744

yes

—

103

C₃×C₄

C₂×#634

yes

—

104

Dic₃

—

105

Dic₃

—

106

C₃²

#4680

yes

characteristic

107

C₉

#4680

yes

complemented

108

Q₈

—

108

—

109

C₂×C₄

—

110

C₂×C₄

C₄⋊(C₃⋊D₉)

yes

—

111

C₂×C₄

C₂²×C₃⋊D₉

yes

characteristic

112

C₈

#6226

yes

complemented

113

C₂×C₃

#63431

yes

—

114

C₂×C₃

C₂×#3664

yes

—

115

C₂×C₃

#64078

yes

characteristic

116

C₂×C₃

C₂×#3744

yes

—

117

C₂²

C₂×C₄⋊(C₃⋊D₉)

yes

characteristic

118

C₄

—

108

—

119

C₄

#88524

yes

—

120

C₄

C₂×#6226

yes

—

121

C₃

#10588360

yes

complemented

122

C₃

#10592379

yes

characteristic

123

C₂

#11721706

yes

characteristic

124

C₂

C₂×#88524

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₃⋊D₉)
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	#498
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊(C₃⋊D₉)

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₃×C₄×C₈×C₉

C₂

socle, prime radical

C₂²×C₃²

C₂×D₁₂

hypercenter

C₄×C₈

C₃⋊D₉

hypocenter

C₃×C₉

#498

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
21. C₃×C₈×C₉	Q₈	semidirect
53. C₈×C₉	Dic₆	semidirect
78. C₃×C₉	#498	semidirect
87. C₃×C₈	Dic₁₈	semidirect
107. C₉	#4680	semidirect
112. C₈	#6226	semidirect
121. C₃	#10588360	semidirect

H ≅ G/N

type

21. C₃×C₈×C₉

Q₈

semidirect

53. C₈×C₉

Dic₆

semidirect

78. C₃×C₉

#498

semidirect

87. C₃×C₈

Dic₁₈

semidirect

107. C₉

#4680

semidirect

112. C₈

#6226

semidirect

121. C₃

#10588360

semidirect

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

subgroup	index
#11721706	2
C₃×C₄×C₈×C₉	2
#11720540	2
#10592379	3
#10588360	3

subgroup

index

C₃×C₄×C₈×C₉

Subgroups — #13143742

Conjugacy classes of subgroups

Complements

Maximal subgroups