Group #13135740 — subgroups

Subgroups — #13135740

Order 1728.

270 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 187 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	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₉⋊C₁₆)	C₂	1	2	yes	maximal
3	#10589327	—	3	1	no	maximal
4	#10588138	—	3	3	no	maximal
5	C₂×C₃×C₈×C₉	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
8	C₃⋊(C₉⋊C₁₆)	C₄	1	4	yes	complemented
9	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	—	3	2	no	—
11	C₄×C₈×C₉	D₃	1	3	yes	—
12	C₂×C₉⋊C₁₆	—	3	6	no	—
13	C₂×C₃×C₄×C₉	C₈	1	2	yes	—
14	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
15	C₃×C₈×C₉	C₈	1	2	yes	—
16	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
17	#4461	—	9	3	no	—
18	#4461	—	9	1	no	—
19	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
21	C₃²×C₄²	Dic₃	1	1	yes	characteristic
22	C₃⋊(C₃⋊C₁₆)	—	3	4	no	—
23	C₂×C₈×C₉	Dic₃	1	3	yes	—
24	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
25	C₄²×C₉	Dic₃	1	3	yes	—
26	C₉⋊C₁₆	—	3	12	no	—
27	C₂²×C₃×C₉	C₂×C₈	1	1	yes	characteristic
28	C₃×C₄×C₉	—	2	2	no	—
29	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
30	C₃×C₄×C₉	C₂×C₈	1	1	yes	characteristic
31	C₃×C₄×C₈	D₉	1	3	yes	—
32	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
33	C₂×C₃⋊C₁₆	—	9	6	no	—
34	C₂×C₃⋊C₁₆	—	9	2	no	—
35	C₂×C₃²×C₄	C₃⋊C₈	1	2	yes	—
36	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
37	C₃²×C₈	C₃⋊C₈	1	2	yes	—
38	C₃²×C₈	C₄×D₃	1	2	yes	—
39	C₂×C₄×C₉	C₃⋊C₈	1	6	yes	—
40	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
41	C₈×C₉	C₃⋊C₈	1	6	yes	—
42	C₈×C₉	C₄×D₃	1	6	yes	—
43	#346	—	27	1	no	Sylow
44	C₂×C₃×C₉	C₄×C₈	1	1	yes	characteristic
45	C₂×C₃×C₉	M₃₂	1	2	yes	—
46	C₂×C₃×C₈	Dic₉	1	3	yes	—
47	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
48	C₂×C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
49	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₃×C₄²	Dic₉	1	3	yes	—
51	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
52	C₃⋊C₁₆	—	9	12	no	—
53	C₃⋊C₁₆	—	9	4	no	—
54	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
55	C₃²×C₄	—	2	2	no	—
56	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
57	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
58	C₂²×C₉	C₂×C₃⋊C₈	1	3	yes	—
59	C₄×C₉	—	2	6	no	—
60	C₄×C₉	C₂×C₃⋊C₈	1	3	yes	—
61	C₄×C₉	C₄×Dic₃	1	3	yes	—
62	C₂×C₁₆	—	27	2	no	—
63	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic
64	C₃×C₉	#346	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃×C₄	C₉⋊C₈	1	6	yes	—
66	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
67	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
68	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
69	C₃×C₈	C₉⋊C₈	1	6	yes	—
70	C₃×C₈	C₄×D₉	1	6	yes	—
71	C₃×C₈	C₃⋊(C₃⋊C₈)	1	2	yes	—
72	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
73	C₂×C₃²	C₄×C₃⋊C₈	1	1	yes	characteristic
74	C₂×C₃²	#812	1	2	yes	—
75	C₂×C₉	C₄×C₃⋊C₈	1	3	yes	—
76	C₂×C₉	#812	1	6	yes	—
77	C₂×C₈	C₃⋊Dic₉	1	1	yes	characteristic
78	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
79	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
80	C₁₆	—	27	4	no	—
81	C₂²×C₃	C₂×C₉⋊C₈	1	3	yes	—
82	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
83	C₃×C₄	—	2	6	no	—
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	C₂×C₉⋊C₈	1	3	yes	—
86	C₃×C₄	C₄×Dic₉	1	3	yes	—
87	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
88	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
89	C₃²	#4461	1	1	yes	characteristic
90	C₉	#4461	1	3	yes	complemented
91	C₂×C₄	C₃⋊(C₉⋊C₈)	1	2	yes	—
92	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
93	C₈	C₃⋊(C₉⋊C₈)	1	2	yes	—
94	C₈	C₄×C₃⋊D₉	1	2	yes	—
95	C₂×C₃	C₄×C₉⋊C₈	1	3	yes	—
96	C₂×C₃	#63371	1	6	yes	—
97	C₂×C₃	C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
98	C₂×C₃	#63639	1	2	yes	—
99	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
102	C₄	C₄×C₃⋊Dic₉	1	1	yes	characteristic
103	C₃	#10588138	1	3	yes	complemented
104	C₃	#10589327	1	1	yes	characteristic
105	C₂	C₄×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
106	C₂	#11720534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×C₈×C₉

C₂

yes

maximal, characteristic

C₂×C₃⋊(C₉⋊C₁₆)

C₂

yes

—

—

C₂×C₃×C₈×C₉

C₄

yes

characteristic

C₂×C₃×C₈×C₉

C₂²

yes

characteristic

C₃×C₄²×C₉

C₄

yes

characteristic

C₃⋊(C₉⋊C₁₆)

C₄

yes

complemented

C₃²×C₄×C₈

D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

—

C₄×C₈×C₉

D₃

yes

—

C₂×C₉⋊C₁₆

—

C₂×C₃×C₄×C₉

C₈

yes

—

C₂×C₃×C₄×C₉

C₂×C₄

yes

characteristic

C₃×C₈×C₉

C₈

yes

—

C₃×C₈×C₉

C₂×C₄

yes

—

#4461

—

#4461

—

C₂×C₃²×C₈

Dic₃

yes

characteristic

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₃²×C₄²

Dic₃

yes

characteristic

C₃⋊(C₃⋊C₁₆)

—

C₂×C₈×C₉

Dic₃

yes

—

C₂×C₈×C₉

C₂×D₃

yes

—

C₄²×C₉

Dic₃

yes

—

C₉⋊C₁₆

—

C₂²×C₃×C₉

C₂×C₈

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₄²

yes

characteristic

C₃×C₄×C₉

C₂×C₈

yes

characteristic

C₃×C₄×C₈

D₉

yes

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃²×C₄

C₃⋊C₈

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₃²×C₈

C₃⋊C₈

yes

—

C₃²×C₈

C₄×D₃

yes

—

C₂×C₄×C₉

C₃⋊C₈

yes

—

C₂×C₄×C₉

C₂×Dic₃

yes

—

C₈×C₉

C₃⋊C₈

yes

—

C₈×C₉

C₄×D₃

yes

—

#346

—

Sylow

C₂×C₃×C₉

C₄×C₈

yes

characteristic

C₂×C₃×C₉

M₃₂

yes

—

C₂×C₃×C₈

Dic₉

yes

—

C₂×C₃×C₈

C₂×D₉

yes

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄²

Dic₉

yes

—

C₃×C₄²

C₃⋊Dic₃

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊C₈

yes

characteristic

C₃²×C₄

C₄×Dic₃

yes

characteristic

C₂²×C₉

C₂×C₃⋊C₈

yes

—

C₄×C₉

—

C₄×C₉

C₂×C₃⋊C₈

yes

—

C₄×C₉

C₄×Dic₃

yes

—

C₂×C₁₆

—

C₄×C₈

C₃⋊D₉

yes

characteristic

C₃×C₉

#346

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

C₉⋊C₈

yes

—

C₂×C₃×C₄

C₂×Dic₉

yes

—

C₂×C₃×C₄

C₃⋊(C₃⋊C₈)

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₃×C₈

C₉⋊C₈

yes

—

C₃×C₈

C₄×D₉

yes

—

C₃×C₈

C₃⋊(C₃⋊C₈)

yes

—

C₃×C₈

C₄×C₃⋊D₃

yes

—

C₂×C₃²

C₄×C₃⋊C₈

yes

characteristic

C₂×C₃²

#812

yes

—

C₂×C₉

C₄×C₃⋊C₈

yes

—

C₂×C₉

#812

yes

—

C₂×C₈

C₃⋊Dic₉

yes

characteristic

C₂×C₈

C₂×C₃⋊D₉

yes

characteristic

C₄²

C₃⋊Dic₉

yes

characteristic

C₁₆

—

C₂²×C₃

C₂×C₉⋊C₈

yes

—

C₂²×C₃

C₂×C₃⋊(C₃⋊C₈)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₉⋊C₈

yes

—

C₃×C₄

C₄×Dic₉

yes

—

C₃×C₄

C₂×C₃⋊(C₃⋊C₈)

yes

characteristic

C₃×C₄

C₄×C₃⋊Dic₃

yes

C₃²

yes

C₉

yes

C₂×C₄

C₃⋊(C₉⋊C₈)

yes

—

C₂×C₄

C₂×C₃⋊Dic₉

yes

characteristic

C₈

C₃⋊(C₉⋊C₈)

yes

—

C₈

C₄×C₃⋊D₉

yes

—

C₂×C₃

C₄×C₉⋊C₈

yes

—

C₂×C₃

#63371

yes

—

C₂×C₃

C₄×C₃⋊(C₃⋊C₈)

yes

characteristic

C₂×C₃

#63639

yes

—

C₂²

C₂×C₃⋊(C₉⋊C₈)

yes

characteristic

100

C₄

—

101

C₄

C₂×C₃⋊(C₉⋊C₈)

yes

characteristic

102

C₄

C₄×C₃⋊Dic₉

yes

characteristic

103

C₃

#10588138

yes

complemented

104

C₃

#10589327

yes

characteristic

105

C₂

C₄×C₃⋊(C₉⋊C₈)

yes

characteristic

106

C₂

#11720534

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₄×C₈
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃⋊C₈
cosocle	C₂×C₃×C₈×C₉	C₂²
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	#346
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×C₃⋊D₉

commutator

C₂×C₃×C₉

C₄×C₈

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting

C₃×C₄×C₈×C₉

C₂

socle, prime radical

C₂²×C₃²

C₂×C₃⋊C₈

cosocle

C₂×C₃×C₈×C₉

C₂²

hypercenter

C₄×C₈

C₃⋊D₉

C₃×C₉

G ▷ C₂×C₃×C₉ ▷ C₃×C₉

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. C₃⋊(C₉⋊C₁₆)	C₄	semidirect
64. C₃×C₉	#346	semidirect
90. C₉	#4461	semidirect
103. C₃	#10588138	semidirect

H ≅ G/N

type

8. C₃⋊(C₉⋊C₁₆)

C₄

semidirect

64. C₃×C₉

#346

semidirect

90. C₉

#4461

semidirect

103. C₃

#10588138

semidirect

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

subgroup	index
C₃×C₄×C₈×C₉	2
C₂×C₃⋊(C₉⋊C₁₆)	2
#10589327	3
#10588138	3

subgroup

index

C₃×C₄×C₈×C₉

C₂×C₃⋊(C₉⋊C₁₆)

#10589327

#10588138

Subgroups — #13135740

Conjugacy classes of subgroups

Complements

Maximal subgroups