Group #13142444 — subgroups

Subgroups — #13142444

Order 1728.

468 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 172 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×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²⋊#95	C₂	1	6	yes	maximal
3	C₃×#4937	—	3	1	no	maximal
4	#10588617	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
6	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
7	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
9	C₃×C₃⋊#95	—	3	6	no	—
10	C₂×C₄²×C₉	—	2	1	no	—
11	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
12	C₉⋊#95	C₂×C₃	1	6	yes	—
13	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
14	C₂×C₄×3₋¹⁺²	—	2	6	no	—
15	C₂×C₄×3₋¹⁺²	C₂×C₄	1	6	yes	—
16	C₂×3₋¹⁺²⋊C₄	—	2	12	no	—
17	C₃×#383	—	9	1	no	—
18	#4937	—	3	1	no	—
19	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
20	C₂²×C₃×Dic₃	—	3	4	no	—
21	C₃²×C₄²	Dic₃	1	4	yes	—
22	C₂²×C₄×C₉	—	2	3	no	—
23	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
24	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
25	C₄²×C₉	—	2	4	no	—
26	C₄²×C₉	C₃×C₄	1	4	yes	complemented
27	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
28	C₂²×3₋¹⁺²	#41	1	6	yes	—
29	C₄×3₋¹⁺²	—	2	6	no	—
30	3₋¹⁺²⋊C₄	—	4	4	no	—
31	C₂×C₃×C₄²	—	3	1	no	—
32	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
33	C₃×#95	—	9	6	no	—
34	C₃⋊#95	—	3	6	no	—
35	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
36	C₂×C₃²×C₄	—	2	6	no	—
37	C₂×C₃²×C₄	C₂×Dic₃	1	6	yes	—
38	C₂×C₃×Dic₃	—	6	12	no	—
39	C₂³×C₉	—	2	1	no	—
40	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
41	C₂×C₄×C₉	—	2	12	no	—
42	C₂×C₄×C₉	—	2	6	no	—
43	C₂×C₄×C₉	—	2	6	no	—
44	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
45	C₂×Dic₉	—	2	12	no	—
46	#383	—	9	1	no	Sylow
47	C₂×3₋¹⁺²	#117	1	3	yes	—
48	C₂×3₋¹⁺²	#126	1	4	yes	—
49	C₂²×C₃×C₄	—	3	3	no	—
50	C₂²×C₃×C₄	—	9	4	no	—
51	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
52	C₂²×Dic₃	—	3	4	no	—
53	C₃×C₄²	—	3	4	no	—
54	C₃×C₄²	C₃×Dic₃	1	4	yes	—
55	C₂²×C₃²	#235	1	6	yes	—
56	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
57	C₃²×C₄	—	2	6	no	—
58	C₃×Dic₃	—	12	4	no	—
59	C₂²×C₉	—	2	6	no	—
60	C₂²×C₉	—	2	1	no	—
61	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
62	C₂²×C₉	C₃×#41	1	6	yes	—
63	C₄×C₉	—	2	12	no	—
64	C₄×C₉	—	2	6	no	—
65	Dic₉	—	4	4	no	—
66	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
67	#95	—	9	6	no	—
68	3₋¹⁺²	#383	1	1	yes	characteristic, complemented, Sylow
69	C₂³×C₃	—	3	1	no	—
70	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
71	C₂×C₃×C₄	—	2	6	no	—
72	C₂×C₃×C₄	—	3	6	no	—
73	C₂×C₃×C₄	—	6	6	no	—
74	C₂×C₃×C₄	—	18	12	no	—
75	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
76	C₂×Dic₃	—	6	12	no	—
77	C₂×C₃²	#876	1	4	yes	—
78	C₂×C₃²	#926	1	3	yes	—
79	C₂×C₉	—	2	3	no	—
80	C₂×C₉	—	2	4	no	—
81	C₂×C₉	C₃×#117	1	3	yes	—
82	C₂×C₉	C₃×#126	1	4	yes	—
83	C₂²×C₄	—	9	4	no	—
84	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
85	C₄²	3₋¹⁺²⋊C₄	1	4	yes	complemented
86	C₂²×C₃	—	3	6	no	—
87	C₂²×C₃	—	3	1	no	—
88	C₂²×C₃	C₃×#235	1	6	yes	—
89	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
90	C₃×C₄	—	2	6	no	—
91	C₃×C₄	—	6	6	no	—
92	C₃×C₄	—	36	4	no	—
93	Dic₃	—	12	4	no	—
94	C₃²	#4937	1	1	yes	characteristic
95	C₉	—	2	1	no	—
96	C₉	C₃×#383	1	1	yes	characteristic, complemented
97	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
98	C₂×C₄	—	2	6	no	—
99	C₂×C₄	—	18	12	no	—
100	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	6	yes	—
101	C₂×C₃	—	3	3	no	—
102	C₂×C₃	—	3	4	no	—
103	C₂×C₃	C₃×#876	1	4	yes	—
104	C₂×C₃	C₃×#926	1	3	yes	—
105	C₂²	#3697⋊C₃	1	6	yes	—
106	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	36	4	no	—
109	C₃	—	3	1	no	—
110	C₃	C₃×#4937	1	1	yes	characteristic
111	C₂	#11721396	1	4	yes	—
112	C₂	#63488⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²⋊#95

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄²×3₋¹⁺²

C₄

yes

complemented

C₂×C₃²×C₄²

D₃

yes

characteristic

C₃×C₃⋊#95

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

C₂×C₃

yes

characteristic

C₉⋊#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₃×#383

—

#4937

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₃²×C₄²

Dic₃

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×3₋¹⁺²

#41

yes

—

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₃×D₃

yes

characteristic

C₃×#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×Dic₉

—

#383

—

Sylow

C₂×3₋¹⁺²

#117

yes

—

C₂×3₋¹⁺²

#126

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₃×C₄²

—

C₃×C₄²

C₃×Dic₃

yes

—

C₂²×C₃²

#235

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

characteristic

C₃²×C₄

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

characteristic

C₂²×C₉

C₃×#41

yes

—

C₄×C₉

—

C₄×C₉

—

Dic₉

—

C₂×C₄²

AGL(1,ℤ/9)

yes

characteristic

#95

—

3₋¹⁺²

#383

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×Dic₃

—

C₂×C₃²

#876

yes

—

C₂×C₃²

#926

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃×#117

yes

—

C₂×C₉

C₃×#126

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

C₄²

3₋¹⁺²⋊C₄

yes

complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃×#235

yes

—

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

C₃²

#4937

yes

characteristic

C₉

—

C₉

C₃×#383

yes

characteristic, complemented

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

C₃×#876

yes

—

104

C₂×C₃

C₃×#926

yes

—

105

C₂²

#3697⋊C₃

yes

—

106

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

107

C₄

—

108

C₄

—

109

C₃

—

110

C₃

C₃×#4937

yes

characteristic

111

C₂

#11721396

yes

—

112

C₂

#63488⋊C₃

yes

—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#383
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator

C₂²×C₉

C₂²×C₃×C₄

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂×C₄²×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂³×C₉

C₂³×C₃

C₂×C₄²

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
4. #10588617	C₃	semidirect
7. C₄²×3₋¹⁺²	C₄	semidirect
26. C₄²×C₉	C₃×C₄	semidirect
68. 3₋¹⁺²	#383	semidirect
85. C₄²	3₋¹⁺²⋊C₄	semidirect
96. C₉	C₃×#383	semidirect

H ≅ G/N

type

4. #10588617

C₃

semidirect

7. C₄²×3₋¹⁺²

C₄

semidirect

26. C₄²×C₉

C₃×C₄

semidirect

68. 3₋¹⁺²

#383

semidirect

85. C₄²

3₋¹⁺²⋊C₄

semidirect

96. C₉

C₃×#383

semidirect

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²⋊#95	2
C₃×#4937	3
#10588617	3

subgroup

index

C₂×C₄²×3₋¹⁺²

3₋¹⁺²⋊#95

C₃×#4937

#10588617

Subgroups — #13142444

Conjugacy classes of subgroups

Complements

Maximal subgroups