Group #28353553 — subgroups

Subgroups — #28353553

Order 13851.

608 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₃³×#4294	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal
3	#17987308	C₃	1	9	yes	maximal, complemented
4	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
5	C₃²×#4294	—	3	3	no	—
6	C₃²×#4294	—	3	9	no	—
7	C₃²×#4294	C₃²	1	6	yes	—
8	C₃²×#4294	C₃²	1	3	yes	—
9	C₃²⋊#4294	—	3	6	no	—
10	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
11	C₃×C₁₉×3₋¹⁺²	—	3	3	no	—
12	C₃×C₁₉×3₋¹⁺²	—	3	6	no	—
13	C₃×C₁₉×3₋¹⁺²	C₉	1	9	yes	complemented
14	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
15	#10624558	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	3	3	no	—
17	C₃³×C₁₉	—	3	6	no	—
18	C₃³×C₁₉	C₃×C₉	1	3	yes	—
19	C₃³×C₁₉	H(3,𝔽₃)	1	3	yes	—
20	C₃³×C₁₉	3₋¹⁺²	1	6	yes	—
21	C₃³×C₁₉	C₃³	1	1	yes	characteristic
22	C₁₉×3₋¹⁺²	—	3	27	no	—
23	C₃×C₉×C₁₉	—	3	3	no	—
24	C₃×C₉×C₁₉	—	3	6	no	—
25	C₃×C₉×C₁₉	C₃×C₉	1	9	yes	complemented
26	C₃×#4294	—	3	27	no	—
27	C₃×#4294	—	3	3	no	—
28	C₃×#4294	—	3	6	no	—
29	C₃×#4294	—	3	6	no	—
30	C₃×#4294	—	3	18	no	—
31	C₃×#4294	3₋¹⁺²	1	9	yes	complemented
32	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
33	C₃³×C₉	—	19	1	no	—
34	C₃×C₃²⋊C₉	—	19	2	no	—
35	#6855	—	19	9	no	—
36	C₃²×C₁₉	—	3	9	no	—
37	C₃²×C₁₉	—	3	3	no	—
38	C₃²×C₁₉	—	3	6	no	—
39	C₃²×C₁₉	—	3	18	no	—
40	C₃²×C₁₉	C₃²⋊C₉	1	9	yes	—
41	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
42	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
44	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
45	C₃²×C₁₉	#737	1	6	yes	—
46	C₉×C₁₉	—	3	9	no	—
47	#4294	—	3	9	no	—
48	#4294	—	9	6	no	—
49	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	6	no	—
51	C₃×3₋¹⁺²	—	3	3	no	—
52	C₃×3₋¹⁺²	#4294	1	9	yes	complemented
53	C₃²×C₉	—	19	3	no	—
54	C₃²×C₉	—	19	6	no	—
55	C₃²×C₉	—	57	9	no	—
56	C₃²×C₉	—	57	3	no	—
57	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
58	C₃²⋊C₉	—	57	6	no	—
59	C₃×C₁₉	—	3	9	no	—
60	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
61	C₃×C₁₉	C₉×3₋¹⁺²	1	3	yes	—
62	C₃×C₁₉	#6891	1	1	yes	characteristic
63	C₃×C₁₉	#6893	1	2	yes	—
64	C₃×C₁₉	#6894	1	6	yes	—
65	C₃³	—	3	3	no	—
66	C₃³	—	3	6	no	—
67	C₃³	C₃×#4294	1	3	yes	—
68	C₃³	#10587025	1	6	yes	—
69	C₃³	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
70	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
71	3₋¹⁺²	—	3	27	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	—	19	9	no	—
75	C₃×C₉	—	57	27	no	—
76	C₃×C₉	—	57	3	no	—
77	C₃×C₉	—	57	6	no	—
78	C₃×C₉	—	57	6	no	—
79	C₃×C₉	—	57	18	no	—
80	C₃×C₉	C₃×#4294	1	9	yes	complemented
81	C₁₉	#10624558	1	1	yes	characteristic, complemented, Sylow
82	C₃²	—	3	9	no	—
83	C₃²	—	3	3	no	—
84	C₃²	—	3	6	no	—
85	C₃²	—	3	18	no	—
86	C₃²	C₃²⋊#4294	1	9	yes	—
87	C₃²	C₃²×#4294	1	1	yes	characteristic
88	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
89	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
90	C₃²	C₃×#10587025	1	2	yes	—
91	C₃²	#13095333	1	6	yes	—
92	C₉	—	3	9	no	—
93	C₉	—	57	9	no	—
94	C₉	—	171	6	no	—
95	C₃	—	3	9	no	—
96	C₃	C₃×C₃²⋊#4294	1	1	yes	characteristic
97	C₃	3₋¹⁺²×#4294	1	3	yes	—
98	C₃	#17987418	1	1	yes	characteristic
99	C₃	#17987432	1	2	yes	—
100	C₃	#17987435	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#4294

C₃

yes

maximal, characteristic

C₃×C₃²⋊#4294

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃²×#4294

—

C₃²×#4294

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃²⋊#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

—

#10624558

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

3₋¹⁺²

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic

C₃³×C₉

—

C₃×C₃²⋊C₉

—

#6855

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#4294

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

—

C₃×C₁₉

#6891

yes

characteristic

C₃×C₁₉

#6893

yes

—

C₃×C₁₉

#6894

yes

—

C₃³

—

C₃³

—

C₃³

C₃×#4294

yes

—

C₃³

#10587025

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#4294

yes

complemented

C₁₉

#10624558

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊#4294

yes

—

C₃²

C₃²×#4294

yes

characteristic

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

C₃²

C₃×#10587025

yes

—

C₃²

#13095333

yes

—

C₉

—

C₉

—

C₉

—

171

—

C₃

—

C₃

C₃×C₃²⋊#4294

yes

characteristic

C₃

3₋¹⁺²×#4294

yes

—

C₃

#17987418

yes

characteristic

C₃

#17987432

yes

—

100

C₃

#17987435

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle, cosocle	C₃³×C₁₉	C₃³
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624558
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator

C₃²×C₁₉

C₃²×C₉

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle, cosocle

C₃³×C₁₉

C₃³

prime radical

C₃⁴×C₁₉

C₃²

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
3. #17987308	C₃	semidirect
13. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
25. C₃×C₉×C₁₉	C₃×C₉	semidirect
31. C₃×#4294	3₋¹⁺²	semidirect
52. C₃×3₋¹⁺²	#4294	semidirect
80. C₃×C₉	C₃×#4294	semidirect
81. C₁₉	#10624558	semidirect

H ≅ G/N

type

3. #17987308

C₃

semidirect

13. C₃×C₁₉×3₋¹⁺²

C₉

semidirect

25. C₃×C₉×C₁₉

C₃×C₉

semidirect

31. C₃×#4294

3₋¹⁺²

semidirect

52. C₃×3₋¹⁺²

#4294

semidirect

80. C₃×C₉

C₃×#4294

semidirect

81. C₁₉

#10624558

semidirect

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

subgroup	index
C₃³×#4294	3
C₃×C₃²⋊#4294	3
#17987308	3
C₃²×C₁₉×3₋¹⁺²	3
#10624558	19

subgroup

index

C₃³×#4294

C₃×C₃²⋊#4294

#17987308

C₃²×C₁₉×3₋¹⁺²

#10624558

Subgroups — #28353553

Conjugacy classes of subgroups

Complements

Maximal subgroups