Group #28353196 — subgroups

Subgroups — #28353196

Order 13851.

230 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₃²×#10587018	C₃	1	2	yes	maximal
2	C₃×C₉×#4294	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₁₉×C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4294	C₉	1	3	yes	—
5	C₃×#10587018	—	3	2	no	—
6	C₃×#10587018	—	3	2	no	—
7	C₃×#10587018	—	3	2	no	—
8	C₃×#10587018	C₉	1	6	yes	complemented
9	C₉×#4294	—	3	1	no	—
10	C₉×#4294	—	3	1	no	—
11	C₉×#4294	—	3	1	no	—
12	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
13	C₃×C₁₉×C₂₇	—	3	1	no	—
14	C₃×C₁₉×C₂₇	—	3	2	no	—
15	C₃×C₁₉×C₂₇	C₉	1	3	yes	complemented
16	#10624463	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₁₉	C₃×C₉	1	2	yes	—
19	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₁₉	3₋¹⁺²	1	2	yes	—
22	C₃×C₉×C₁₉	3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₉×C₁₉	3₋¹⁺²	1	2	yes	—
24	C₃×C₉×C₁₉	3₋¹⁺²	1	1	yes	characteristic
25	C₃×C₉×C₁₉	3₋¹⁺²	1	2	yes	—
26	C₃×#4294	—	3	3	no	—
27	C₃×#4294	—	3	3	no	—
28	C₃×#4294	—	3	3	no	—
29	C₃×#4294	C₂₇	1	9	yes	complemented
30	C₁₉×C₂₇	—	3	3	no	—
31	#10587018	—	3	6	no	—
32	C₃²×C₂₇	—	19	2	no	—
33	C₃²×C₂₇	C₁₉⋊C₃	1	1	yes	characteristic
34	C₃×C₉²	—	19	1	no	—
35	C₃²×C₁₉	C₉²	1	1	yes	characteristic
36	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
37	C₃²×C₁₉	C₉⋊C₉	1	2	yes	—
38	C₃²×C₁₉	C₃×C₂₇	1	3	yes	—
39	C₃²×C₁₉	C₂₇⋊C₃	1	3	yes	—
40	C₃²×C₁₉	C₂₇⋊C₃	1	3	yes	—
41	C₉×C₁₉	C₃²⋊C₉	1	2	yes	—
42	C₉×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
43	C₉×C₁₉	C₉⋊C₉	1	2	yes	—
44	C₉×C₁₉	C₉⋊C₉	1	2	yes	—
45	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
46	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
47	#4294	—	3	9	no	—
48	C₃²×C₉	—	19	3	no	—
49	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
50	C₃×C₂₇	—	3	2	no	—
51	C₃×C₂₇	—	3	1	no	—
52	C₃×C₂₇	—	19	6	no	—
53	C₃×C₂₇	—	57	2	no	—
54	C₃×C₂₇	—	57	2	no	—
55	C₃×C₂₇	—	57	2	no	—
56	C₃×C₂₇	#4294	1	3	yes	complemented
57	C₉²	—	57	1	no	—
58	C₉²	—	57	1	no	—
59	C₉²	—	57	1	no	—
60	C₃×C₁₉	#6855	1	1	yes	characteristic
61	C₃×C₁₉	C₉×C₂₇	1	1	yes	characteristic
62	C₃×C₁₉	#6864	1	1	yes	characteristic
63	C₃×C₁₉	#6864	1	1	yes	characteristic
64	C₃×C₁₉	C₃²⋊C₂₇	1	3	yes	—
65	C₃×C₁₉	C₉⋊C₂₇	1	6	yes	—
66	C₃³	C₉×C₁₉⋊C₃	1	1	yes	characteristic
67	C₃×C₉	—	19	9	no	—
68	C₃×C₉	—	57	3	no	—
69	C₃×C₉	—	57	3	no	—
70	C₃×C₉	—	57	3	no	—
71	C₃×C₉	C₉×C₁₉⋊C₃	1	2	yes	—
72	C₃×C₉	#10587022	1	2	yes	—
73	C₃×C₉	#10587022	1	1	yes	characteristic
74	C₃×C₉	#10587023	1	2	yes	—
75	C₃×C₉	#10587023	1	1	yes	characteristic
76	C₃×C₉	C₃×#4294	1	1	yes	characteristic
77	C₃×C₉	#10587025	1	2	yes	—
78	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
79	C₂₇	—	3	3	no	—
80	C₂₇	—	57	6	no	—
81	C₁₉	#10624463	1	1	yes	characteristic, complemented, Sylow
82	C₃²	C₉×#4294	1	1	yes	characteristic
83	C₃²	#13095303	1	1	yes	characteristic
84	C₃²	#13095305	1	2	yes	—
85	C₃²	C₂₇×C₁₉⋊C₃	1	3	yes	—
86	C₃²	#13095310	1	3	yes	—
87	C₃²	#13095311	1	3	yes	—
88	C₉	—	57	9	no	—
89	C₉	#13095303	1	2	yes	—
90	C₉	C₃²⋊#4294	1	1	yes	characteristic
91	C₉	#13095305	1	2	yes	—
92	C₉	#13095305	1	2	yes	—
93	C₉	#13095306	1	1	yes	characteristic
94	C₉	#13095307	1	1	yes	characteristic
95	C₃	#17987308	1	1	yes	characteristic
96	C₃	C₂₇×#4294	1	1	yes	characteristic
97	C₃	#17987329	1	1	yes	characteristic
98	C₃	#17987330	1	1	yes	characteristic
99	C₃	#17987332	1	3	yes	—
100	C₃	#17987355	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587018

C₃

yes

maximal

C₃×C₉×#4294

C₃

yes

maximal, characteristic

C₃²×C₁₉×C₂₇

C₃

yes

maximal, characteristic

C₃²×#4294

C₉

yes

—

—

—

—

C₉

yes

—

—

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×C₂₇

—

C₃×C₁₉×C₂₇

—

C₃×C₁₉×C₂₇

C₉

yes

complemented

#10624463

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

characteristic

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

characteristic

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

—

—

—

C₂₇

yes

complemented

C₁₉×C₂₇

—

#10587018

—

C₃²×C₂₇

—

C₃²×C₂₇

C₁₉⋊C₃

yes

characteristic

C₃×C₉²

—

C₃²×C₁₉

C₉²

yes

characteristic

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃²×C₁₉

C₉⋊C₉

yes

—

C₃²×C₁₉

C₃×C₂₇

yes

—

C₃²×C₁₉

C₂₇⋊C₃

yes

—

C₃²×C₁₉

C₂₇⋊C₃

yes

—

C₉×C₁₉

C₃²⋊C₉

yes

—

C₉×C₁₉

C₃²⋊C₉

yes

characteristic

C₉×C₁₉

C₉⋊C₉

yes

—

C₉×C₁₉

C₉⋊C₉

yes

—

C₉×C₁₉

C₉⋊C₉

yes

characteristic

C₉×C₁₉

C₉⋊C₉

yes

characteristic

#4294

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

#4294

yes

complemented

C₉²

—

C₉²

—

C₉²

—

C₃×C₁₉

#6855

yes

characteristic

C₃×C₁₉

C₉×C₂₇

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

C₃²⋊C₂₇

yes

—

C₃×C₁₉

C₉⋊C₂₇

yes

—

C₃³

C₉×C₁₉⋊C₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₉×C₁₉⋊C₃

yes

—

C₃×C₉

#10587022

yes

—

C₃×C₉

#10587022

yes

characteristic

C₃×C₉

#10587023

yes

—

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

—

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₂₇

—

C₂₇

—

C₁₉

#10624463

yes

characteristic, complemented, Sylow

C₃²

yes

C₃²

yes

C₃²

yes

—

C₃²

C₂₇×C₁₉⋊C₃

yes

—

C₃²

#13095310

yes

—

C₃²

#13095311

yes

—

C₉

—

C₉

#13095303

yes

—

C₉

C₃²⋊#4294

yes

characteristic

C₉

#13095305

yes

—

C₉

#13095305

yes

—

C₉

yes

C₉

yes

C₃

yes

C₃

C₂₇×#4294

yes

C₃

yes

C₃

yes

C₃

yes

—

100

C₃

#17987355

yes

—

	subgroup	G/H
center, Frattini	C₃²×C₉	C₃×C₁₉⋊C₃
commutator	C₃×C₁₉	C₉×C₂₇
Fitting	C₃²×C₁₉×C₂₇	C₃
socle, prime radical	C₃³×C₁₉	C₃×C₉
cosocle	C₃²×C₉×C₁₉	C₃²
hypercenter	C₃²×C₂₇	C₁₉⋊C₃
hypocenter	C₁₉	#10624463
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃²×C₉ ◁ C₃²×C₂₇

subgroup

G/H

center, Frattini

C₃²×C₉

C₃×C₁₉⋊C₃

commutator

C₃×C₁₉

C₉×C₂₇

Fitting

C₃²×C₁₉×C₂₇

C₃

socle, prime radical

C₃³×C₁₉

C₃×C₉

cosocle

C₃²×C₉×C₁₉

C₃²

hypercenter

C₃²×C₂₇

C₁₉⋊C₃

C₁₉

G ▷ C₃×C₁₉ ▷ C₁₉

C₁ ◁ C₃²×C₉ ◁ C₃²×C₂₇

N	H ≅ G/N	type
8. C₃×#10587018	C₉	semidirect
15. C₃×C₁₉×C₂₇	C₉	semidirect
29. C₃×#4294	C₂₇	semidirect
56. C₃×C₂₇	#4294	semidirect
81. C₁₉	#10624463	semidirect

H ≅ G/N

type

8. C₃×#10587018

C₉

semidirect

15. C₃×C₁₉×C₂₇

C₉

semidirect

29. C₃×#4294

C₂₇

semidirect

56. C₃×C₂₇

#4294

semidirect

81. C₁₉

#10624463

semidirect

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

subgroup	index
C₃²×#10587018	3
C₃×C₉×#4294	3
C₃²×C₁₉×C₂₇	3
#10624463	19

subgroup

index

C₃²×#10587018

C₃×C₉×#4294

C₃²×C₁₉×C₂₇

#10624463

Subgroups — #28353196

Conjugacy classes of subgroups

Complements

Maximal subgroups