Group #28353302 — subgroups

Subgroups — #28353302

Order 13851.

124 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 28 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×#13095320	C₃	1	1	yes	maximal, characteristic
2	#17987339	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987308	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#6868	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
6	C₃²×#4294	—	3	1	no	—
7	#13095320	—	3	3	no	—
8	C₃²⋊#4294	—	3	1	no	—
9	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
10	C₁₉×C₃²⋊C₉	—	3	1	no	—
11	#10624489	—	19	1	no	maximal, Sylow
12	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃²×C₁₉⋊C₃	—	9	1	no	—
14	C₃×C₉×C₁₉	—	3	1	no	—
15	C₃×C₉×C₁₉	—	3	1	no	—
16	C₃×C₉×C₁₉	—	3	1	no	—
17	C₃×C₉×C₁₉	—	3	1	no	—
18	C₃×C₉×C₁₉	—	9	1	no	—
19	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
20	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
21	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
22	C₃×#4294	—	3	3	no	—
23	C₃×#4294	—	9	1	no	—
24	C₃×#4294	—	9	1	no	—
25	C₃×#4294	—	9	1	no	—
26	C₃×#4294	—	9	1	no	—
27	C₃×#732	—	19	1	no	—
28	#6868	—	19	1	no	—
29	#6868	C₁₉⋊C₃	1	1	yes	characteristic, complemented
30	#6855	—	19	1	no	—
31	C₃²×C₁₉	—	3	1	no	—
32	C₃²×C₁₉	—	3	1	no	—
33	C₃²×C₁₉	—	3	1	no	—
34	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₃²×C₁₉	#731	1	1	yes	characteristic
36	C₃²×C₁₉	#731	1	1	yes	characteristic
37	C₃²×C₁₉	#732	1	1	yes	characteristic
38	C₃×C₁₉⋊C₃	—	9	3	no	—
39	C₃×C₁₉⋊C₃	—	27	1	no	—
40	C₃×C₁₉⋊C₃	—	27	1	no	—
41	C₃×C₁₉⋊C₃	—	27	1	no	—
42	C₉×C₁₉	—	9	1	no	—
43	C₉×C₁₉	—	27	1	no	—
44	#4294	—	9	3	no	—
45	#4294	—	27	1	no	—
46	C₃×H(3,𝔽₃)	—	57	1	no	—
47	C₃²×C₉	—	57	1	no	—
48	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	#732	—	57	3	no	—
50	C₃²⋊C₉	—	3	1	no	—
51	C₃²⋊C₉	—	57	1	no	—
52	C₃×C₁₉	—	3	1	no	—
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	—	3	1	no	—
55	C₃×C₁₉	#6859	1	1	yes	characteristic
56	C₃×C₁₉	#6879	1	1	yes	characteristic
57	C₃×C₁₉	#6883	1	1	yes	characteristic
58	C₃×C₁₉	#6883	1	1	yes	characteristic
59	C₁₉⋊C₃	—	27	3	no	—
60	C₃³	—	171	1	no	—
61	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
62	H(3,𝔽₃)	—	57	3	no	—
63	H(3,𝔽₃)	—	171	1	no	—
64	H(3,𝔽₃)	—	171	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	9	1	no	—
70	C₃×C₉	—	57	3	no	—
71	C₃×C₉	—	171	1	no	—
72	C₃×C₉	—	171	1	no	—
73	C₃×C₉	—	171	1	no	—
74	C₃×C₉	—	171	1	no	—
75	C₁₉	#10624489	1	1	yes	characteristic, complemented, Sylow
76	C₃²	—	513	1	no	—
77	C₃²	—	513	1	no	—
78	C₃²	—	513	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	171	3	no	—
83	C₃²	#13095315	1	1	yes	characteristic
84	C₃²	#13095317	1	1	yes	characteristic
85	C₃²	#13095318	1	1	yes	characteristic
86	C₃²	#13095319	1	1	yes	characteristic
87	C₉	—	513	1	no	—
88	C₉	—	9	1	no	—
89	C₉	—	27	1	no	—
90	C₉	—	171	3	no	—
91	C₃	—	513	3	no	—
92	C₃	—	3	1	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	1	no	—
95	C₃	#17987320	1	1	yes	characteristic
96	C₃	#17987373	1	1	yes	characteristic
97	C₃	#17987388	1	1	yes	characteristic
98	C₃	#17987389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095320

C₃

yes

maximal, characteristic

#17987339

C₃

yes

maximal, characteristic, complemented

#17987308

C₃

yes

maximal, characteristic, complemented

C₁₉×#6868

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃²×#4294

—

#13095320

—

C₃²⋊#4294

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

—

—

C₃³×C₁₉

yes

C₃²×C₁₉⋊C₃

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

#6855

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

C₃²×C₁₉

yes

C₃²×C₁₉

yes

C₃²×C₁₉

yes

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

#732

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₃³

—

171

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₁₉

#10624489

yes

characteristic, complemented, Sylow

C₃²

—

513

—

C₃²

—

513

—

C₃²

—

513

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

yes

C₃²

yes

C₃²

yes

C₃²

yes

C₉

—

513

—

C₉

—

C₉

—

C₉

—

171

—

C₃

—

513

—

C₃

—

C₃

—

C₃

—

C₃

yes

C₃

yes

C₃

yes

C₃

yes

	subgroup	G/H
center	C₃²	#13095319
commutator, cosocle	C₃²×C₉×C₁₉	C₃²
Frattini	C₃²×C₉	C₃×C₁₉⋊C₃
Fitting	C₁₉×#6868	C₃
socle	C₃²×C₁₉	#732
prime radical	C₃×#13095320	C₃
hypercenter	#6868	C₁₉⋊C₃
hypocenter	C₁₉	#10624489
LCS	G ▷ C₃²×C₉×C₁₉ ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

subgroup

G/H

center

C₃²

#13095319

commutator, cosocle

C₃²×C₉×C₁₉

C₃²

Frattini

C₃²×C₉

C₃×C₁₉⋊C₃

Fitting

C₁₉×#6868

C₃

C₃²×C₁₉

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₉×C₁₉ ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

N	H ≅ G/N	type
2. #17987339	C₃	semidirect
3. #17987308	C₃	semidirect
4. C₁₉×#6868	C₃	semidirect
29. #6868	C₁₉⋊C₃	semidirect
75. C₁₉	#10624489	semidirect

H ≅ G/N

type

2. #17987339

C₃

semidirect

3. #17987308

C₃

semidirect

4. C₁₉×#6868

C₃

semidirect

29. #6868

C₁₉⋊C₃

semidirect

75. C₁₉

#10624489

semidirect

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

subgroup	index
C₃×#13095320	3
#17987339	3
#17987308	3
C₁₉×#6868	3
#10624489	19

subgroup

index

C₁₉×#6868

Subgroups — #28353302

Conjugacy classes of subgroups

Complements

Maximal subgroups