Group #28356814 — subgroups

Subgroups — #28356814

Order 13851.

142 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 27 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×#13095339	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987560	C₃	1	2	yes	maximal, complemented
3	C₁₉⋊C₃×#732	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4296	—	3	1	no	—
5	#13095339	—	3	3	no	—
6	C₃²⋊#4296	—	3	2	no	—
7	H(3,𝔽₃)×C₁₉⋊C₃	—	3	1	no	—
8	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	#13095320	C₉	1	2	yes	complemented
10	#13095319	—	3	2	no	—
11	C₁₉×#732	C₉	1	1	yes	characteristic, complemented
12	#10624527	—	19	1	no	maximal, Sylow
13	C₃²×C₁₉⋊C₃	—	9	1	no	—
14	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃×#4296	—	3	3	no	—
16	C₃×#4296	—	9	2	no	—
17	C₃×#4296	—	9	1	no	—
18	C₃×#4296	—	9	2	no	—
19	C₁₉×H(3,𝔽₃)	—	3	1	no	—
20	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
21	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
22	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
23	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
24	C₃×#4294	3₋¹⁺²	1	2	yes	—
25	C₉×C₁₉⋊C₃	—	3	1	no	—
26	C₉×C₁₉⋊C₃	—	3	1	no	—
27	C₉×C₁₉⋊C₃	—	3	1	no	—
28	C₃×#732	—	19	1	no	—
29	C₃×C₉⋊C₉	—	19	1	no	—
30	#6868	—	19	2	no	—
31	C₃²×C₁₉	—	9	1	no	—
32	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
33	C₃×C₁₉⋊C₃	—	3	2	no	—
34	C₃×C₁₉⋊C₃	—	3	1	no	—
35	C₃×C₁₉⋊C₃	—	9	2	no	—
36	C₃×C₁₉⋊C₃	—	27	2	no	—
37	C₃×C₁₉⋊C₃	—	27	1	no	—
38	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
39	C₃×C₁₉⋊C₃	#731	1	1	yes	characteristic
40	C₃×C₁₉⋊C₃	#732	1	1	yes	characteristic
41	#4296	—	9	3	no	—
42	#4296	—	27	2	no	—
43	C₉×C₁₉	—	3	1	no	—
44	#4294	—	3	2	no	—
45	C₃×H(3,𝔽₃)	—	57	1	no	—
46	C₃²×C₉	—	19	1	no	—
47	C₃²×C₉	—	57	1	no	—
48	#732	—	19	2	no	—
49	#732	—	57	2	no	—
50	#732	#4296	1	1	yes	characteristic, complemented
51	C₉⋊C₉	—	57	3	no	—
52	C₃²⋊C₉	—	57	2	no	—
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	—	27	1	no	—
55	C₃×C₁₉	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
56	C₁₉⋊C₃	—	3	1	no	—
57	C₁₉⋊C₃	—	3	1	no	—
58	C₁₉⋊C₃	—	27	2	no	—
59	C₁₉⋊C₃	#6881	1	2	yes	—
60	C₁₉⋊C₃	#6883	1	1	yes	characteristic
61	C₃³	—	19	1	no	—
62	C₃³	—	171	1	no	—
63	H(3,𝔽₃)	—	3	1	no	—
64	H(3,𝔽₃)	—	57	2	no	—
65	H(3,𝔽₃)	—	171	1	no	—
66	H(3,𝔽₃)	—	171	1	no	—
67	C₃×C₉	—	19	2	no	—
68	C₃×C₉	—	57	1	no	—
69	C₃×C₉	—	57	1	no	—
70	C₃×C₉	—	57	1	no	—
71	C₃×C₉	—	57	3	no	—
72	C₃×C₉	—	171	2	no	—
73	C₃×C₉	—	171	2	no	—
74	C₃×C₉	—	171	1	no	—
75	C₃×C₉	C₃×#4296	1	1	yes	characteristic
76	C₁₉	#10624527	1	1	yes	characteristic, complemented, Sylow
77	C₃²	—	513	2	no	—
78	C₃²	—	513	1	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	19	1	no	—
81	C₃²	—	19	1	no	—
82	C₃²	—	19	1	no	—
83	C₃²	—	57	2	no	—
84	C₃²	—	57	1	no	—
85	C₃²	—	171	2	no	—
86	C₃²	C₃²⋊#4296	1	1	yes	characteristic
87	C₉	—	513	2	no	—
88	C₉	—	3	1	no	—
89	C₉	—	57	2	no	—
90	C₉	—	171	3	no	—
91	C₃	—	513	2	no	—
92	C₃	—	3	1	no	—
93	C₃	—	19	2	no	—
94	C₃	—	19	1	no	—
95	C₃	—	27	1	no	—
96	C₃	—	57	1	no	—
97	C₃	—	57	1	no	—
98	C₃	H(3,𝔽₃)⋊#4296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095339

C₃

yes

maximal, characteristic, complemented

#17987560

C₃

yes

maximal, complemented

C₁₉⋊C₃×#732

C₃

yes

maximal, characteristic

C₃²×#4296

—

#13095339

—

C₃²⋊#4296

—

H(3,𝔽₃)×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

#13095320

C₉

yes

complemented

#13095319

—

C₁₉×#732

C₉

yes

characteristic, complemented

#10624527

—

maximal, Sylow

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

yes

—

—

—

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃×#4294

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₃×#732

—

C₃×C₉⋊C₉

—

#6868

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

AΣL(1,𝔽₂₇)

yes

C₃×C₁₉⋊C₃

yes

C₃×C₁₉⋊C₃

yes

—

—

C₉×C₁₉

—

#4294

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

—

—

yes

characteristic, complemented

C₉⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

H(3,𝔽₃)⋊C₉

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

#6881

yes

—

C₁₉⋊C₃

#6883

yes

characteristic

C₃³

—

C₃³

—

171

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

C₃×#4296

yes

characteristic

C₁₉

#10624527

yes

characteristic, complemented, Sylow

C₃²

—

513

—

C₃²

—

513

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

C₃²⋊#4296

yes

characteristic

C₉

—

513

—

C₉

—

C₉

—

C₉

—

171

—

C₃

—

513

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

H(3,𝔽₃)⋊#4296

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)⋊#4296
commutator	C₃×C₉×C₁₉	C₃×C₉
Frattini	C₃×C₉	C₃×#4296
Fitting	C₁₉×#732	C₉
socle	C₃×C₁₉	H(3,𝔽₃)⋊C₉
prime radical	C₁₉⋊C₃×#732	C₃
cosocle	C₃×C₉×C₁₉⋊C₃	C₃²
hypercenter	#732	#4296
hypocenter	C₁₉	#10624527
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #732

subgroup

G/H

center

C₃

H(3,𝔽₃)⋊#4296

commutator

C₃×C₉×C₁₉

C₃×C₉

Frattini

C₃×C₉

C₃×#4296

Fitting

C₁₉×#732

C₉

socle

C₃×C₁₉

H(3,𝔽₃)⋊C₉

prime radical

C₁₉⋊C₃×#732

C₃

cosocle

C₃×C₉×C₁₉⋊C₃

C₃²

C₁₉

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

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #732

N	H ≅ G/N	type
1. C₃×#13095339	C₃	semidirect
2. #17987560	C₃	semidirect
9. #13095320	C₉	semidirect
11. C₁₉×#732	C₉	semidirect
50. #732	#4296	semidirect
76. C₁₉	#10624527	semidirect

H ≅ G/N

type

1. C₃×#13095339

C₃

semidirect

2. #17987560

C₃

semidirect

9. #13095320

C₉

semidirect

11. C₁₉×#732

C₉

semidirect

50. #732

#4296

semidirect

76. C₁₉

#10624527

semidirect

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

subgroup	index
C₃×#13095339	3
#17987560	3
C₁₉⋊C₃×#732	3
#10624527	19

subgroup

index

C₃×#13095339

#17987560

C₁₉⋊C₃×#732

#10624527

Subgroups — #28356814

Conjugacy classes of subgroups

Complements

Maximal subgroups