Group #28356780 — subgroups

Subgroups — #28356780

Order 13851.

350 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 24 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	#17987558	C₃	1	3	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#4296	—	3	3	no	—
4	C₃³×C₁₉⋊C₃	—	3	1	no	—
5	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	2	yes	complemented
9	H(3,𝔽₃)×C₁₉⋊C₃	—	3	9	no	—
10	C₃×C₁₉×H(3,𝔽₃)	C₉	1	1	yes	characteristic, complemented
11	#10624508	—	19	1	no	maximal, Sylow
12	C₃³×C₁₉	—	3	1	no	—
13	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₁₉⋊C₃	—	3	9	no	—
15	C₃²×C₁₉⋊C₃	—	3	2	no	—
16	C₃²×C₁₉⋊C₃	—	3	3	no	—
17	C₃²×C₁₉⋊C₃	—	9	9	no	—
18	C₃²×C₁₉⋊C₃	—	9	3	no	—
19	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
22	C₃×#4296	—	9	3	no	—
23	C₁₉×H(3,𝔽₃)	—	3	3	no	—
24	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
25	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
26	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
27	C₃²×H(3,𝔽₃)	—	19	1	no	—
28	#6866	—	19	3	no	—
29	C₃²×C₁₉	—	3	1	no	—
30	C₃²×C₁₉	—	3	3	no	—
31	C₃²×C₁₉	—	9	1	no	—
32	C₃²×C₁₉	—	9	3	no	—
33	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₁₉⋊C₃	—	3	1	no	—
35	C₃×C₁₉⋊C₃	—	3	3	no	—
36	C₃×C₁₉⋊C₃	—	3	1	no	—
37	C₃×C₁₉⋊C₃	—	3	6	no	—
38	C₃×C₁₉⋊C₃	—	9	1	no	—
39	C₃×C₁₉⋊C₃	—	9	9	no	—
40	C₃×C₁₉⋊C₃	—	9	27	no	—
41	C₃×C₁₉⋊C₃	—	9	6	no	—
42	C₃×C₁₉⋊C₃	—	9	1	no	—
43	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
44	#4296	—	27	3	no	—
45	C₃⁴	—	19	1	no	—
46	C₃⁴	—	57	1	no	—
47	C₃×H(3,𝔽₃)	—	19	2	no	—
48	C₃×H(3,𝔽₃)	—	57	9	no	—
49	C₃×H(3,𝔽₃)	—	57	1	no	—
50	C₃×H(3,𝔽₃)	—	57	1	no	—
51	C₃×H(3,𝔽₃)	#4296	1	1	yes	characteristic, complemented
52	C₃²⋊C₉	—	57	3	no	—
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	—	9	1	no	—
55	C₃×C₁₉	—	9	3	no	—
56	C₃×C₁₉	#6866	1	1	yes	characteristic
57	C₁₉⋊C₃	—	3	3	no	—
58	C₁₉⋊C₃	—	3	3	no	—
59	C₁₉⋊C₃	—	9	1	no	—
60	C₁₉⋊C₃	—	9	6	no	—
61	C₃³	—	3	1	no	—
62	C₃³	—	19	1	no	—
63	C₃³	—	19	1	no	—
64	C₃³	—	19	1	no	—
65	C₃³	—	57	3	no	—
66	C₃³	—	57	9	no	—
67	C₃³	—	57	2	no	—
68	C₃³	—	171	3	no	—
69	C₃³	—	171	9	no	—
70	C₃³	C₃×#4296	1	1	yes	characteristic
71	H(3,𝔽₃)	—	3	3	no	—
72	H(3,𝔽₃)	—	57	9	no	—
73	H(3,𝔽₃)	—	57	6	no	—
74	H(3,𝔽₃)	—	57	9	no	—
75	C₃×C₉	—	171	3	no	—
76	C₁₉	#10624508	1	1	yes	characteristic, complemented, Sylow
77	C₃²	—	3	1	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	9	3	no	—
81	C₃²	—	19	3	no	—
82	C₃²	—	57	3	no	—
83	C₃²	—	57	1	no	—
84	C₃²	—	57	1	no	—
85	C₃²	—	57	6	no	—
86	C₃²	—	171	1	no	—
87	C₃²	—	171	1	no	—
88	C₃²	—	171	6	no	—
89	C₃²	—	171	9	no	—
90	C₃²	—	171	27	no	—
91	C₃²	C₃²⋊#4296	1	1	yes	characteristic
92	C₉	—	513	3	no	—
93	C₃	—	3	1	no	—
94	C₃	—	9	3	no	—
95	C₃	—	9	1	no	—
96	C₃	—	57	3	no	—
97	C₃	—	57	3	no	—
98	C₃	—	171	6	no	—
99	C₃	—	171	1	no	—
100	C₃	#17987558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987558

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃²⋊#4296

—

C₃³×C₁₉⋊C₃

—

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

complemented

H(3,𝔽₃)×C₁₉⋊C₃

—

C₃×C₁₉×H(3,𝔽₃)

C₉

yes

characteristic, complemented

#10624508

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₃×#4296

—

C₁₉×H(3,𝔽₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²×H(3,𝔽₃)

—

#6866

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

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

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

AΣL(1,𝔽₂₇)

yes

—

#4296

—

C₃⁴

—

C₃⁴

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

#4296

yes

characteristic, complemented

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6866

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

—

171

—

C₃³

yes

—

—

—

—

C₃×C₉

—

171

—

C₁₉

#10624508

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

C₃²⋊#4296

yes

characteristic

C₉

—

513

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

171

—

C₃

—

171

—

100

C₃

#17987558

yes

characteristic

	subgroup	G/H
center	C₃	#17987558
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃³	C₃×#4296
Fitting	C₃×C₁₉×H(3,𝔽₃)	C₉
socle	C₃×C₁₉	#6866
prime radical	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃
cosocle	C₃³×C₁₉⋊C₃	C₃²
hypercenter	C₃×H(3,𝔽₃)	#4296
hypocenter	C₁₉	#10624508
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

subgroup

G/H

center

C₃

#17987558

commutator

C₃³×C₁₉

C₃×C₉

Frattini

C₃³

C₃×#4296

Fitting

C₃×C₁₉×H(3,𝔽₃)

C₉

socle

C₃×C₁₉

#6866

prime radical

C₃×H(3,𝔽₃)×C₁₉⋊C₃

C₃

cosocle

C₃³×C₁₉⋊C₃

C₃²

C₃×H(3,𝔽₃)

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

N	H ≅ G/N	type
1. #17987558	C₃	semidirect
8. C₃×C₃²⋊(C₁₉⋊C₃)	C₉	semidirect
10. C₃×C₁₉×H(3,𝔽₃)	C₉	semidirect
51. C₃×H(3,𝔽₃)	#4296	semidirect
76. C₁₉	#10624508	semidirect

H ≅ G/N

type

1. #17987558

C₃

semidirect

8. C₃×C₃²⋊(C₁₉⋊C₃)

C₉

semidirect

10. C₃×C₁₉×H(3,𝔽₃)

C₉

semidirect

51. C₃×H(3,𝔽₃)

#4296

semidirect

76. C₁₉

#10624508

semidirect

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

subgroup	index
#17987558	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
#10624508	19

subgroup

index

#17987558

C₃×H(3,𝔽₃)×C₁₉⋊C₃

#10624508

Subgroups — #28356780

Conjugacy classes of subgroups

Complements

Maximal subgroups