Group #28354469 — subgroups

Subgroups — #28354469

Order 13851.

236 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17987421	C₃	1	1	yes	maximal, characteristic
2	#17987421	C₃	1	2	yes	maximal
3	#17987325	C₃	1	9	yes	maximal, complemented
4	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	2	no	—
6	C₃×#10587025	—	3	1	no	—
7	C₃²⋊#4294	—	3	2	no	—
8	C₃²⋊#4294	—	3	2	no	—
9	C₃²⋊#4294	—	3	2	no	—
10	C₃²⋊#4294	C₃²	1	3	yes	—
11	C₃²⋊#4294	C₃²	1	6	yes	—
12	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
13	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
14	C₁₉×C₃²⋊C₉	—	3	2	no	—
15	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
16	#10624666	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	—	9	1	no	—
18	C₃³×C₁₉	—	9	1	no	—
19	C₃³×C₁₉	—	9	1	no	—
20	C₃³×C₁₉	—	9	1	no	—
21	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
22	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₁₉×3₋¹⁺²	—	9	3	no	—
25	C₃×C₉×C₁₉	—	3	3	no	—
26	#10587025	—	9	3	no	—
27	#10587025	—	9	6	no	—
28	C₃×#4294	—	3	3	no	—
29	C₃×#4294	—	3	6	no	—
30	#6891	—	19	1	no	—
31	#6891	—	19	2	no	—
32	#6891	C₁₉⋊C₃	1	1	yes	characteristic
33	#6861	—	19	9	no	—
34	C₃²×C₁₉	—	3	2	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	—	3	2	no	—
37	C₃²×C₁₉	—	3	1	no	—
38	C₃²×C₁₉	—	3	1	no	—
39	C₃²×C₁₉	—	3	1	no	—
40	C₃²×C₁₉	—	3	1	no	—
41	C₃²×C₁₉	—	3	2	no	—
42	C₃²×C₁₉	—	3	1	no	—
43	C₃²×C₁₉	—	3	2	no	—
44	C₃²×C₁₉	—	3	1	no	—
45	C₃²×C₁₉	—	3	1	no	—
46	C₃²×C₁₉	—	9	9	no	—
47	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₉×C₁₉	—	9	3	no	—
49	#4294	—	9	3	no	—
50	#4294	—	9	6	no	—
51	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	1	no	—
53	C₃×3₋¹⁺²	—	57	2	no	—
54	C₃×3₋¹⁺²	—	57	1	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	19	3	no	—
57	C₃²⋊C₉	—	19	6	no	—
58	C₃²⋊C₉	—	57	2	no	—
59	C₃²⋊C₉	—	57	2	no	—
60	C₃²⋊C₉	—	57	2	no	—
61	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
62	C₃×C₁₉	—	9	1	no	—
63	C₃×C₁₉	—	9	2	no	—
64	C₃×C₁₉	—	9	1	no	—
65	C₃×C₁₉	#6910	1	1	yes	characteristic
66	C₃×C₁₉	#6911	1	1	yes	characteristic
67	C₃×C₁₉	#6912	1	1	yes	characteristic
68	C₃×C₁₉	#6912	1	1	yes	characteristic
69	C₃³	—	9	1	no	—
70	C₃³	—	9	1	no	—
71	C₃³	—	9	1	no	—
72	C₃³	—	9	1	no	—
73	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
74	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
75	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
76	3₋¹⁺²	—	9	3	no	—
77	3₋¹⁺²	—	171	3	no	—
78	3₋¹⁺²	—	171	6	no	—
79	C₃×C₉	—	3	3	no	—
80	C₃×C₉	—	57	3	no	—
81	C₃×C₉	—	57	6	no	—
82	C₁₉	#10624666	1	1	yes	characteristic, complemented, Sylow
83	C₃²	—	3	1	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	1	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	1	no	—
91	C₃²	—	3	1	no	—
92	C₃²	—	3	2	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	9	9	no	—
96	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
97	C₉	—	9	3	no	—
98	C₉	—	171	3	no	—
99	C₉	—	171	6	no	—
100	C₃	—	9	1	no	—
101	C₃	—	9	2	no	—
102	C₃	—	9	1	no	—
103	C₃	#17987511	1	1	yes	characteristic
104	C₃	#17987514	1	1	yes	characteristic
105	C₃	#17987518	1	1	yes	characteristic
106	C₃	#17987519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987421

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal

#17987325

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃×#10587025

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624666

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic

#6861

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624666

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₉

—

C₉

—

171

—

C₉

—

171

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

#17987511

yes

characteristic

104

C₃

#17987514

yes

characteristic

105

C₃

#17987518

yes

characteristic

106

C₃

#17987519

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624666
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6891

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
3. #17987325	C₃	semidirect
82. C₁₉	#10624666	semidirect

H ≅ G/N

type

3. #17987325

C₃

semidirect

82. C₁₉

#10624666

semidirect

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

subgroup	index
#17987421	3
#17987325	3
C₁₉×#6891	3
#10624666	19

subgroup

index

#17987421

#17987325

C₁₉×#6891

#10624666

Subgroups — #28354469

Conjugacy classes of subgroups

Complements

Maximal subgroups