Group #28353591 — subgroups

Subgroups — #28353591

Order 13851.

758 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 121 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×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal, complemented
3	C₃×C₃²⋊#4294	C₃	1	9	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	3	no	—
6	C₃×#10587025	—	3	9	no	—
7	C₃²×#4294	C₃²	1	6	yes	—
8	C₃²×#4294	C₃²	1	3	yes	—
9	C₃²⋊#4294	—	3	6	no	—
10	C₃²⋊#4294	—	3	27	no	—
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₃⁴×C₁₉	C₃²	1	3	yes	—
13	C₃×C₁₉×H(3,𝔽₃)	—	3	3	no	—
14	C₃×C₁₉×H(3,𝔽₃)	—	3	9	no	—
15	#10624563	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	3	3	no	—
17	C₃³×C₁₉	—	3	9	no	—
18	C₃³×C₁₉	—	3	3	no	—
19	C₃³×C₁₉	—	3	3	no	—
20	C₃³×C₁₉	—	3	6	no	—
21	C₃³×C₁₉	—	3	18	no	—
22	C₃³×C₁₉	—	3	6	no	—
23	C₃³×C₁₉	H(3,𝔽₃)	1	3	yes	—
24	C₃³×C₁₉	3₋¹⁺²	1	9	yes	complemented
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₁₉×H(3,𝔽₃)	—	3	27	no	—
27	#10587025	—	3	27	no	—
28	C₃×#4294	—	3	6	no	—
29	C₃×#4294	—	3	3	no	—
30	C₃×#4294	—	3	18	no	—
31	C₃×#4294	—	3	6	no	—
32	C₃×#4294	H(3,𝔽₃)	1	9	yes	complemented
33	C₃²×3₋¹⁺²	—	19	1	no	—
34	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
35	C₃×C₃²⋊C₉	—	19	9	no	—
36	C₃×C₃²⋊C₉	—	19	2	no	—
37	C₃²×C₁₉	—	3	3	no	—
38	C₃²×C₁₉	—	3	3	no	—
39	C₃²×C₁₉	—	3	9	no	—
40	C₃²×C₁₉	—	3	27	no	—
41	C₃²×C₁₉	—	3	6	no	—
42	C₃²×C₁₉	—	9	3	no	—
43	C₃²×C₁₉	—	9	18	no	—
44	C₃²×C₁₉	—	9	9	no	—
45	C₃²×C₁₉	—	9	6	no	—
46	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
47	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
49	C₃²×C₁₉	#737	1	6	yes	—
50	#4294	—	3	9	no	—
51	#4294	—	9	6	no	—
52	C₃⁴	C₃×C₁₉⋊C₃	1	3	yes	—
53	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	57	9	no	—
55	C₃×3₋¹⁺²	—	57	3	no	—
56	C₃×H(3,𝔽₃)	—	3	9	no	—
57	C₃×H(3,𝔽₃)	—	3	3	no	—
58	C₃²×C₉	—	19	3	no	—
59	C₃²×C₉	—	19	6	no	—
60	C₃²⋊C₉	—	57	27	no	—
61	C₃²⋊C₉	—	57	6	no	—
62	C₃×C₁₉	—	9	3	no	—
63	C₃×C₁₉	—	9	9	no	—
64	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×C₁₉	#6891	1	1	yes	characteristic
66	C₃×C₁₉	C₉⋊H(3,𝔽₃)	1	9	yes	—
67	C₃×C₁₉	#6893	1	2	yes	—
68	C₃³	—	3	9	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	6	no	—
72	C₃³	—	3	3	no	—
73	C₃³	—	3	18	no	—
74	C₃³	—	3	6	no	—
75	C₃³	#10587025	1	9	yes	complemented
76	C₃³	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
77	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
78	3₋¹⁺²	—	57	27	no	—
79	H(3,𝔽₃)	—	3	27	no	—
80	C₃×C₉	—	19	9	no	—
81	C₃×C₉	—	57	6	no	—
82	C₃×C₉	—	57	3	no	—
83	C₃×C₉	—	57	18	no	—
84	C₃×C₉	—	57	6	no	—
85	C₁₉	#10624563	1	1	yes	characteristic, complemented, Sylow
86	C₃²	—	3	3	no	—
87	C₃²	—	3	9	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	3	27	no	—
90	C₃²	—	3	6	no	—
91	C₃²	—	9	3	no	—
92	C₃²	—	9	18	no	—
93	C₃²	—	9	9	no	—
94	C₃²	—	9	6	no	—
95	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
96	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
97	C₃²	C₃×#10587025	1	3	yes	—
98	C₃²	#13095331	1	6	yes	—
99	C₉	—	57	9	no	—
100	C₉	—	171	6	no	—
101	C₃	—	9	3	no	—
102	C₃	—	9	9	no	—
103	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
104	C₃	#17987421	1	1	yes	characteristic
105	C₃	#17987423	1	9	yes	—
106	C₃	#17987429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊#4294

C₃

yes

maximal, complemented

C₃×C₃²⋊#4294

C₃

yes

maximal, complemented

C₃²×C₁₉×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃×#10587025

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃⁴×C₁₉

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

#10624563

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

3₋¹⁺²

yes

complemented

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×H(3,𝔽₃)

—

—

—

—

—

—

yes

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

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

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

#6891

yes

characteristic

C₃×C₁₉

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₉

#6893

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

#10587025

yes

complemented

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624563

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃²

C₃×#10587025

yes

—

C₃²

#13095331

yes

—

C₉

—

100

C₉

—

171

—

101

C₃

—

102

C₃

—

103

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

104

C₃

#17987421

yes

characteristic

105

C₃

#17987423

yes

—

106

C₃

#17987429

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃³×C₁₉	C₃³
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624563
LCS	G ▷ C₃³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator, socle, cosocle

C₃³×C₁₉

C₃³

Fitting, prime radical

C₃²×C₁₉×H(3,𝔽₃)

C₃

hypercenter

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
2. C₃×C₃²⋊#4294	C₃	semidirect
3. C₃×C₃²⋊#4294	C₃	semidirect
24. C₃³×C₁₉	3₋¹⁺²	semidirect
32. C₃×#4294	H(3,𝔽₃)	semidirect
75. C₃³	#10587025	semidirect
85. C₁₉	#10624563	semidirect

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

2. C₃×C₃²⋊#4294

C₃

semidirect

3. C₃×C₃²⋊#4294

C₃

semidirect

24. C₃³×C₁₉

3₋¹⁺²

semidirect

32. C₃×#4294

H(3,𝔽₃)

semidirect

75. C₃³

#10587025

semidirect

85. C₁₉

#10624563

semidirect

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

subgroup	index
C₃²×#10587025	3
C₃×C₃²⋊#4294	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624563	19

subgroup

index

C₃²×#10587025

C₃×C₃²⋊#4294

C₃²×C₁₉×H(3,𝔽₃)

#10624563

Subgroups — #28353591

Conjugacy classes of subgroups

Complements

Maximal subgroups