Group #28353588 — subgroups

Subgroups — #28353588

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×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal, complemented
3	C₃×#13095303	C₃	1	9	yes	maximal, complemented
4	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉⋊C₃	C₃²	1	3	yes	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
8	C₃²×#4294	C₃²	1	6	yes	—
9	C₃²⋊#4294	—	3	6	no	—
10	#13095303	—	3	27	no	—
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₃×C₁₉×3₋¹⁺²	—	3	3	no	—
13	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
14	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
15	#10624563	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	3	3	no	—
17	C₃³×C₁₉	—	3	3	no	—
18	C₃³×C₁₉	—	3	6	no	—
19	C₃³×C₁₉	H(3,𝔽₃)	1	3	yes	—
20	C₃³×C₁₉	C₃³	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	—	3	9	no	—
22	C₃²×C₁₉⋊C₃	—	3	3	no	—
23	C₃²×C₁₉⋊C₃	—	3	6	no	—
24	C₃²×C₁₉⋊C₃	—	3	18	no	—
25	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	complemented
26	C₁₉×3₋¹⁺²	—	3	27	no	—
27	C₃×C₉×C₁₉	—	3	3	no	—
28	C₃×C₉×C₁₉	—	3	6	no	—
29	C₃×C₉×C₁₉	H(3,𝔽₃)	1	9	yes	complemented
30	C₃²⋊(C₁₉⋊C₃)	—	3	27	no	—
31	C₃×#4294	—	3	18	no	—
32	C₃×#4294	—	3	6	no	—
33	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
34	C₃²×H(3,𝔽₃)	—	19	1	no	—
35	C₃×C₃²⋊C₉	—	19	2	no	—
36	C₃×C₃²⋊C₉	—	19	9	no	—
37	C₃²×C₁₉	—	3	3	no	—
38	C₃²×C₁₉	—	3	3	no	—
39	C₃²×C₁₉	—	3	6	no	—
40	C₃²×C₁₉	—	9	9	no	—
41	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
43	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
44	C₃²×C₁₉	#737	1	6	yes	—
45	C₃×C₁₉⋊C₃	—	3	9	no	—
46	C₃×C₁₉⋊C₃	—	3	27	no	—
47	C₃×C₁₉⋊C₃	—	9	3	no	—
48	C₃×C₁₉⋊C₃	—	9	18	no	—
49	C₃×C₁₉⋊C₃	—	9	6	no	—
50	C₉×C₁₉	—	3	9	no	—
51	#4294	—	9	6	no	—
52	C₃⁴	—	19	3	no	—
53	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	3	3	no	—
55	C₃×3₋¹⁺²	—	3	9	no	—
56	C₃×H(3,𝔽₃)	—	57	9	no	—
57	C₃×H(3,𝔽₃)	—	57	3	no	—
58	C₃²×C₉	—	19	6	no	—
59	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
60	C₃²⋊C₉	—	57	6	no	—
61	C₃²⋊C₉	—	57	27	no	—
62	C₃×C₁₉	—	9	3	no	—
63	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
64	C₃×C₁₉	#6891	1	1	yes	characteristic
65	C₃×C₁₉	C₉⋊H(3,𝔽₃)	1	9	yes	—
66	C₃×C₁₉	#6893	1	2	yes	—
67	C₁₉⋊C₃	—	9	9	no	—
68	C₃³	—	3	3	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	6	no	—
71	C₃³	—	19	9	no	—
72	C₃³	—	57	9	no	—
73	C₃³	—	57	3	no	—
74	C₃³	—	57	6	no	—
75	C₃³	—	57	18	no	—
76	C₃³	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
77	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
78	3₋¹⁺²	—	3	27	no	—
79	H(3,𝔽₃)	—	57	27	no	—
80	C₃×C₉	—	3	3	no	—
81	C₃×C₉	—	3	6	no	—
82	C₃×C₉	—	57	6	no	—
83	C₃×C₉	—	57	18	no	—
84	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	9	yes	complemented
85	C₁₉	#10624563	1	1	yes	characteristic, complemented, Sylow
86	C₃²	—	3	3	no	—
87	C₃²	—	3	3	no	—
88	C₃²	—	3	6	no	—
89	C₃²	—	9	9	no	—
90	C₃²	—	57	9	no	—
91	C₃²	—	57	27	no	—
92	C₃²	—	171	3	no	—
93	C₃²	—	171	18	no	—
94	C₃²	—	171	6	no	—
95	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
96	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
97	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
98	C₃²	#13095333	1	6	yes	—
99	C₉	—	3	9	no	—
100	C₉	—	171	6	no	—
101	C₃	—	9	3	no	—
102	C₃	—	171	9	no	—
103	C₃	#17987417	1	1	yes	characteristic
104	C₃	#17987418	1	1	yes	characteristic
105	C₃	#17987422	1	9	yes	—
106	C₃	#17987432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, characteristic

C₃×C₃²⋊#4294

C₃

yes

maximal, complemented

C₃×#13095303

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃²×#4294

C₃²

yes

—

C₃²⋊#4294

—

#13095303

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃²×C₉×C₁₉

C₃²

yes

—

#10624563

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

complemented

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

complemented

C₃²⋊(C₁₉⋊C₃)

—

C₃×#4294

—

C₃×#4294

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

C₃⁴

—

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

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

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

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

C₃²⋊(C₁₉⋊C₃)

yes

complemented

C₁₉

#10624563

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

C₃²

#13095333

yes

—

C₉

—

100

C₉

—

171

—

101

C₃

—

102

C₃

—

171

—

103

C₃

#17987417

yes

characteristic

104

C₃

#17987418

yes

characteristic

105

C₃

#17987422

yes

—

106

C₃

#17987432

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃³×C₁₉	C₃³
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
prime radical	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624563
LCS	G ▷ C₃³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator, socle, cosocle

C₃³×C₁₉

C₃³

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

prime radical

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃×C₃²⋊#4294	C₃	semidirect
3. C₃×#13095303	C₃	semidirect
4. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
25. C₃²×C₁₉⋊C₃	3₋¹⁺²	semidirect
29. C₃×C₉×C₁₉	H(3,𝔽₃)	semidirect
33. C₃²×3₋¹⁺²	C₁₉⋊C₃	semidirect
84. C₃×C₉	C₃²⋊(C₁₉⋊C₃)	semidirect
85. C₁₉	#10624563	semidirect

H ≅ G/N

type

2. C₃×C₃²⋊#4294

C₃

semidirect

3. C₃×#13095303

C₃

semidirect

4. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

25. C₃²×C₁₉⋊C₃

3₋¹⁺²

semidirect

29. C₃×C₉×C₁₉

H(3,𝔽₃)

semidirect

33. C₃²×3₋¹⁺²

C₁₉⋊C₃

semidirect

84. C₃×C₉

C₃²⋊(C₁₉⋊C₃)

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×C₃²⋊(C₁₉⋊C₃)	3
C₃×C₃²⋊#4294	3
C₃×#13095303	3
C₃²×C₁₉×3₋¹⁺²	3
#10624563	19

subgroup

index

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃×C₃²⋊#4294

C₃×#13095303

C₃²×C₁₉×3₋¹⁺²

#10624563

Subgroups — #28353588

Conjugacy classes of subgroups

Complements

Maximal subgroups