Group #28353370 — subgroups

Subgroups — #28353370

Order 13851.

350 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 31 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	#17987335	C₃	1	2	yes	maximal, complemented
3	C₁₉×#6866	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₁₉⋊C₃	—	3	1	no	—
5	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
9	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	3	yes	complemented
10	C₃²⋊#4294	—	3	2	no	—
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₁₉×C₃²⋊C₉	—	3	1	no	—
13	#10624508	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	—	3	1	no	—
15	C₃³×C₁₉	—	3	2	no	—
16	C₃³×C₁₉	—	9	3	no	—
17	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
18	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃³×C₁₉	3₋¹⁺²	1	1	yes	characteristic
20	C₃³×C₁₉	3₋¹⁺²	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	—	3	9	no	—
22	C₃²×C₁₉⋊C₃	—	3	3	no	—
23	C₃²×C₁₉⋊C₃	—	9	9	no	—
24	C₃×C₉×C₁₉	—	9	1	no	—
25	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
26	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
27	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
28	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
29	C₃×#4294	—	9	2	no	—
30	C₃²×H(3,𝔽₃)	—	19	1	no	—
31	#6866	—	19	2	no	—
32	#6866	C₁₉⋊C₃	1	1	yes	characteristic, complemented
33	C₃²×C₁₉	—	3	3	no	—
34	C₃²×C₁₉	—	3	1	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	—	3	1	no	—
37	C₃²×C₁₉	—	9	1	no	—
38	C₃²×C₁₉	—	9	6	no	—
39	C₃²×C₁₉	—	9	3	no	—
40	C₃²×C₁₉	—	9	1	no	—
41	C₃²×C₁₉	—	9	1	no	—
42	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
43	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
44	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	2	yes	—
45	C₃×C₁₉⋊C₃	—	3	9	no	—
46	C₃×C₁₉⋊C₃	—	9	27	no	—
47	C₃×C₁₉⋊C₃	—	9	9	no	—
48	C₉×C₁₉	—	27	1	no	—
49	#4294	—	27	2	no	—
50	C₃⁴	—	57	1	no	—
51	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
52	C₃×H(3,𝔽₃)	—	19	3	no	—
53	C₃×H(3,𝔽₃)	—	57	1	no	—
54	C₃×H(3,𝔽₃)	—	57	1	no	—
55	C₃×H(3,𝔽₃)	—	57	3	no	—
56	C₃×H(3,𝔽₃)	—	57	6	no	—
57	C₃²⋊C₉	—	3	1	no	—
58	C₃²⋊C₉	—	57	2	no	—
59	C₃×C₁₉	—	3	2	no	—
60	C₃×C₁₉	—	3	1	no	—
61	C₃×C₁₉	—	3	3	no	—
62	C₃×C₁₉	—	3	1	no	—
63	C₃×C₁₉	—	9	1	no	—
64	C₃×C₁₉	—	9	1	no	—
65	C₃×C₁₉	#6866	1	1	yes	characteristic
66	C₁₉⋊C₃	—	9	9	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	9	3	no	—
70	C₃³	—	57	9	no	—
71	C₃³	—	57	3	no	—
72	C₃³	—	171	9	no	—
73	C₃³	C₉×C₁₉⋊C₃	1	1	yes	characteristic
74	C₃³	#10587022	1	1	yes	characteristic
75	C₃³	#10587023	1	1	yes	characteristic
76	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
77	H(3,𝔽₃)	—	57	9	no	—
78	H(3,𝔽₃)	—	57	9	no	—
79	H(3,𝔽₃)	—	57	3	no	—
80	H(3,𝔽₃)	—	57	6	no	—
81	C₃×C₉	—	9	1	no	—
82	C₃×C₉	—	171	2	no	—
83	C₁₉	#10624508	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	3	3	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	9	1	no	—
89	C₃²	—	9	6	no	—
90	C₃²	—	9	3	no	—
91	C₃²	—	9	1	no	—
92	C₃²	—	9	1	no	—
93	C₃²	—	57	9	no	—
94	C₃²	—	171	9	no	—
95	C₃²	—	171	27	no	—
96	C₃²	#13095303	1	1	yes	characteristic
97	C₃²	#13095313	1	1	yes	characteristic
98	C₃²	#13095315	1	2	yes	—
99	C₉	—	513	2	no	—
100	C₉	—	27	1	no	—
101	C₃	—	3	2	no	—
102	C₃	—	3	1	no	—
103	C₃	—	3	1	no	—
104	C₃	—	3	3	no	—
105	C₃	—	9	1	no	—
106	C₃	—	9	1	no	—
107	C₃	—	171	9	no	—
108	C₃	#17987334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, characteristic

#17987335

C₃

yes

maximal, complemented

C₁₉×#6866

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

complemented

C₃²⋊#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

—

#10624508

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

3₋¹⁺²

yes

characteristic

C₃³×C₁₉

3₋¹⁺²

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃×#4294

—

C₃²×H(3,𝔽₃)

—

#6866

—

#6866

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6866

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

C₉×C₁₉⋊C₃

yes

C₃³

yes

C₃³

yes

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

—

—

—

C₃×C₉

—

C₃×C₉

—

171

—

C₁₉

#10624508

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

513

—

100

C₉

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

—

171

—

108

C₃

#17987334

yes

characteristic

	subgroup	G/H
center	C₃	#17987334
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃⁴	C₃×C₁₉⋊C₃
Fitting	C₁₉×#6866	C₃
socle	C₃×C₁₉	#6866
prime radical	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
cosocle	C₃⁴×C₁₉	C₃²
hypercenter	#6866	C₁₉⋊C₃
hypocenter	C₁₉	#10624508
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

subgroup

G/H

center

C₃

#17987334

commutator

C₃³×C₁₉

C₃×C₉

Frattini

C₃⁴

C₃×C₁₉⋊C₃

Fitting

C₁₉×#6866

C₃

socle

C₃×C₁₉

#6866

prime radical

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

cosocle

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

N	H ≅ G/N	type
2. #17987335	C₃	semidirect
3. C₁₉×#6866	C₃	semidirect
9. C₃×C₃²⋊(C₁₉⋊C₃)	C₉	semidirect
32. #6866	C₁₉⋊C₃	semidirect
83. C₁₉	#10624508	semidirect

H ≅ G/N

type

2. #17987335

C₃

semidirect

3. C₁₉×#6866

C₃

semidirect

9. C₃×C₃²⋊(C₁₉⋊C₃)

C₉

semidirect

32. #6866

C₁₉⋊C₃

semidirect

83. C₁₉

#10624508

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
#17987335	3
C₁₉×#6866	3
#10624508	19

subgroup

index

C₃²×C₃²⋊(C₁₉⋊C₃)

#17987335

C₁₉×#6866

#10624508

Subgroups — #28353370

Conjugacy classes of subgroups

Complements

Maximal subgroups