Group #19103195 — subgroups

Subgroups — #19103195

Order 5103.

824 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 301 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×C₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
2	#13133568	C₃	1	6	yes	maximal, complemented
3	#13133559	C₃	1	18	yes	maximal, complemented
4	#13133545	C₃	1	3	yes	maximal
5	C₉×#4427	C₃	1	9	yes	maximal, complemented
6	#13133530	C₃	1	2	yes	maximal, complemented
7	C₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624872	—	7	1	no	maximal, Sylow
9	C₃³×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
10	C₃²×C₇×C₉	C₃²	1	1	yes	characteristic
11	C₃²×C₇⋊C₉	C₃²	1	2	yes	—
12	C₃×#4427	C₃²	1	9	yes	—
13	C₃×C₉×ΓL(1,𝔽₈)	—	3	9	no	—
14	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
15	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	6	yes	—
16	C₇×C₉⋊C₉	C₃²	1	6	yes	complemented
17	C₇×C₃²⋊C₉	C₃²	1	3	yes	—
18	C₇×C₉²	C₃²	1	3	yes	complemented
19	C₃²⋊(C₇⋊C₉)	C₃²	1	6	yes	—
20	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	18	yes	—
21	#10588036	C₃²	1	6	yes	complemented
22	#10588035	C₃²	1	18	yes	complemented
23	#10588035	C₃²	1	6	yes	complemented
24	(C₇⋊C₉)⋊C₉	C₃²	1	36	yes	complemented
25	C₉×C₇⋊C₉	C₃²	1	6	yes	complemented
26	C₃³×C₉	—	7	1	no	—
27	#6895	—	7	6	no	—
28	#6894	—	7	18	no	—
29	#6891	—	7	3	no	—
30	C₉×3₋¹⁺²	—	7	9	no	—
31	#6887	—	7	2	no	—
32	#6887	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₃³×C₇	C₃³	1	1	yes	characteristic
34	C₃²×ΓL(1,𝔽₈)	—	3	3	no	—
35	C₃²×ΓL(1,𝔽₈)	—	3	9	no	—
36	C₃²×ΓL(1,𝔽₈)	C₃³	1	3	yes	—
37	C₃×C₇×C₉	3₋¹⁺²	1	9	yes	complemented
38	C₃×C₇×C₉	C₃³	1	9	yes	—
39	C₃×C₇×C₉	C₃³	1	2	yes	—
40	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
41	C₃×C₇⋊C₉	—	3	6	no	—
42	C₃×C₇⋊C₉	C₃³	1	18	yes	—
43	C₃×C₇⋊C₉	C₃³	1	6	yes	—
44	#4427	—	3	27	no	—
45	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
46	C₉×ΓL(1,𝔽₈)	—	3	18	no	—
47	C₃⁴	—	7	1	no	—
48	C₃×3₋¹⁺²	—	7	9	no	—
49	C₃²×C₉	—	7	2	no	—
50	C₃²×C₉	—	7	6	no	—
51	C₃²×C₉	—	7	3	no	—
52	C₃²×C₉	—	21	9	no	—
53	C₃²×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₉⋊C₉	—	7	36	no	—
55	C₉⋊C₉	—	7	6	no	—
56	C₉⋊C₉	—	7	18	no	—
57	C₉⋊C₉	—	7	6	no	—
58	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	6	yes	complemented
59	C₃²⋊C₉	—	7	6	no	—
60	C₃²⋊C₉	—	7	18	no	—
61	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
62	C₉²	—	7	6	no	—
63	C₉²	C₃×ΓL(1,𝔽₈)	1	3	yes	complemented
64	C₃²×C₇	—	3	3	no	—
65	C₃²×C₇	C₃×3₋¹⁺²	1	3	yes	—
66	C₃²×C₇	C₃⁴	1	1	yes	characteristic
67	C₃×ΓL(1,𝔽₈)	—	3	27	no	—
68	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
69	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
70	C₃×ΓL(1,𝔽₈)	#737	1	9	yes	—
71	C₇×C₉	—	3	9	no	—
72	C₇×C₉	C₃×3₋¹⁺²	1	6	yes	—
73	C₇×C₉	C₃×3₋¹⁺²	1	3	yes	—
74	C₇⋊C₉	—	3	6	no	—
75	C₇⋊C₉	—	3	18	no	—
76	C₃³	—	7	3	no	—
77	C₃³	—	21	9	no	—
78	C₃³	—	21	3	no	—
79	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
80	3₋¹⁺²	—	21	27	no	—
81	C₃×C₉	—	7	18	no	—
82	C₃×C₉	—	7	6	no	—
83	C₃×C₉	—	21	6	no	—
84	C₃×C₉	—	21	18	no	—
85	C₃×C₉	—	21	9	no	—
86	C₃×C₉	#4427	1	9	yes	complemented
87	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	9	yes	—
88	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	2	yes	—
89	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
90	C₃×C₇	—	3	3	no	—
91	C₃×C₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
92	C₃×C₇	C₃×#737	1	3	yes	—
93	ΓL(1,𝔽₈)	—	3	9	no	—
94	C₃²	—	3	3	no	—
95	C₃²	—	7	9	no	—
96	C₃²	—	21	27	no	—
97	C₃²	—	21	3	no	—
98	C₃²	—	21	6	no	—
99	C₃²	C₃×#4427	1	3	yes	—
100	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
101	C₉	—	3	9	no	—
102	C₉	—	21	6	no	—
103	C₉	—	21	18	no	—
104	C₉	C₃×#4427	1	6	yes	—
105	C₉	C₃×#4427	1	3	yes	—
106	C₇	#10624872	1	1	yes	characteristic, complemented, Sylow
107	C₃	—	3	3	no	—
108	C₃	—	21	9	no	—
109	C₃	C₃²×#4427	1	1	yes	characteristic
110	C₃	ΓL(1,𝔽₈)×#737	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₉×ΓL(1,𝔽₈)

C₃

yes

maximal, characteristic

#13133568

C₃

yes

maximal, complemented

#13133559

C₃

yes

maximal, complemented

#13133545

C₃

yes

maximal

C₉×#4427

C₃

yes

maximal, complemented

#13133530

C₃

yes

maximal, complemented

C₇×#6887

C₃

yes

maximal, characteristic, complemented

#10624872

—

maximal, Sylow

C₃³×ΓL(1,𝔽₈)

C₃²

yes

characteristic

C₃²×C₇×C₉

C₃²

yes

characteristic

C₃²×C₇⋊C₉

C₃²

yes

—

C₃×#4427

C₃²

yes

—

C₃×C₉×ΓL(1,𝔽₈)

—

C₃×C₉×ΓL(1,𝔽₈)

C₃²

yes

—

C₃×C₉×ΓL(1,𝔽₈)

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

complemented

C₇×C₃²⋊C₉

C₃²

yes

—

C₇×C₉²

C₃²

yes

complemented

C₃²⋊(C₇⋊C₉)

C₃²

yes

—

(C₃×ΓL(1,𝔽₈))⋊C₉

C₃²

yes

—

#10588036

C₃²

yes

complemented

#10588035

C₃²

yes

complemented

#10588035

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

C₉×C₇⋊C₉

C₃²

yes

complemented

C₃³×C₉

—

#6895

—

#6894

—

#6891

—

C₉×3₋¹⁺²

—

#6887

—

#6887

ΓL(1,𝔽₈)

yes

characteristic, complemented

C₃³×C₇

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

C₃³

yes

—

C₃×C₇×C₉

3₋¹⁺²

yes

complemented

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

characteristic

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

#4427

—

C₉×ΓL(1,𝔽₈)

—

C₉×ΓL(1,𝔽₈)

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×ΓL(1,𝔽₈)

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×ΓL(1,𝔽₈)

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×ΓL(1,𝔽₈)

yes

—

C₉²

—

C₉²

C₃×ΓL(1,𝔽₈)

yes

complemented

C₃²×C₇

—

C₃²×C₇

C₃×3₋¹⁺²

yes

—

C₃²×C₇

C₃⁴

yes

—

—

—

yes

—

C₇×C₉

—

C₇×C₉

C₃×3₋¹⁺²

yes

—

C₇×C₉

C₃×3₋¹⁺²

yes

—

C₇⋊C₉

—

C₇⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#4427

yes

complemented

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇

—

C₃×C₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₇

C₃×#737

yes

—

ΓL(1,𝔽₈)

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×#4427

yes

—

100

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

C₃×#4427

yes

—

105

C₉

C₃×#4427

yes

—

106

C₇

#10624872

yes

characteristic, complemented, Sylow

107

C₃

—

108

C₃

—

109

C₃

C₃²×#4427

yes

characteristic

110

C₃

ΓL(1,𝔽₈)×#737

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
Fitting	C₇×#6887	C₃
hypercenter	#6887	ΓL(1,𝔽₈)
hypocenter	C₇	#10624872
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₉ ◁ #6887

subgroup

G/H

center

C₃×C₉

C₃²×ΓL(1,𝔽₈)

commutator, socle

C₃²×C₇

C₃⁴

Frattini

C₃²

C₃³×ΓL(1,𝔽₈)

C₃

C₇

G ▷ C₃²×C₇ ▷ C₇

C₁ ◁ C₃×C₉ ◁ #6887

N	H ≅ G/N	type
2. #13133568	C₃	semidirect
3. #13133559	C₃	semidirect
5. C₉×#4427	C₃	semidirect
6. #13133530	C₃	semidirect
7. C₇×#6887	C₃	semidirect
16. C₇×C₉⋊C₉	C₃²	semidirect
18. C₇×C₉²	C₃²	semidirect
21. #10588036	C₃²	semidirect
22. #10588035	C₃²	semidirect
23. #10588035	C₃²	semidirect
24. (C₇⋊C₉)⋊C₉	C₃²	semidirect
25. C₉×C₇⋊C₉	C₃²	semidirect
32. #6887	ΓL(1,𝔽₈)	semidirect
37. C₃×C₇×C₉	3₋¹⁺²	semidirect
58. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
63. C₉²	C₃×ΓL(1,𝔽₈)	semidirect
86. C₃×C₉	#4427	semidirect
106. C₇	#10624872	semidirect

H ≅ G/N

type

2. #13133568

C₃

semidirect

3. #13133559

C₃

semidirect

5. C₉×#4427

C₃

semidirect

6. #13133530

C₃

semidirect

7. C₇×#6887

C₃

semidirect

16. C₇×C₉⋊C₉

C₃²

semidirect

18. C₇×C₉²

C₃²

semidirect

21. #10588036

C₃²

semidirect

22. #10588035

C₃²

semidirect

23. #10588035

C₃²

semidirect

24. (C₇⋊C₉)⋊C₉

C₃²

semidirect

25. C₉×C₇⋊C₉

C₃²

semidirect

32. #6887

ΓL(1,𝔽₈)

semidirect

37. C₃×C₇×C₉

3₋¹⁺²

semidirect

58. C₉⋊C₉

C₃×ΓL(1,𝔽₈)

semidirect

63. C₉²

C₃×ΓL(1,𝔽₈)

semidirect

86. C₃×C₉

#4427

semidirect

106. C₇

#10624872

semidirect

Subgroups — #19103195

Conjugacy classes of subgroups

Complements