Group #19103197 — subgroups

Subgroups — #19103197

Order 5103.

824 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 283 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	#13133570	C₃	1	2	yes	maximal, complemented
3	#13133568	C₃	1	3	yes	maximal, complemented
4	#13133561	C₃	1	18	yes	maximal, complemented
5	#13133544	C₃	1	3	yes	maximal
6	#4428⋊C₉	C₃	1	9	yes	maximal, complemented
7	#13133527	C₃	1	3	yes	maximal, complemented
8	C₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624872	—	7	1	no	maximal, Sylow
10	C₃³×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
11	C₃²×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃²×C₇⋊C₉	C₃²	1	1	yes	characteristic
13	C₃²×C₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×#4428	C₃²	1	9	yes	—
15	C₃×C₉×ΓL(1,𝔽₈)	—	3	9	no	—
16	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
17	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	6	yes	—
18	C₇×C₉⋊C₉	C₃²	1	9	yes	complemented
19	C₇×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₃²⋊(C₇⋊C₉)	C₃²	1	6	yes	—
21	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	18	yes	—
22	#10588035	C₃²	1	9	yes	complemented
23	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
24	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
25	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
26	C₉×C₇⋊C₉	C₃²	1	9	yes	complemented
27	C₃³×C₉	—	7	1	no	—
28	#6895	—	7	2	no	—
29	#6895	—	7	3	no	—
30	#6895	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	#6894	—	7	18	no	—
32	#6891	—	7	3	no	—
33	C₉×3₋¹⁺²	—	7	9	no	—
34	#6887	—	7	3	no	—
35	C₃³×C₇	C₃³	1	1	yes	characteristic
36	C₃²×ΓL(1,𝔽₈)	—	3	3	no	—
37	C₃²×ΓL(1,𝔽₈)	—	3	9	no	—
38	C₃²×ΓL(1,𝔽₈)	C₃³	1	3	yes	—
39	C₃×C₇×C₉	—	3	3	no	—
40	C₃×C₇×C₉	C₃³	1	9	yes	—
41	C₃×C₇×C₉	C₃³	1	3	yes	—
42	C₃×C₇⋊C₉	—	3	3	no	—
43	C₃×C₇⋊C₉	3₋¹⁺²	1	9	yes	complemented
44	C₃×C₇⋊C₉	C₃³	1	18	yes	—
45	C₃×C₇⋊C₉	C₃³	1	3	yes	—
46	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
47	C₃×C₇⋊C₉	C₃³	1	2	yes	—
48	#4428	—	3	27	no	—
49	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
50	C₉×ΓL(1,𝔽₈)	—	3	18	no	—
51	C₃⁴	—	7	1	no	—
52	C₃×3₋¹⁺²	—	7	9	no	—
53	C₃²×C₉	—	7	1	no	—
54	C₃²×C₉	—	7	1	no	—
55	C₃²×C₉	—	7	3	no	—
56	C₃²×C₉	—	7	6	no	—
57	C₃²×C₉	—	21	9	no	—
58	C₃²×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₉⋊C₉	—	7	18	no	—
60	C₉⋊C₉	—	7	18	no	—
61	C₉⋊C₉	—	7	18	no	—
62	C₉⋊C₉	—	7	9	no	—
63	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	9	yes	complemented
64	C₃²⋊C₉	—	7	6	no	—
65	C₃²⋊C₉	—	7	18	no	—
66	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
67	C₉²	—	7	9	no	—
68	C₃²×C₇	—	3	3	no	—
69	C₃²×C₇	C₃×3₋¹⁺²	1	3	yes	—
70	C₃²×C₇	C₃⁴	1	1	yes	characteristic
71	C₃×ΓL(1,𝔽₈)	—	3	27	no	—
72	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
73	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
74	C₃×ΓL(1,𝔽₈)	#737	1	9	yes	—
75	C₇×C₉	—	3	3	no	—
76	C₇×C₉	—	3	9	no	—
77	C₇⋊C₉	—	3	3	no	—
78	C₇⋊C₉	—	3	18	no	—
79	C₇⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
80	C₇⋊C₉	C₃×3₋¹⁺²	1	6	yes	—
81	C₃³	—	7	3	no	—
82	C₃³	—	21	9	no	—
83	C₃³	—	21	3	no	—
84	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
85	3₋¹⁺²	—	21	27	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	7	9	no	—
88	C₃×C₉	—	7	1	no	—
89	C₃×C₉	—	7	2	no	—
90	C₃×C₉	—	7	18	no	—
91	C₃×C₉	—	7	3	no	—
92	C₃×C₉	—	21	9	no	—
93	C₃×C₉	—	21	18	no	—
94	C₃×C₉	—	21	3	no	—
95	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	9	yes	—
96	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
97	C₃×C₇	—	3	3	no	—
98	C₃×C₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
99	C₃×C₇	C₃×#737	1	3	yes	—
100	ΓL(1,𝔽₈)	—	3	9	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	7	9	no	—
103	C₃²	—	21	27	no	—
104	C₃²	—	21	3	no	—
105	C₃²	—	21	6	no	—
106	C₃²	ΓL(1,𝔽₈)×3₋¹⁺²	1	3	yes	—
107	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
108	C₉	—	3	3	no	—
109	C₉	—	3	9	no	—
110	C₉	—	7	3	no	—
111	C₉	—	7	6	no	—
112	C₉	—	21	3	no	—
113	C₉	—	21	18	no	—
114	C₇	#10624872	1	1	yes	characteristic, complemented, Sylow
115	C₃	—	3	3	no	—
116	C₃	—	21	9	no	—
117	C₃	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
118	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

#13133570

C₃

yes

maximal, complemented

#13133568

C₃

yes

maximal, complemented

#13133561

C₃

yes

maximal, complemented

#13133544

C₃

yes

maximal

#4428⋊C₉

C₃

yes

maximal, complemented

#13133527

C₃

yes

maximal, complemented

C₇×#6895

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

characteristic

C₃²×C₇⋊C₉

C₃²

yes

characteristic

C₃×#4428

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

—

C₃²⋊(C₇⋊C₉)

C₃²

yes

—

(C₃×ΓL(1,𝔽₈))⋊C₉

C₃²

yes

—

#10588035

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

C₉×C₇⋊C₉

C₃²

yes

complemented

C₃³×C₉

—

—

—

yes

characteristic, complemented

#6894

—

#6891

—

C₉×3₋¹⁺²

—

#6887

—

C₃³×C₇

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

C₃³

yes

—

C₃×C₇×C₉

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇⋊C₉

—

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

C₃³

yes

—

#4428

—

C₉×ΓL(1,𝔽₈)

—

C₉×ΓL(1,𝔽₈)

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

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

—

C₃²×C₇

—

C₃²×C₇

C₃×3₋¹⁺²

yes

—

C₃²×C₇

C₃⁴

yes

—

—

—

yes

—

C₇×C₉

—

C₇×C₉

—

C₇⋊C₉

—

C₇⋊C₉

—

C₇⋊C₉

C₃×3₋¹⁺²

yes

—

C₇⋊C₉

C₃×3₋¹⁺²

yes

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₇

—

C₃×C₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₇

C₃×#737

yes

—

100

ΓL(1,𝔽₈)

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

ΓL(1,𝔽₈)×3₋¹⁺²

yes

—

107

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₇

#10624872

yes

characteristic, complemented, Sylow

115

C₃

—

116

C₃

—

117

C₃

C₃×ΓL(1,𝔽₈)×3₋¹⁺²

yes

characteristic

118

C₃

ΓL(1,𝔽₈)×#737

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6895	C₃
hypercenter	#6895	ΓL(1,𝔽₈)
hypocenter	C₇	#10624872
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6895

subgroup

G/H

center, Frattini

C₃²

C₃³×ΓL(1,𝔽₈)

commutator, socle

C₃²×C₇

C₃⁴

C₃

C₇

G ▷ C₃²×C₇ ▷ C₇

C₁ ◁ C₃² ◁ #6895

N	H ≅ G/N	type
2. #13133570	C₃	semidirect
3. #13133568	C₃	semidirect
4. #13133561	C₃	semidirect
6. #4428⋊C₉	C₃	semidirect
7. #13133527	C₃	semidirect
8. C₇×#6895	C₃	semidirect
18. C₇×C₉⋊C₉	C₃²	semidirect
22. #10588035	C₃²	semidirect
23. (C₇⋊C₉)⋊C₉	C₃²	semidirect
24. (C₇⋊C₉)⋊C₉	C₃²	semidirect
25. (C₇⋊C₉)⋊C₉	C₃²	semidirect
26. C₉×C₇⋊C₉	C₃²	semidirect
30. #6895	ΓL(1,𝔽₈)	semidirect
43. C₃×C₇⋊C₉	3₋¹⁺²	semidirect
63. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
114. C₇	#10624872	semidirect

H ≅ G/N

type

2. #13133570

C₃

semidirect

3. #13133568

C₃

semidirect

4. #13133561

C₃

semidirect

6. #4428⋊C₉

C₃

semidirect

7. #13133527

C₃

semidirect

8. C₇×#6895

C₃

semidirect

18. C₇×C₉⋊C₉

C₃²

semidirect

22. #10588035

C₃²

semidirect

23. (C₇⋊C₉)⋊C₉

C₃²

semidirect

24. (C₇⋊C₉)⋊C₉

C₃²

semidirect

25. (C₇⋊C₉)⋊C₉

C₃²

semidirect

26. C₉×C₇⋊C₉

C₃²

semidirect

30. #6895

ΓL(1,𝔽₈)

semidirect

43. C₃×C₇⋊C₉

3₋¹⁺²

semidirect

63. C₉⋊C₉

C₃×ΓL(1,𝔽₈)

semidirect

114. C₇

#10624872

semidirect

Subgroups — #19103197

Conjugacy classes of subgroups

Complements