Group #19103171 — subgroups

Subgroups — #19103171

Order 5103.

1166 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 292 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	C₃×#10588062	C₃	1	3	yes	maximal, complemented
2	C₃³×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	#13133554	C₃	1	6	yes	maximal, complemented
4	#13133548	C₃	1	18	yes	maximal, complemented
5	#13133546	C₃	1	2	yes	maximal, complemented
6	H(3,𝔽₃)×C₇⋊C₉	C₃	1	9	yes	maximal, complemented
7	C₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
8	#10624869	—	7	1	no	maximal, Sylow
9	C₃⁴×C₇	C₃²	1	1	yes	characteristic
10	C₃×C₇×H(3,𝔽₃)	C₃²	1	3	yes	—
11	C₃×C₇×H(3,𝔽₃)	C₃²	1	9	yes	—
12	C₃×#4430	C₃²	1	6	yes	—
13	C₃×#4430	C₃²	1	18	yes	complemented
14	C₃²×C₇⋊C₉	—	3	9	no	—
15	C₃²×C₇⋊C₉	C₃²	1	3	yes	—
16	C₃²×C₇⋊C₉	C₃²	1	6	yes	—
17	C₃²×C₇⋊C₉	C₃²	1	2	yes	—
18	C₃²×C₇⋊C₉	C₃²	1	1	yes	characteristic
19	C₃²×C₇⋊C₉	C₃²	1	9	yes	complemented
20	#10588062	—	3	27	no	—
21	C₃²⋊(C₇⋊C₉)	C₃²	1	18	yes	—
22	C₃²⋊(C₇⋊C₉)	C₃²	1	54	yes	complemented
23	C₃×#737	—	7	3	no	—
24	C₃³×C₉	—	7	1	no	—
25	#6893	—	7	6	no	—
26	C₉⋊H(3,𝔽₃)	—	7	18	no	—
27	#6891	—	7	2	no	—
28	C₃²⋊H(3,𝔽₃)	ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₉×H(3,𝔽₃)	—	7	9	no	—
30	C₃³×C₇	—	3	3	no	—
31	C₃³×C₇	—	3	9	no	—
32	C₃³×C₇	C₃³	1	9	yes	—
33	C₃³×C₇	C₃³	1	1	yes	characteristic
34	C₃³×C₇	C₃³	1	3	yes	—
35	C₇×H(3,𝔽₃)	—	3	9	no	—
36	C₇×H(3,𝔽₃)	—	3	27	no	—
37	#4430	—	3	18	no	—
38	#4430	—	3	54	no	—
39	C₃×C₇⋊C₉	—	3	6	no	—
40	C₃×C₇⋊C₉	—	3	27	no	—
41	C₃×C₇⋊C₉	—	3	9	no	—
42	C₃×C₇⋊C₉	—	3	18	no	—
43	C₃×C₇⋊C₉	H(3,𝔽₃)	1	9	yes	complemented
44	C₃×C₇⋊C₉	C₃³	1	18	yes	—
45	C₃×C₇⋊C₉	C₃³	1	6	yes	—
46	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
47	C₃×C₇⋊C₉	C₃³	1	2	yes	—
48	C₃⁴	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
49	#737	—	21	27	no	—
50	C₃×3₋¹⁺²	—	7	6	no	—
51	C₃×3₋¹⁺²	—	7	18	no	—
52	C₃×H(3,𝔽₃)	C₃×ΓL(1,𝔽₈)	1	3	yes	—
53	C₃×H(3,𝔽₃)	C₃×ΓL(1,𝔽₈)	1	9	yes	—
54	C₃²×C₉	—	7	1	no	—
55	C₃²×C₉	—	7	3	no	—
56	C₃²×C₉	—	7	6	no	—
57	C₃²×C₉	—	7	2	no	—
58	C₃²×C₉	—	7	9	no	—
59	C₃²×C₉	—	21	9	no	—
60	C₃²⋊C₉	—	7	18	no	—
61	C₃²⋊C₉	—	7	54	no	—
62	C₃²×C₇	—	3	27	no	—
63	C₃²×C₇	—	3	3	no	—
64	C₃²×C₇	—	3	3	no	—
65	C₃²×C₇	—	3	6	no	—
66	C₃²×C₇	—	3	9	no	—
67	C₃²×C₇	—	3	27	no	—
68	C₃²×C₇	C₃×H(3,𝔽₃)	1	3	yes	—
69	C₃²×C₇	#737	1	9	yes	complemented
70	C₃²×C₇	C₃⁴	1	1	yes	characteristic
71	C₇⋊C₉	—	3	6	no	—
72	C₇⋊C₉	—	3	18	no	—
73	C₇⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
74	C₇⋊C₉	C₃×H(3,𝔽₃)	1	6	yes	—
75	C₃³	—	3	9	no	—
76	C₃³	—	3	3	no	—
77	C₃³	C₃²×ΓL(1,𝔽₈)	1	9	yes	—
78	C₃³	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
79	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
80	3₋¹⁺²	—	21	54	no	—
81	3₋¹⁺²	—	21	18	no	—
82	H(3,𝔽₃)	—	3	27	no	—
83	H(3,𝔽₃)	—	3	9	no	—
84	C₃×C₉	—	7	9	no	—
85	C₃×C₉	—	7	1	no	—
86	C₃×C₉	—	7	2	no	—
87	C₃×C₉	—	7	6	no	—
88	C₃×C₉	—	7	18	no	—
89	C₃×C₉	—	21	9	no	—
90	C₃×C₉	—	21	18	no	—
91	C₃×C₉	—	21	27	no	—
92	C₃×C₉	—	21	6	no	—
93	C₃×C₇	—	3	3	no	—
94	C₃×C₇	—	3	9	no	—
95	C₃×C₇	—	9	9	no	—
96	C₃×C₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
97	C₃×C₇	C₃×#737	1	3	yes	—
98	C₃²	—	3	27	no	—
99	C₃²	—	3	3	no	—
100	C₃²	—	3	6	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	3	9	no	—
103	C₃²	—	3	27	no	—
104	C₃²	#10588062	1	9	yes	complemented
105	C₃²	ΓL(1,𝔽₈)×H(3,𝔽₃)	1	3	yes	—
106	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
107	C₉	—	7	3	no	—
108	C₉	—	7	6	no	—
109	C₉	—	21	6	no	—
110	C₉	—	21	18	no	—
111	C₇	#10624869	1	1	yes	characteristic, complemented, Sylow
112	C₃	—	3	9	no	—
113	C₃	—	3	3	no	—
114	C₃	—	9	9	no	—
115	C₃	C₃×ΓL(1,𝔽₈)×H(3,𝔽₃)	1	1	yes	characteristic
116	C₃	C₃×#10588062	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#10588062

C₃

yes

maximal, complemented

C₃³×C₇⋊C₉

C₃

yes

maximal, characteristic, complemented

#13133554

C₃

yes

maximal, complemented

#13133548

C₃

yes

maximal, complemented

#13133546

C₃

yes

maximal, complemented

H(3,𝔽₃)×C₇⋊C₉

C₃

yes

maximal, complemented

C₇×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10624869

—

maximal, Sylow

C₃⁴×C₇

C₃²

yes

characteristic

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃×#4430

C₃²

yes

—

C₃×#4430

C₃²

yes

complemented

C₃²×C₇⋊C₉

—

C₃²×C₇⋊C₉

C₃²

yes

—

C₃²×C₇⋊C₉

C₃²

yes

—

C₃²×C₇⋊C₉

C₃²

yes

—

C₃²×C₇⋊C₉

C₃²

yes

characteristic

C₃²×C₇⋊C₉

C₃²

yes

complemented

#10588062

—

C₃²⋊(C₇⋊C₉)

C₃²

yes

—

C₃²⋊(C₇⋊C₉)

C₃²

yes

complemented

C₃×#737

—

C₃³×C₉

—

#6893

—

C₉⋊H(3,𝔽₃)

—

#6891

—

C₃²⋊H(3,𝔽₃)

ΓL(1,𝔽₈)

yes

characteristic

C₉×H(3,𝔽₃)

—

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

C₃³

yes

—

C₃³×C₇

C₃³

yes

characteristic

C₃³×C₇

C₃³

yes

—

C₇×H(3,𝔽₃)

—

C₇×H(3,𝔽₃)

—

#4430

—

#4430

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

H(3,𝔽₃)

yes

complemented

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

characteristic

C₃×C₇⋊C₉

C₃³

yes

—

C₃⁴

C₃×ΓL(1,𝔽₈)

yes

characteristic

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×ΓL(1,𝔽₈)

yes

—

C₃×H(3,𝔽₃)

C₃×ΓL(1,𝔽₈)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₇

#737

yes

complemented

C₃²×C₇

C₃⁴

yes

characteristic

C₇⋊C₉

—

C₇⋊C₉

—

C₇⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₇⋊C₉

C₃×H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₇

C₃×#737

yes

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

#10588062

yes

complemented

105

C₃²

ΓL(1,𝔽₈)×H(3,𝔽₃)

yes

—

106

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₇

#10624869

yes

characteristic, complemented, Sylow

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

C₃×ΓL(1,𝔽₈)×H(3,𝔽₃)

yes

characteristic

116

C₃

C₃×#10588062

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×C₃²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊H(3,𝔽₃)	ΓL(1,𝔽₈)
hypocenter	C₇	#10624869
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₃²

C₃³×ΓL(1,𝔽₈)

commutator, socle

C₃²×C₇

C₃⁴

Fitting

C₇×C₃²⋊H(3,𝔽₃)

C₃

C₃²⋊H(3,𝔽₃)

C₇

G ▷ C₃²×C₇ ▷ C₇

C₁ ◁ C₃² ◁ C₃²⋊H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃×#10588062	C₃	semidirect
2. C₃³×C₇⋊C₉	C₃	semidirect
3. #13133554	C₃	semidirect
4. #13133548	C₃	semidirect
5. #13133546	C₃	semidirect
6. H(3,𝔽₃)×C₇⋊C₉	C₃	semidirect
13. C₃×#4430	C₃²	semidirect
19. C₃²×C₇⋊C₉	C₃²	semidirect
22. C₃²⋊(C₇⋊C₉)	C₃²	semidirect
43. C₃×C₇⋊C₉	H(3,𝔽₃)	semidirect
69. C₃²×C₇	#737	semidirect
104. C₃²	#10588062	semidirect
111. C₇	#10624869	semidirect

H ≅ G/N

type

1. C₃×#10588062

C₃

semidirect

2. C₃³×C₇⋊C₉

C₃

semidirect

3. #13133554

C₃

semidirect

4. #13133548

C₃

semidirect

5. #13133546

C₃

semidirect

6. H(3,𝔽₃)×C₇⋊C₉

C₃

semidirect

13. C₃×#4430

C₃²

semidirect

19. C₃²×C₇⋊C₉

C₃²

semidirect

22. C₃²⋊(C₇⋊C₉)

C₃²

semidirect

43. C₃×C₇⋊C₉

H(3,𝔽₃)

semidirect

69. C₃²×C₇

#737

semidirect

104. C₃²

#10588062

semidirect

111. C₇

#10624869

semidirect

Subgroups — #19103171

Conjugacy classes of subgroups

Complements