Group #19103188 — subgroups

Subgroups — ΓL(1,𝔽₈)×#6895 #19103188

Order 5103.

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

normal subgroups only

This group has 836 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	C₃²×C₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
2	#13133570	C₃	1	2	yes	maximal, complemented
3	#13133569	C₃	1	18	yes	maximal, complemented
4	#13133568	C₃	1	3	yes	maximal, complemented
5	#13133567	C₃	1	3	yes	maximal, complemented
6	ΓL(1,𝔽₈)×C₉⋊C₉	C₃	1	9	yes	maximal, complemented
7	ΓL(1,𝔽₈)×C₃²⋊C₉	C₃	1	3	yes	maximal
8	C₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃×#6895	—	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×C₉×ΓL(1,𝔽₈)	—	3	9	no	—
15	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	6	yes	—
16	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
17	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	9	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	#10588036	C₃²	1	9	yes	complemented
23	#10588035	C₃²	1	9	yes	complemented
24	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
25	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
26	(C₇⋊C₉)⋊C₉	C₃²	1	18	yes	complemented
27	C₃³×C₉	—	7	1	no	—
28	#6895	—	7	2	no	—
29	#6895	—	7	18	no	—
30	#6895	—	7	3	no	—
31	#6895	—	7	3	no	—
32	#6895	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
33	C₃×C₉⋊C₉	—	7	9	no	—
34	C₃×C₃²⋊C₉	—	7	3	no	—
35	C₃³×C₇	C₃³	1	1	yes	characteristic
36	C₃²×ΓL(1,𝔽₈)	—	3	9	no	—
37	C₃²×ΓL(1,𝔽₈)	3₋¹⁺²	1	9	yes	—
38	C₃²×ΓL(1,𝔽₈)	C₃³	1	2	yes	—
39	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
40	C₃×C₇×C₉	—	3	3	no	—
41	C₃×C₇×C₉	C₃³	1	3	yes	—
42	C₃×C₇×C₉	C₃³	1	9	yes	—
43	C₃×C₇⋊C₉	—	3	3	no	—
44	C₃×C₇⋊C₉	—	3	3	no	—
45	C₃×C₇⋊C₉	C₃³	1	18	yes	—
46	C₃×C₇⋊C₉	C₃³	1	3	yes	—
47	C₃×C₇⋊C₉	C₃³	1	3	yes	—
48	C₉×ΓL(1,𝔽₈)	—	3	18	no	—
49	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
50	C₉×ΓL(1,𝔽₈)	—	3	27	no	—
51	C₃⁴	—	7	1	no	—
52	C₃²×C₉	—	7	1	no	—
53	C₃²×C₉	—	7	1	no	—
54	C₃²×C₉	—	7	3	no	—
55	C₃²×C₉	—	7	6	no	—
56	C₃²×C₉	—	7	9	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	—	7	9	no	—
64	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	9	yes	complemented
65	C₃²⋊C₉	—	7	6	no	—
66	C₃²⋊C₉	—	7	18	no	—
67	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
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	6	no	—
72	C₃×ΓL(1,𝔽₈)	—	3	27	no	—
73	C₃×ΓL(1,𝔽₈)	C₃×3₋¹⁺²	1	6	yes	—
74	C₃×ΓL(1,𝔽₈)	C₃×3₋¹⁺²	1	3	yes	—
75	C₃×ΓL(1,𝔽₈)	#737	1	9	yes	—
76	C₇×C₉	—	3	3	no	—
77	C₇×C₉	—	3	9	no	—
78	C₇⋊C₉	—	3	3	no	—
79	C₇⋊C₉	—	3	3	no	—
80	C₇⋊C₉	—	3	18	no	—
81	C₃³	—	7	1	no	—
82	C₃³	—	7	9	no	—
83	C₃³	—	7	2	no	—
84	C₃³	—	21	9	no	—
85	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	7	18	no	—
88	C₃×C₉	—	7	3	no	—
89	C₃×C₉	—	7	3	no	—
90	C₃×C₉	—	21	27	no	—
91	C₃×C₉	—	21	9	no	—
92	C₃×C₉	—	21	18	no	—
93	C₃×C₉	—	21	3	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	6	no	—
101	ΓL(1,𝔽₈)	#6895	1	9	yes	direct product factor
102	C₃²	—	3	3	no	—
103	C₃²	—	7	3	no	—
104	C₃²	—	7	9	no	—
105	C₃²	—	7	6	no	—
106	C₃²	—	21	27	no	—
107	C₃²	—	21	6	no	—
108	C₃²	ΓL(1,𝔽₈)×3₋¹⁺²	1	3	yes	—
109	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
110	C₉	—	3	3	no	—
111	C₉	—	3	9	no	—
112	C₉	—	21	3	no	—
113	C₉	—	21	3	no	—
114	C₉	—	21	18	no	—
115	C₇	C₃×#6895	1	1	yes	characteristic, complemented, Sylow
116	C₃	—	3	3	no	—
117	C₃	—	7	9	no	—
118	C₃	—	21	6	no	—
119	C₃	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
120	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

#13133569

C₃

yes

maximal, complemented

#13133568

C₃

yes

maximal, complemented

#13133567

C₃

yes

maximal, complemented

ΓL(1,𝔽₈)×C₉⋊C₉

C₃

yes

maximal, complemented

ΓL(1,𝔽₈)×C₃²⋊C₉

C₃

yes

maximal

C₇×#6895

C₃

yes

maximal, characteristic, complemented

C₃×#6895

—

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×C₉×ΓL(1,𝔽₈)

—

C₃×C₉×ΓL(1,𝔽₈)

C₃²

yes

—

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

—

#10588036

C₃²

yes

complemented

#10588035

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

C₃³×C₉

—

—

—

—

—

yes

characteristic, direct product factor

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃³×C₇

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

3₋¹⁺²

yes

—

C₃²×ΓL(1,𝔽₈)

C₃³

yes

—

C₃²×ΓL(1,𝔽₈)

C₃³

yes

characteristic

C₃×C₇×C₉

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₉×ΓL(1,𝔽₈)

—

C₉×ΓL(1,𝔽₈)

—

C₉×ΓL(1,𝔽₈)

—

C₃⁴

—

C₃²×C₉

—

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

—

C₉⋊C₉

C₃×ΓL(1,𝔽₈)

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×ΓL(1,𝔽₈)

yes

—

C₃²×C₇

—

C₃²×C₇

C₃×3₋¹⁺²

yes

—

C₃²×C₇

C₃⁴

yes

—

—

C₃×3₋¹⁺²

yes

—

C₃×ΓL(1,𝔽₈)

C₃×3₋¹⁺²

yes

—

C₃×ΓL(1,𝔽₈)

#737

yes

—

C₇×C₉

—

C₇×C₉

—

C₇⋊C₉

—

C₇⋊C₉

—

C₇⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

characteristic

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

ΓL(1,𝔽₈)

#6895

yes

direct product factor

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

ΓL(1,𝔽₈)×3₋¹⁺²

yes

—

109

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₇

C₃×#6895

yes

characteristic, complemented, Sylow

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

C₃×ΓL(1,𝔽₈)×3₋¹⁺²

yes

characteristic

120

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	C₃×#6895
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. #13133569	C₃	semidirect
4. #13133568	C₃	semidirect
5. #13133567	C₃	semidirect
6. ΓL(1,𝔽₈)×C₉⋊C₉	C₃	semidirect
8. C₇×#6895	C₃	semidirect
18. C₇×C₉⋊C₉	C₃²	semidirect
22. #10588036	C₃²	semidirect
23. #10588035	C₃²	semidirect
24. (C₇⋊C₉)⋊C₉	C₃²	semidirect
25. (C₇⋊C₉)⋊C₉	C₃²	semidirect
26. (C₇⋊C₉)⋊C₉	C₃²	semidirect
32. #6895	ΓL(1,𝔽₈)	direct (N × H)
64. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
101. ΓL(1,𝔽₈)	#6895	direct (N × H)
115. C₇	C₃×#6895	semidirect

H ≅ G/N

type

2. #13133570

C₃

semidirect

3. #13133569

C₃

semidirect

4. #13133568

C₃

semidirect

5. #13133567

C₃

semidirect

6. ΓL(1,𝔽₈)×C₉⋊C₉

C₃

semidirect

8. C₇×#6895

C₃

semidirect

18. C₇×C₉⋊C₉

C₃²

semidirect

22. #10588036

C₃²

semidirect

23. #10588035

C₃²

semidirect

24. (C₇⋊C₉)⋊C₉

C₃²

semidirect

25. (C₇⋊C₉)⋊C₉

C₃²

semidirect

26. (C₇⋊C₉)⋊C₉

C₃²

semidirect

32. #6895

ΓL(1,𝔽₈)

direct (N × H)

64. C₉⋊C₉

C₃×ΓL(1,𝔽₈)

semidirect

101. ΓL(1,𝔽₈)

#6895

direct (N × H)

115. C₇

C₃×#6895

semidirect

Subgroups — ΓL(1,𝔽₈)×#6895 #19103188

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽8)×#6895 #19103188

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽₈)×#6895 #19103188