Group #19103208 — subgroups

Subgroups — ΓL(1,𝔽₈)×#6897 #19103208

Order 5103.

692 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 271 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	#13133581	C₃	1	8	yes	maximal, complemented
2	#13133580	C₃	1	8	yes	maximal, complemented
3	#13133579	C₃	1	1	yes	maximal, characteristic, complemented
4	#13133578	C₃	1	1	yes	maximal, characteristic, complemented
5	#13133577	C₃	1	8	yes	maximal, complemented
6	ΓL(1,𝔽₈)×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
7	ΓL(1,𝔽₈)×C₃²⋊C₉	C₃	1	4	yes	maximal
8	C₉²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
9	C₇×#6897	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#6897	—	7	1	no	maximal, Sylow
11	C₃³×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
12	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	4	yes	—
13	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	8	yes	—
14	C₇×C₉⋊C₉	C₃²	1	8	yes	complemented
15	C₇×C₃²⋊C₉	C₃²	1	4	yes	—
16	C₇×C₉²	C₃²	1	1	yes	characteristic, complemented
17	C₃²⋊(C₇⋊C₉)	C₃²	1	8	yes	—
18	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	4	yes	—
19	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	4	yes	—
20	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	8	yes	—
21	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	8	yes	—
22	#10588036	C₃²	1	8	yes	complemented
23	#10588035	C₃²	1	8	yes	complemented
24	(C₇⋊C₉)⋊C₉	C₃²	1	8	yes	complemented
25	(C₇⋊C₉)⋊C₉	C₃²	1	8	yes	complemented
26	(C₇⋊C₉)⋊C₉	C₃²	1	32	yes	complemented
27	C₉×C₇⋊C₉	C₃²	1	8	yes	complemented
28	#6897	—	7	8	no	—
29	#6897	—	7	8	no	—
30	#6897	—	7	8	no	—
31	#6897	—	7	1	no	—
32	#6897	—	7	1	no	—
33	#6897	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
34	C₃×C₉⋊C₉	—	7	8	no	—
35	C₃×C₃²⋊C₉	—	7	4	no	—
36	C₃×C₉²	—	7	1	no	—
37	C₃³×C₇	C₃³	1	1	yes	characteristic
38	C₃²×ΓL(1,𝔽₈)	—	3	12	no	—
39	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
40	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
41	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
42	C₃×C₇×C₉	C₃³	1	4	yes	—
43	C₃×C₇×C₉	C₃³	1	8	yes	—
44	C₃×C₇⋊C₉	C₃³	1	8	yes	—
45	C₃×C₇⋊C₉	C₃³	1	8	yes	—
46	C₃×C₇⋊C₉	C₃³	1	8	yes	—
47	C₉×ΓL(1,𝔽₈)	—	3	12	no	—
48	C₉×ΓL(1,𝔽₈)	—	3	24	no	—
49	C₃⁴	—	7	1	no	—
50	C₃²×C₉	—	7	4	no	—
51	C₃²×C₉	—	7	8	no	—
52	C₉⋊C₉	—	7	32	no	—
53	C₉⋊C₉	—	7	8	no	—
54	C₉⋊C₉	—	7	8	no	—
55	C₉⋊C₉	—	7	8	no	—
56	C₉⋊C₉	—	7	8	no	—
57	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
58	C₃²⋊C₉	—	7	8	no	—
59	C₃²⋊C₉	—	7	8	no	—
60	C₃²⋊C₉	—	7	8	no	—
61	C₃²⋊C₉	—	7	4	no	—
62	C₃²⋊C₉	—	7	4	no	—
63	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	4	yes	—
64	C₉²	—	7	8	no	—
65	C₉²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
66	C₃²×C₇	—	3	4	no	—
67	C₃²×C₇	C₃⁴	1	1	yes	characteristic
68	C₃×ΓL(1,𝔽₈)	—	3	4	no	—
69	C₃×ΓL(1,𝔽₈)	—	3	4	no	—
70	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
71	C₃×ΓL(1,𝔽₈)	#737	1	12	yes	—
72	C₇×C₉	—	3	4	no	—
73	C₇×C₉	—	3	8	no	—
74	C₇⋊C₉	—	3	8	no	—
75	C₇⋊C₉	—	3	8	no	—
76	C₇⋊C₉	—	3	8	no	—
77	C₃³	—	7	1	no	—
78	C₃³	—	7	1	no	—
79	C₃³	—	7	1	no	—
80	C₃³	—	21	12	no	—
81	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
82	C₃×C₉	—	7	8	no	—
83	C₃×C₉	—	7	8	no	—
84	C₃×C₉	—	7	8	no	—
85	C₃×C₉	—	21	12	no	—
86	C₃×C₉	—	21	24	no	—
87	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	4	yes	—
88	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	8	yes	—
89	C₃×C₇	—	9	1	no	—
90	C₃×C₇	C₃×#737	1	4	yes	—
91	ΓL(1,𝔽₈)	—	9	1	no	—
92	ΓL(1,𝔽₈)	—	9	1	no	—
93	ΓL(1,𝔽₈)	#6897	1	9	yes	direct product factor
94	C₃²	—	3	4	no	—
95	C₃²	—	7	12	no	—
96	C₃²	—	21	4	no	—
97	C₃²	—	21	4	no	—
98	C₃²	—	63	9	no	—
99	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
100	C₉	—	3	4	no	—
101	C₉	—	3	8	no	—
102	C₉	—	21	8	no	—
103	C₉	—	21	8	no	—
104	C₉	—	21	8	no	—
105	C₇	C₃×#6897	1	1	yes	characteristic, complemented, Sylow
106	C₃	—	7	9	no	—
107	C₃	—	9	1	no	—
108	C₃	—	63	1	no	—
109	C₃	—	63	1	no	—
110	C₃	ΓL(1,𝔽₈)×#737	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13133581

C₃

yes

maximal, complemented

#13133580

C₃

yes

maximal, complemented

#13133579

C₃

yes

maximal, characteristic, complemented

#13133578

C₃

yes

maximal, characteristic, complemented

#13133577

C₃

yes

maximal, complemented

ΓL(1,𝔽₈)×C₉⋊C₉

C₃

yes

maximal, complemented

ΓL(1,𝔽₈)×C₃²⋊C₉

C₃

yes

maximal

C₉²×ΓL(1,𝔽₈)

C₃

yes

maximal, characteristic, complemented

C₇×#6897

C₃

yes

maximal, characteristic, complemented

C₃×#6897

—

maximal, Sylow

C₃³×ΓL(1,𝔽₈)

C₃²

yes

characteristic

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

characteristic, complemented

C₃²⋊(C₇⋊C₉)

C₃²

yes

—

(C₃×ΓL(1,𝔽₈))⋊C₉

C₃²

yes

—

(C₃×ΓL(1,𝔽₈))⋊C₉

C₃²

yes

—

(C₃×ΓL(1,𝔽₈))⋊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

C₉×C₇⋊C₉

C₃²

yes

—

—

—

—

—

yes

characteristic, direct product factor

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₉²

—

C₃³×C₇

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

—

C₃²×ΓL(1,𝔽₈)

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

C₃³

yes

characteristic

C₃²×ΓL(1,𝔽₈)

C₃³

yes

characteristic

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₉×ΓL(1,𝔽₈)

—

C₉×ΓL(1,𝔽₈)

—

C₃⁴

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×ΓL(1,𝔽₈)

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×ΓL(1,𝔽₈)

yes

—

C₉²

—

C₉²

C₃×ΓL(1,𝔽₈)

yes

characteristic, complemented

C₃²×C₇

—

C₃²×C₇

C₃⁴

yes

—

—

—

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

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₉

C₃²×ΓL(1,𝔽₈)

yes

—

C₃×C₇

—

C₃×C₇

C₃×#737

yes

—

—

—

yes

direct product factor

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₇

C₃×#6897

yes

characteristic, complemented, Sylow

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

ΓL(1,𝔽₈)×#737

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6897	C₃
hypercenter	#6897	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#6897
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6897

subgroup

G/H

center, Frattini

C₃²

C₃³×ΓL(1,𝔽₈)

commutator, socle

C₃²×C₇

C₃⁴

C₃

C₇

G ▷ C₃²×C₇ ▷ C₇

C₁ ◁ C₃² ◁ #6897

N	H ≅ G/N	type
1. #13133581	C₃	semidirect
2. #13133580	C₃	semidirect
3. #13133579	C₃	semidirect
4. #13133578	C₃	semidirect
5. #13133577	C₃	semidirect
6. ΓL(1,𝔽₈)×C₉⋊C₉	C₃	semidirect
8. C₉²×ΓL(1,𝔽₈)	C₃	semidirect
9. C₇×#6897	C₃	semidirect
14. C₇×C₉⋊C₉	C₃²	semidirect
16. C₇×C₉²	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
27. C₉×C₇⋊C₉	C₃²	semidirect
33. #6897	ΓL(1,𝔽₈)	direct (N × H)
57. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
65. C₉²	C₃×ΓL(1,𝔽₈)	semidirect
93. ΓL(1,𝔽₈)	#6897	direct (N × H)
105. C₇	C₃×#6897	semidirect

H ≅ G/N

type

1. #13133581

C₃

semidirect

2. #13133580

C₃

semidirect

3. #13133579

C₃

semidirect

4. #13133578

C₃

semidirect

5. #13133577

C₃

semidirect

6. ΓL(1,𝔽₈)×C₉⋊C₉

C₃

semidirect

8. C₉²×ΓL(1,𝔽₈)

C₃

semidirect

9. C₇×#6897

C₃

semidirect

14. C₇×C₉⋊C₉

C₃²

semidirect

16. C₇×C₉²

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

27. C₉×C₇⋊C₉

C₃²

semidirect

33. #6897

ΓL(1,𝔽₈)

direct (N × H)

57. C₉⋊C₉

C₃×ΓL(1,𝔽₈)

semidirect

65. C₉²

C₃×ΓL(1,𝔽₈)

semidirect

93. ΓL(1,𝔽₈)

#6897

direct (N × H)

105. C₇

C₃×#6897

semidirect

Subgroups — ΓL(1,𝔽₈)×#6897 #19103208

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽8)×#6897 #19103208

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽₈)×#6897 #19103208