Group #19103241 — subgroups

Subgroups — ΓL(1,𝔽₈)×#6900 #19103241

Order 5103.

1076 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 355 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×ΓL(1,𝔽₈)×3₋¹⁺²	C₃	1	4	yes	maximal
2	#13133597	C₃	1	8	yes	maximal, complemented
3	#13133596	C₃	1	8	yes	maximal, complemented
4	#13133595	C₃	1	1	yes	maximal, characteristic, complemented
5	#13133594	C₃	1	1	yes	maximal, characteristic, complemented
6	#13133593	C₃	1	8	yes	maximal, complemented
7	ΓL(1,𝔽₈)×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
8	C₉²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
9	C₇×#6900	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#6900	—	7	1	no	maximal, Sylow
11	C₃³×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
12	C₃×C₇×3₋¹⁺²	C₃²	1	4	yes	—
13	C₃×#4430	C₃²	1	8	yes	—
14	ΓL(1,𝔽₈)×3₋¹⁺²	—	3	36	no	—
15	C₃×#4428	C₃²	1	4	yes	—
16	C₃×#4428	C₃²	1	8	yes	—
17	C₃×#4427	C₃²	1	4	yes	—
18	C₃×#4427	C₃²	1	8	yes	—
19	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	8	yes	—
20	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	4	yes	—
21	C₇×C₉⋊C₉	C₃²	1	8	yes	complemented
22	C₇×C₉²	C₃²	1	1	yes	characteristic, complemented
23	#10588036	C₃²	1	8	yes	complemented
24	#10588035	C₃²	1	8	yes	complemented
25	(C₇⋊C₉)⋊C₉	C₃²	1	48	yes	complemented
26	C₉×C₇⋊C₉	C₃²	1	8	yes	complemented
27	C₃²×3₋¹⁺²	—	7	4	no	—
28	#6900	—	7	8	no	—
29	#6900	—	7	8	no	—
30	#6900	—	7	8	no	—
31	#6900	—	7	1	no	—
32	#6900	—	7	1	no	—
33	#6900	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
34	C₃×C₉⋊C₉	—	7	8	no	—
35	C₃×C₉²	—	7	1	no	—
36	C₃³×C₇	C₃³	1	1	yes	characteristic
37	C₃²×ΓL(1,𝔽₈)	—	3	12	no	—
38	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
39	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
40	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
41	C₇×3₋¹⁺²	—	3	12	no	—
42	C₃×C₇×C₉	C₃³	1	8	yes	—
43	C₃×C₇×C₉	C₃³	1	4	yes	—
44	#4430	—	3	24	no	—
45	C₃×C₇⋊C₉	C₃³	1	8	yes	—
46	C₃×C₇⋊C₉	C₃³	1	8	yes	—
47	C₃×C₇⋊C₉	C₃³	1	8	yes	—
48	#4428	—	3	12	no	—
49	#4428	—	3	24	no	—
50	#4427	—	3	12	no	—
51	#4427	—	3	24	no	—
52	C₉×ΓL(1,𝔽₈)	—	3	24	no	—
53	C₉×ΓL(1,𝔽₈)	3₋¹⁺²	1	36	yes	complemented
54	C₃⁴	—	7	1	no	—
55	C₃×3₋¹⁺²	—	7	8	no	—
56	C₃×3₋¹⁺²	—	7	8	no	—
57	C₃×3₋¹⁺²	—	7	4	no	—
58	C₃×3₋¹⁺²	—	7	8	no	—
59	C₃×3₋¹⁺²	—	7	4	no	—
60	C₃×3₋¹⁺²	—	21	36	no	—
61	C₃×3₋¹⁺²	C₃×ΓL(1,𝔽₈)	1	4	yes	—
62	C₃²×C₉	—	7	4	no	—
63	C₃²×C₉	—	7	8	no	—
64	C₉⋊C₉	—	7	48	no	—
65	C₉⋊C₉	—	7	8	no	—
66	C₉⋊C₉	—	7	8	no	—
67	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
68	C₉²	—	7	8	no	—
69	C₉²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
70	C₃²×C₇	—	3	4	no	—
71	C₃²×C₇	C₃⁴	1	1	yes	characteristic
72	C₃×ΓL(1,𝔽₈)	—	3	4	no	—
73	C₃×ΓL(1,𝔽₈)	—	3	4	no	—
74	C₃×ΓL(1,𝔽₈)	—	9	9	no	—
75	C₃×ΓL(1,𝔽₈)	C₃×3₋¹⁺²	1	12	yes	—
76	C₇×C₉	—	3	8	no	—
77	C₇×C₉	C₃×3₋¹⁺²	1	12	yes	complemented
78	C₇⋊C₉	—	3	8	no	—
79	C₇⋊C₉	—	3	8	no	—
80	C₇⋊C₉	C₃×3₋¹⁺²	1	24	yes	complemented
81	C₃³	—	7	1	no	—
82	C₃³	—	7	1	no	—
83	C₃³	—	7	1	no	—
84	C₃³	—	21	12	no	—
85	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
86	3₋¹⁺²	—	3	12	no	—
87	3₋¹⁺²	—	21	24	no	—
88	3₋¹⁺²	—	21	24	no	—
89	3₋¹⁺²	—	21	24	no	—
90	3₋¹⁺²	—	21	12	no	—
91	3₋¹⁺²	—	21	12	no	—
92	C₃×C₉	—	7	8	no	—
93	C₃×C₉	—	7	36	no	—
94	C₃×C₉	—	7	8	no	—
95	C₃×C₉	—	7	8	no	—
96	C₃×C₉	—	21	24	no	—
97	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	8	yes	—
98	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	4	yes	—
99	C₃×C₇	—	9	1	no	—
100	C₃×C₇	C₃²×3₋¹⁺²	1	4	yes	—
101	ΓL(1,𝔽₈)	—	9	1	no	—
102	ΓL(1,𝔽₈)	—	9	1	no	—
103	ΓL(1,𝔽₈)	#6900	1	9	yes	direct product factor
104	C₃²	—	3	4	no	—
105	C₃²	—	7	12	no	—
106	C₃²	—	21	4	no	—
107	C₃²	—	21	4	no	—
108	C₃²	—	63	9	no	—
109	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
110	C₉	—	3	8	no	—
111	C₉	—	7	24	no	—
112	C₉	—	21	8	no	—
113	C₉	—	21	8	no	—
114	C₉	ΓL(1,𝔽₈)×3₋¹⁺²	1	12	yes	complemented
115	C₇	C₃×#6900	1	1	yes	characteristic, complemented, Sylow
116	C₃	—	7	9	no	—
117	C₃	—	9	1	no	—
118	C₃	—	63	1	no	—
119	C₃	—	63	1	no	—
120	C₃	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×ΓL(1,𝔽₈)×3₋¹⁺²

C₃

yes

maximal

#13133597

C₃

yes

maximal, complemented

#13133596

C₃

yes

maximal, complemented

#13133595

C₃

yes

maximal, characteristic, complemented

#13133594

C₃

yes

maximal, characteristic, complemented

#13133593

C₃

yes

maximal, complemented

ΓL(1,𝔽₈)×C₉⋊C₉

C₃

yes

maximal, complemented

C₉²×ΓL(1,𝔽₈)

C₃

yes

maximal, characteristic, complemented

C₇×#6900

C₃

yes

maximal, characteristic, complemented

C₃×#6900

—

maximal, Sylow

C₃³×ΓL(1,𝔽₈)

C₃²

yes

characteristic

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×#4430

C₃²

yes

—

ΓL(1,𝔽₈)×3₋¹⁺²

—

C₃×#4428

C₃²

yes

—

C₃×#4428

C₃²

yes

—

C₃×#4427

C₃²

yes

—

C₃×#4427

C₃²

yes

—

C₃×C₉×ΓL(1,𝔽₈)

C₃²

yes

—

C₃×C₉×ΓL(1,𝔽₈)

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

complemented

C₇×C₉²

C₃²

yes

characteristic, complemented

#10588036

C₃²

yes

complemented

#10588035

C₃²

yes

complemented

(C₇⋊C₉)⋊C₉

C₃²

yes

complemented

C₉×C₇⋊C₉

C₃²

yes

complemented

C₃²×3₋¹⁺²

—

—

—

—

—

—

yes

characteristic, direct product factor

C₃×C₉⋊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

C₇×3₋¹⁺²

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

#4430

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

C₃×C₇⋊C₉

C₃³

yes

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×ΓL(1,𝔽₈)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×ΓL(1,𝔽₈)

yes

complemented

C₉²

—

C₉²

C₃×ΓL(1,𝔽₈)

yes

characteristic, complemented

C₃²×C₇

—

C₃²×C₇

C₃⁴

yes

—

—

—

C₃×3₋¹⁺²

yes

—

C₇×C₉

—

C₇×C₉

C₃×3₋¹⁺²

yes

complemented

C₇⋊C₉

—

C₇⋊C₉

—

C₇⋊C₉

C₃×3₋¹⁺²

yes

complemented

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×ΓL(1,𝔽₈)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

—

100

C₃×C₇

C₃²×3₋¹⁺²

yes

—

101

ΓL(1,𝔽₈)

—

102

ΓL(1,𝔽₈)

—

103

ΓL(1,𝔽₈)

#6900

yes

direct product factor

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

C₃³×ΓL(1,𝔽₈)

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

ΓL(1,𝔽₈)×3₋¹⁺²

yes

complemented

115

C₇

C₃×#6900

yes

characteristic, complemented, Sylow

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

C₃×ΓL(1,𝔽₈)×3₋¹⁺²

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6900	C₃
hypercenter	#6900	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#6900
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6900

subgroup

G/H

center, Frattini

C₃²

C₃³×ΓL(1,𝔽₈)

commutator, socle

C₃²×C₇

C₃⁴

C₃

C₇

G ▷ C₃²×C₇ ▷ C₇

C₁ ◁ C₃² ◁ #6900

N	H ≅ G/N	type
2. #13133597	C₃	semidirect
3. #13133596	C₃	semidirect
4. #13133595	C₃	semidirect
5. #13133594	C₃	semidirect
6. #13133593	C₃	semidirect
7. ΓL(1,𝔽₈)×C₉⋊C₉	C₃	semidirect
8. C₉²×ΓL(1,𝔽₈)	C₃	semidirect
9. C₇×#6900	C₃	semidirect
21. C₇×C₉⋊C₉	C₃²	semidirect
22. C₇×C₉²	C₃²	semidirect
23. #10588036	C₃²	semidirect
24. #10588035	C₃²	semidirect
25. (C₇⋊C₉)⋊C₉	C₃²	semidirect
26. C₉×C₇⋊C₉	C₃²	semidirect
33. #6900	ΓL(1,𝔽₈)	direct (N × H)
53. C₉×ΓL(1,𝔽₈)	3₋¹⁺²	semidirect
67. C₉⋊C₉	C₃×ΓL(1,𝔽₈)	semidirect
69. C₉²	C₃×ΓL(1,𝔽₈)	semidirect
77. C₇×C₉	C₃×3₋¹⁺²	semidirect
80. C₇⋊C₉	C₃×3₋¹⁺²	semidirect
103. ΓL(1,𝔽₈)	#6900	direct (N × H)
114. C₉	ΓL(1,𝔽₈)×3₋¹⁺²	semidirect
115. C₇	C₃×#6900	semidirect

H ≅ G/N

type

2. #13133597

C₃

semidirect

3. #13133596

C₃

semidirect

4. #13133595

C₃

semidirect

5. #13133594

C₃

semidirect

6. #13133593

C₃

semidirect

7. ΓL(1,𝔽₈)×C₉⋊C₉

C₃

semidirect

8. C₉²×ΓL(1,𝔽₈)

C₃

semidirect

9. C₇×#6900

C₃

semidirect

21. C₇×C₉⋊C₉

C₃²

semidirect

22. C₇×C₉²

C₃²

semidirect

23. #10588036

C₃²

semidirect

24. #10588035

C₃²

semidirect

25. (C₇⋊C₉)⋊C₉

C₃²

semidirect

26. C₉×C₇⋊C₉

C₃²

semidirect

33. #6900

ΓL(1,𝔽₈)

direct (N × H)

53. C₉×ΓL(1,𝔽₈)

3₋¹⁺²

semidirect

67. C₉⋊C₉

C₃×ΓL(1,𝔽₈)

semidirect

69. C₉²

C₃×ΓL(1,𝔽₈)

semidirect

77. C₇×C₉

C₃×3₋¹⁺²

semidirect

80. C₇⋊C₉

C₃×3₋¹⁺²

semidirect

103. ΓL(1,𝔽₈)

#6900

direct (N × H)

114. C₉

ΓL(1,𝔽₈)×3₋¹⁺²

semidirect

115. C₇

C₃×#6900

semidirect

Subgroups — ΓL(1,𝔽₈)×#6900 #19103241

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽8)×#6900 #19103241

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽₈)×#6900 #19103241