Subgroups — ΓL(1,𝔽₈)×C₃⋊(D₂₉⋊C₇) #38784651

Order 25578.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₂₉⋊C₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₃⋊(D₂₉⋊C₇)	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×D₂₉⋊C₇	—	3	1	no	maximal
4	C₃×C₇×C₂₉⋊C₇	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₂₉⋊C₇	—	2	1	no	—
6	ΓL(1,𝔽₈)×C₂₉⋊C₇	D₃	1	1	yes	characteristic, complemented
7	ΓL(1,𝔽₈)×D₈₇	C₇	1	1	yes	maximal, characteristic, complemented
8	C₃×C₃⋊(D₂₉⋊C₇)	—	7	1	no	maximal
9	C₇×D₂₉⋊C₇	—	3	1	no	Hall
10	C₃×ΓL(1,𝔽₈)×C₂₉	C₂×C₇	1	1	yes	characteristic, complemented
11	C₃²×C₂₉⋊C₇	—	7	1	no	—
12	C₇×C₂₉⋊C₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₇×D₈₇	C₃×C₇	1	1	yes	characteristic, complemented
14	ΓL(1,𝔽₈)×D₂₉	—	3	1	no	—
15	C₃⋊(D₂₉⋊C₇)	—	3	2	no	—
16	C₃⋊(D₂₉⋊C₇)	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃×D₂₉⋊C₇	—	21	1	no	—
18	D₃×C₇×ΓL(1,𝔽₈)	—	29	1	no	maximal, Hall
19	C₃×C₇×C₂₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₈)×C₂₉	—	2	1	no	—
21	ΓL(1,𝔽₈)×C₂₉	D₃×C₇	1	1	yes	characteristic, complemented
22	C₃×C₂₉⋊C₇	—	3	2	no	—
23	C₃×C₂₉⋊C₇	—	7	1	no	—
24	C₃×C₂₉⋊C₇	—	14	1	no	—
25	C₃×C₂₉⋊C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₃×D₈₇	—	7	1	no	Hall
27	C₃×C₇×ΓL(1,𝔽₈)	—	58	1	no	Hall
28	C₇×D₂₉	—	3	1	no	—
29	D₂₉⋊C₇	—	3	1	no	—
30	D₂₉⋊C₇	—	9	2	no	—
31	D₃×C₇²	—	29	1	no	—
32	C₂×C₇×ΓL(1,𝔽₈)	—	87	1	no	—
33	C₃²×C₂₉	—	7	1	no	Hall
34	C₇×C₂₉	C₃×D₃×C₇	1	1	yes	characteristic, complemented
35	C₂₉⋊C₇	—	3	2	no	—
36	C₂₉⋊C₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
37	D₈₇	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	C₃×D₂₉	—	21	1	no	—
39	C₃×C₇²	—	58	1	no	—
40	C₇×ΓL(1,𝔽₈)	—	58	1	no	—
41	C₇×ΓL(1,𝔽₈)	—	58	1	no	—
42	C₃×D₃×C₇	—	203	1	no	—
43	D₃×ΓL(1,𝔽₈)	—	29	1	no	—
44	C₂×C₇²	—	87	1	no	Hall
45	C₃×C₂₉	—	7	1	no	—
46	C₃×C₂₉	—	14	1	no	—
47	C₃×C₂₉	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
48	C₃²×C₇	—	406	1	no	—
49	C₃×ΓL(1,𝔽₈)	D₂₉⋊C₇	1	1	yes	characteristic, complemented
50	D₂₉	—	3	1	no	Hall
51	C₇²	—	58	1	no	Sylow
52	C₂×C₃×C₇	—	609	1	no	—
53	D₃×C₇	—	29	1	no	—
54	D₃×C₇	—	29	1	no	—
55	D₃×C₇	—	87	2	no	—
56	C₂×ΓL(1,𝔽₈)	—	87	1	no	—
57	C₂₉	D₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
58	C₃×C₇	—	58	1	no	—
59	C₃×C₇	—	406	1	no	—
60	C₃×C₇	—	406	1	no	—
61	C₃×C₇	—	174	2	no	—
62	C₃×C₇	C₃×D₂₉⋊C₇	1	1	yes	characteristic, complemented
63	ΓL(1,𝔽₈)	—	2	1	no	—
64	ΓL(1,𝔽₈)	C₃⋊(D₂₉⋊C₇)	1	1	yes	characteristic, direct product factor
65	C₃×D₃	—	203	1	no	Hall
66	C₂×C₇	—	261	2	no	—
67	C₂×C₇	—	87	1	no	—
68	C₂×C₇	—	87	1	no	—
69	C₃²	—	7	1	no	Sylow
70	C₇	—	58	1	no	—
71	C₇	—	174	2	no	—
72	C₇	C₃×C₃⋊(D₂₉⋊C₇)	1	1	yes	characteristic, complemented
73	C₂×C₃	—	609	1	no	—
74	D₃	—	29	1	no	—
75	C₃	—	7	1	no	—
76	C₃	—	14	1	no	—
77	C₃	ΓL(1,𝔽₈)×D₂₉⋊C₇	1	1	yes	characteristic, complemented
78	C₂	—	87	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₂₉	C₂×C₃×C₇
LCS	G ▷ C₃×C₇×C₂₉
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₂₉⋊C₇	C₂	semidirect
2. C₇×C₃⋊(D₂₉⋊C₇)	C₃	semidirect
4. C₃×C₇×C₂₉⋊C₇	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₂₉⋊C₇	D₃	semidirect
7. ΓL(1,𝔽₈)×D₈₇	C₇	semidirect
10. C₃×ΓL(1,𝔽₈)×C₂₉	C₂×C₇	semidirect
12. C₇×C₂₉⋊C₇	C₃×D₃	semidirect
13. C₇×D₈₇	C₃×C₇	semidirect
16. C₃⋊(D₂₉⋊C₇)	ΓL(1,𝔽₈)	direct (N × H)
19. C₃×C₇×C₂₉	C₂×C₃×C₇	semidirect
21. ΓL(1,𝔽₈)×C₂₉	D₃×C₇	semidirect
25. C₃×C₂₉⋊C₇	C₂×ΓL(1,𝔽₈)	semidirect
34. C₇×C₂₉	C₃×D₃×C₇	semidirect
36. C₂₉⋊C₇	D₃×ΓL(1,𝔽₈)	semidirect
37. D₈₇	C₇×ΓL(1,𝔽₈)	semidirect
47. C₃×C₂₉	C₂×C₇×ΓL(1,𝔽₈)	semidirect
49. C₃×ΓL(1,𝔽₈)	D₂₉⋊C₇	semidirect
57. C₂₉	D₃×C₇×ΓL(1,𝔽₈)	semidirect
62. C₃×C₇	C₃×D₂₉⋊C₇	semidirect
64. ΓL(1,𝔽₈)	C₃⋊(D₂₉⋊C₇)	direct (N × H)
72. C₇	C₃×C₃⋊(D₂₉⋊C₇)	semidirect
77. C₃	ΓL(1,𝔽₈)×D₂₉⋊C₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×ΓL(1,𝔽₈)×C₂₉⋊C₇	2
C₇×C₃⋊(D₂₉⋊C₇)	3
ΓL(1,𝔽₈)×D₂₉⋊C₇	3
ΓL(1,𝔽₈)×D₈₇	7
C₃×C₃⋊(D₂₉⋊C₇)	7
D₃×C₇×ΓL(1,𝔽₈)	29

Subgroups — ΓL(1,𝔽8)×C3⋊(D29⋊C7) #38784651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₃⋊(D₂₉⋊C₇) #38784651