Subgroups — ΓL(1,𝔽₈)×D₇₄₁ #44527993

Order 31122.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₂₄₇	—	3	1	no	maximal
3	C₇×D₇₄₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×D₇₄₁	—	7	1	no	maximal, Hall
8	C₇×D₂₄₇	—	3	1	no	Hall
9	ΓL(1,𝔽₈)×D₅₇	—	13	1	no	maximal, Hall
10	C₃²×C₁₃×C₁₉	—	7	1	no	Hall
11	C₇×C₁₃×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)×D₃₉	—	19	1	no	maximal, Hall
13	D₇₄₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×D₂₄₇	—	21	1	no	—
15	C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₃×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
17	C₇×D₅₇	—	13	1	no	—
18	ΓL(1,𝔽₈)×D₁₉	—	39	1	no	—
19	C₃×C₁₃×C₁₉	—	7	1	no	—
20	C₃×C₁₃×C₁₉	—	14	1	no	—
21	C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₇×D₃₉	—	19	1	no	—
23	ΓL(1,𝔽₈)×D₁₃	—	57	1	no	—
24	D₂₄₇	—	3	1	no	Hall
25	C₃×C₇×C₁₉	C₃×D₁₃	1	1	yes	characteristic, complemented
26	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
27	C₁₉×ΓL(1,𝔽₈)	D₃₉	1	1	yes	characteristic, complemented
28	C₃×D₅₇	—	91	1	no	Hall
29	C₃×C₇×C₁₃	C₃×D₁₉	1	1	yes	characteristic, complemented
30	C₁₃×ΓL(1,𝔽₈)	—	2	1	no	—
31	C₁₃×ΓL(1,𝔽₈)	D₅₇	1	1	yes	characteristic, complemented
32	C₇×D₁₉	—	39	1	no	Hall
33	C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
34	C₃×D₃₉	—	133	1	no	Hall
35	C₇×D₁₃	—	57	1	no	Hall
36	C₃²×C₁₉	—	7	1	no	Hall
37	C₇×C₁₉	C₃×D₃₉	1	1	yes	characteristic, complemented, Hall
38	D₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
39	C₃²×C₁₃	—	7	1	no	Hall
40	D₅₇	—	13	1	no	—
41	C₃×D₁₉	—	273	1	no	—
42	C₇×C₁₃	C₃×D₅₇	1	1	yes	characteristic, complemented, Hall
43	D₃₉	—	19	1	no	—
44	C₃×D₁₃	—	399	1	no	—
45	C₃×ΓL(1,𝔽₈)	D₂₄₇	1	1	yes	characteristic, complemented, Hall
46	C₃×C₁₉	—	7	1	no	—
47	C₃×C₁₉	—	14	1	no	—
48	C₃×C₁₉	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
49	D₃×C₇	—	247	1	no	—
50	C₂×ΓL(1,𝔽₈)	—	741	1	no	—
51	C₃×C₁₃	—	7	1	no	—
52	C₃×C₁₃	—	14	1	no	—
53	C₃×C₁₃	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented
54	D₁₉	—	39	1	no	Hall
55	D₁₃	—	57	1	no	Hall
56	C₃×C₇	C₃×D₂₄₇	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	2	1	no	—
58	ΓL(1,𝔽₈)	D₇₄₁	1	1	yes	characteristic, direct product factor
59	C₁₉	ΓL(1,𝔽₈)×D₃₉	1	1	yes	characteristic, complemented, Sylow
60	C₃×D₃	—	1729	1	no	Hall
61	C₂×C₇	—	741	1	no	Hall
62	C₁₃	ΓL(1,𝔽₈)×D₅₇	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	7	1	no	Sylow
64	C₇	C₃×D₇₄₁	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	5187	1	no	—
66	D₃	—	247	1	no	—
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	ΓL(1,𝔽₈)×D₂₄₇	1	1	yes	characteristic, complemented
70	C₂	—	741	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₃×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₃×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×D₇₄₁	C₃	semidirect
5. C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
11. C₇×C₁₃×C₁₉	C₃×D₃	semidirect
13. D₇₄₁	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₉×ΓL(1,𝔽₈)	D₁₃	semidirect
16. C₃×C₁₃×ΓL(1,𝔽₈)	D₁₉	semidirect
21. C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
25. C₃×C₇×C₁₉	C₃×D₁₃	semidirect
27. C₁₉×ΓL(1,𝔽₈)	D₃₉	semidirect
29. C₃×C₇×C₁₃	C₃×D₁₉	semidirect
31. C₁₃×ΓL(1,𝔽₈)	D₅₇	semidirect
33. C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	semidirect
37. C₇×C₁₉	C₃×D₃₉	semidirect
42. C₇×C₁₃	C₃×D₅₇	semidirect
45. C₃×ΓL(1,𝔽₈)	D₂₄₇	semidirect
48. C₃×C₁₉	ΓL(1,𝔽₈)×D₁₃	semidirect
53. C₃×C₁₃	ΓL(1,𝔽₈)×D₁₉	semidirect
56. C₃×C₇	C₃×D₂₄₇	semidirect
58. ΓL(1,𝔽₈)	D₇₄₁	direct (N × H)
59. C₁₉	ΓL(1,𝔽₈)×D₃₉	semidirect
62. C₁₃	ΓL(1,𝔽₈)×D₅₇	semidirect
64. C₇	C₃×D₇₄₁	semidirect
69. C₃	ΓL(1,𝔽₈)×D₂₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×D₂₄₇	3
C₇×D₇₄₁	3
C₃×D₇₄₁	7
ΓL(1,𝔽₈)×D₅₇	13
ΓL(1,𝔽₈)×D₃₉	19

Subgroups — ΓL(1,𝔽8)×D741 #44527993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₇₄₁ #44527993