Subgroups — ΓL(1,𝔽₈)×D₁₃×D₂₉ #44539869

Order 31668.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₁₃×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×ΓL(1,𝔽₈)×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₁₃×D₂₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₃₇₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₁₃×C₂₉	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₃×D₂₉	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₁₃×D₂₉	—	7	1	no	maximal, Hall
10	C₇×C₁₃×C₂₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×ΓL(1,𝔽₈)×D₂₉	—	13	1	no	maximal, Hall
12	C₃×D₃₇₇	—	7	1	no	—
13	C₃×D₁₃×C₂₉	—	7	1	no	—
14	C₃×C₁₃×D₂₉	—	7	1	no	—
15	D₁₃×D₂₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	C₂×ΓL(1,𝔽₈)×C₂₉	—	13	1	no	—
17	ΓL(1,𝔽₈)×D₂₉	—	13	1	no	—
18	ΓL(1,𝔽₈)×D₂₉	D₁₃	1	1	yes	characteristic, direct product factor
19	C₃×C₁₃×C₂₉	—	7	1	no	Hall
20	C₂×ΓL(1,𝔽₈)×D₁₃	—	29	1	no	maximal, Hall
21	C₂×C₇×D₂₉	—	13	1	no	Hall
22	D₃₇₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₁₃×D₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	D₁₃×C₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	ΓL(1,𝔽₈)×C₂₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₁₃×ΓL(1,𝔽₈)	—	29	1	no	—
27	ΓL(1,𝔽₈)×D₁₃	—	29	1	no	—
28	ΓL(1,𝔽₈)×D₁₃	D₂₉	1	1	yes	characteristic, direct product factor
29	C₂×C₇×C₂₉	—	13	1	no	—
30	C₇×D₂₉	—	13	1	no	—
31	C₇×D₂₉	C₃×D₁₃	1	1	yes	characteristic, complemented
32	C₁₃×C₂₉	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
33	C₂×C₇×D₁₃	—	29	1	no	Hall
34	C₂×C₃×D₂₉	—	91	1	no	Hall
35	C₁₃×ΓL(1,𝔽₈)	C₂×D₂₉	1	1	yes	characteristic, complemented, Hall
36	C₇×C₂₉	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₇×C₁₃	—	29	1	no	—
38	C₇×D₁₃	—	29	1	no	—
39	C₇×D₁₃	C₃×D₂₉	1	1	yes	characteristic, complemented
40	C₂×C₃×C₂₉	—	91	1	no	—
41	C₃×D₂₉	—	7	1	no	—
42	C₃×D₂₉	—	91	1	no	—
43	C₂×C₃×D₁₃	—	203	1	no	Hall
44	C₂×D₂₉	—	13	1	no	Hall
45	C₇×C₁₃	C₂×C₃×D₂₉	1	1	yes	characteristic, complemented, Hall
46	C₃×C₂₉	—	7	1	no	Hall
47	C₂²×ΓL(1,𝔽₈)	—	377	1	no	Hall
48	C₂×C₃×C₁₃	—	203	1	no	—
49	C₃×D₁₃	—	7	1	no	—
50	C₃×D₁₃	—	203	1	no	—
51	C₂×C₂₉	—	13	1	no	—
52	D₂₉	—	13	1	no	—
53	D₂₉	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, direct product factor
54	C₂×D₁₃	—	29	1	no	Hall
55	C₂×ΓL(1,𝔽₈)	—	13	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	29	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	377	1	no	—
58	C₃×C₁₃	—	7	1	no	Hall
59	C₂₉	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₇	—	377	1	no	Hall
61	C₂×C₁₃	—	29	1	no	—
62	D₁₃	—	29	1	no	—
63	D₁₃	ΓL(1,𝔽₈)×D₂₉	1	1	yes	characteristic, direct product factor
64	ΓL(1,𝔽₈)	D₁₃×D₂₉	1	1	yes	characteristic, direct product factor, Hall
65	C₂×C₇	—	13	1	no	—
66	C₂×C₇	—	29	1	no	—
67	C₂×C₇	—	377	1	no	—
68	C₁₃	C₂×ΓL(1,𝔽₈)×D₂₉	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₃	—	2639	1	no	Hall
70	C₇	C₃×D₁₃×D₂₉	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	2639	1	no	—
72	C₂×C₃	—	91	1	no	—
73	C₂×C₃	—	203	1	no	—
74	C₂²	—	377	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	13	1	no	—
77	C₂	—	29	1	no	—
78	C₂	—	377	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₂₉	C₂²×C₃
LCS	G ▷ C₇×C₁₃×C₂₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₃₇₇	C₂	semidirect
2. ΓL(1,𝔽₈)×D₁₃×C₂₉	C₂	semidirect
3. C₁₃×ΓL(1,𝔽₈)×D₂₉	C₂	semidirect
4. C₇×D₁₃×D₂₉	C₃	semidirect
5. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂²	semidirect
6. C₇×D₃₇₇	C₂×C₃	semidirect
7. C₇×D₁₃×C₂₉	C₂×C₃	semidirect
8. C₇×C₁₃×D₂₉	C₂×C₃	semidirect
10. C₇×C₁₃×C₂₉	C₂²×C₃	semidirect
15. D₁₃×D₂₉	ΓL(1,𝔽₈)	direct (N × H)
18. ΓL(1,𝔽₈)×D₂₉	D₁₃	direct (N × H)
22. D₃₇₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₁₃×D₂₉	C₂×ΓL(1,𝔽₈)	semidirect
24. D₁₃×C₂₉	C₂×ΓL(1,𝔽₈)	semidirect
25. ΓL(1,𝔽₈)×C₂₉	C₂×D₁₃	semidirect
28. ΓL(1,𝔽₈)×D₁₃	D₂₉	direct (N × H)
31. C₇×D₂₉	C₃×D₁₃	semidirect
32. C₁₃×C₂₉	C₂²×ΓL(1,𝔽₈)	semidirect
35. C₁₃×ΓL(1,𝔽₈)	C₂×D₂₉	semidirect
36. C₇×C₂₉	C₂×C₃×D₁₃	semidirect
39. C₇×D₁₃	C₃×D₂₉	semidirect
45. C₇×C₁₃	C₂×C₃×D₂₉	semidirect
53. D₂₉	ΓL(1,𝔽₈)×D₁₃	direct (N × H)
59. C₂₉	C₂×ΓL(1,𝔽₈)×D₁₃	semidirect
63. D₁₃	ΓL(1,𝔽₈)×D₂₉	direct (N × H)
64. ΓL(1,𝔽₈)	D₁₃×D₂₉	direct (N × H)
68. C₁₃	C₂×ΓL(1,𝔽₈)×D₂₉	semidirect
70. C₇	C₃×D₁₃×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₃₇₇	2
ΓL(1,𝔽₈)×D₁₃×C₂₉	2
C₁₃×ΓL(1,𝔽₈)×D₂₉	2
C₇×D₁₃×D₂₉	3
C₃×D₁₃×D₂₉	7
C₂×ΓL(1,𝔽₈)×D₂₉	13
C₂×ΓL(1,𝔽₈)×D₁₃	29

Subgroups — ΓL(1,𝔽8)×D13×D29 #44539869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₃×D₂₉ #44539869