Subgroups — ΓL(1,𝔽₈)×C₂₉×A₅ #49141698

Order 36540.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₉×A₅	C₃	1	1	yes	maximal, characteristic, complemented
2	A₄×ΓL(1,𝔽₈)×C₂₉	—	5	1	no	maximal, Hall
3	D₅×ΓL(1,𝔽₈)×C₂₉	—	6	1	no	maximal
4	C₃×C₂₉×A₅	—	7	1	no	maximal, Hall
5	D₃×ΓL(1,𝔽₈)×C₂₉	—	10	1	no	maximal
6	C₅×ΓL(1,𝔽₈)×C₂₉	—	6	1	no	—
7	C₂₉×C₂²⋊ΓL(1,𝔽₈)	—	5	2	no	—
8	C₇×A₄×C₂₉	—	5	1	no	—
9	C₂²×ΓL(1,𝔽₈)×C₂₉	—	5	1	no	—
10	C₇×D₅×C₂₉	—	6	1	no	—
11	C₃×ΓL(1,𝔽₈)×C₂₉	—	10	1	no	Hall
12	C₂₉×A₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
13	ΓL(1,𝔽₈)×A₅	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	D₃×C₇×C₂₉	—	10	1	no	—
15	C₂×ΓL(1,𝔽₈)×C₂₉	—	15	1	no	—
16	C₃×A₄×C₂₉	—	35	1	no	Hall
17	C₅×C₇×C₂₉	—	6	1	no	Hall
18	C₃×D₅×C₂₉	—	42	1	no	—
19	C₂²×C₇×C₂₉	—	5	1	no	Hall
20	C₃×C₇×C₂₉	—	10	1	no	—
21	ΓL(1,𝔽₈)×C₂₉	—	20	1	no	—
22	ΓL(1,𝔽₈)×C₂₉	A₅	1	1	yes	maximal normal, characteristic, direct product factor
23	C₃×D₃×C₂₉	—	70	1	no	—
24	C₃×C₅×C₂₉	—	42	1	no	—
25	C₇×A₅	C₃×C₂₉	1	1	yes	characteristic, complemented
26	C₂×C₇×C₂₉	—	15	1	no	—
27	C₂²×C₃×C₂₉	—	35	1	no	—
28	A₄×C₂₉	—	5	1	no	—
29	A₄×C₂₉	—	35	2	no	—
30	D₅×C₂₉	—	6	1	no	—
31	C₃²×C₂₉	—	70	1	no	Hall
32	A₄×ΓL(1,𝔽₈)	—	5	1	no	Hall
33	D₅×ΓL(1,𝔽₈)	—	6	1	no	—
34	C₇×C₂₉	C₃×A₅	1	1	yes	characteristic, complemented, Hall
35	C₃×A₅	—	7	1	no	Hall
36	C₂×C₃×C₂₉	—	105	1	no	—
37	D₃×C₂₉	—	10	1	no	—
38	C₅×C₂₉	—	6	1	no	Hall
39	D₃×ΓL(1,𝔽₈)	—	10	1	no	—
40	C₂²×C₂₉	—	5	1	no	Hall
41	C₅×ΓL(1,𝔽₈)	—	6	1	no	—
42	C₃×C₂₉	—	7	1	no	—
43	C₃×C₂₉	—	10	1	no	—
44	C₃×C₂₉	—	140	1	no	—
45	C₂²⋊ΓL(1,𝔽₈)	—	5	2	no	—
46	C₇×A₄	—	5	1	no	—
47	C₂²×ΓL(1,𝔽₈)	—	5	1	no	—
48	C₇×D₅	—	6	1	no	—
49	C₃×ΓL(1,𝔽₈)	—	10	1	no	Hall
50	A₅	ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, direct product factor
51	C₂×C₂₉	—	15	1	no	—
52	D₃×C₇	—	10	1	no	—
53	C₂×ΓL(1,𝔽₈)	—	15	1	no	—
54	C₃×A₄	—	35	1	no	Hall
55	C₅×C₇	—	6	1	no	Hall
56	C₃×D₅	—	42	1	no	—
57	C₂₉	ΓL(1,𝔽₈)×A₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₇	—	5	1	no	Hall
59	C₃×C₇	—	10	1	no	—
60	ΓL(1,𝔽₈)	—	20	1	no	—
61	ΓL(1,𝔽₈)	C₂₉×A₅	1	1	yes	characteristic, direct product factor
62	C₃×D₃	—	70	1	no	—
63	C₃×C₅	—	42	1	no	—
64	C₂×C₇	—	15	1	no	—
65	C₂²×C₃	—	35	1	no	—
66	A₄	—	5	1	no	—
67	A₄	—	35	2	no	—
68	D₅	—	6	1	no	—
69	C₃²	—	70	1	no	Sylow
70	C₇	C₃×C₂₉×A₅	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	105	1	no	—
72	D₃	—	10	1	no	—
73	C₅	—	6	1	no	Sylow
74	C₂²	—	5	1	no	Sylow
75	C₃	—	7	1	no	—
76	C₃	—	10	1	no	—
77	C₃	—	140	1	no	—
78	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₂₉	ΓL(1,𝔽₈)×A₅
commutator, hypocenter	C₇×A₅	C₃×C₂₉
Frattini	C₁	G
Fitting	C₇×C₂₉	C₃×A₅
socle	C₇×C₂₉×A₅	C₃
cosocle	C₇	C₃×C₂₉×A₅
LCS	G ▷ C₇×A₅
UCS	C₁ ◁ C₂₉

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₉×A₅	C₃	semidirect
12. C₂₉×A₅	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×A₅	C₂₉	direct (N × H)
22. ΓL(1,𝔽₈)×C₂₉	A₅	direct (N × H)
25. C₇×A₅	C₃×C₂₉	semidirect
34. C₇×C₂₉	C₃×A₅	semidirect
50. A₅	ΓL(1,𝔽₈)×C₂₉	direct (N × H)
57. C₂₉	ΓL(1,𝔽₈)×A₅	direct (N × H)
61. ΓL(1,𝔽₈)	C₂₉×A₅	direct (N × H)
70. C₇	C₃×C₂₉×A₅	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₂₉×A₅	3
A₄×ΓL(1,𝔽₈)×C₂₉	5
D₅×ΓL(1,𝔽₈)×C₂₉	6
C₃×C₂₉×A₅	7
D₃×ΓL(1,𝔽₈)×C₂₉	10
ΓL(1,𝔽₈)×A₅	29

Subgroups — ΓL(1,𝔽8)×C29×A5 #49141698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₉×A₅ #49141698