Subgroups — #3862×#63122 #50303940

Order 38220.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₅×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇²×#63122	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	#261×#3862	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)×#63122	—	7	2	no	maximal
7	D₁₃×#3862	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	AGL(1,𝔽₅)×#3862	—	13	1	no	maximal, Hall
10	ΓL(1,𝔽₈)×D₆₅	—	7	2	no	—
11	C₇²×#261	—	5	1	no	Hall
12	C₁₃×#3862	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇×#63122	—	3	2	no	—
14	C₇×#63122	ΓL(1,𝔽₈)	1	2	yes	complemented
15	D₅×#3862	—	13	1	no	—
16	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
17	C₇²×D₁₃	—	5	1	no	—
18	ΓL(1,𝔽₈)×#261	—	35	2	no	—
19	C₇²×AGL(1,𝔽₅)	—	13	1	no	Hall
20	C₇×D₆₅	—	3	2	no	—
21	C₇×D₆₅	C₂×ΓL(1,𝔽₈)	1	2	yes	—
22	C₃×#63122	—	49	1	no	Hall
23	C₅×#3862	#261	1	1	yes	characteristic, complemented, Hall
24	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
25	C₄×#3862	—	65	1	no	Hall
26	ΓL(1,𝔽₈)×D₁₃	—	35	2	no	—
27	C₇²×D₅	—	13	1	no	—
28	C₅×C₇×C₁₃	—	3	2	no	—
29	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
30	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	91	2	no	—
31	C₃×D₆₅	—	49	1	no	—
32	C₇×#261	—	5	2	no	—
33	C₇×#261	—	15	2	no	—
34	C₂×#3862	—	65	1	no	—
35	C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
36	#63122	#3862	1	1	yes	characteristic, direct product factor, Hall
37	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
38	D₅×ΓL(1,𝔽₈)	—	91	2	no	—
39	C₄×C₇²	—	65	1	no	Hall
40	C₃×C₅×C₁₃	—	49	1	no	Hall
41	C₇×D₁₃	—	5	2	no	—
42	C₇×D₁₃	—	15	2	no	—
43	C₃×#261	—	245	1	no	Hall
44	#3862	#63122	1	1	yes	characteristic, direct product factor, Hall
45	C₇×AGL(1,𝔽₅)	—	13	2	no	—
46	C₇×AGL(1,𝔽₅)	—	39	2	no	—
47	D₆₅	C₂×#3862	1	1	yes	characteristic
48	C₅×ΓL(1,𝔽₈)	—	7	2	no	—
49	C₂×C₇²	—	65	1	no	—
50	C₇×C₁₃	—	3	2	no	—
51	C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	2	yes	complemented
52	C₄×ΓL(1,𝔽₈)	—	455	2	no	—
53	C₃×D₁₃	—	245	1	no	—
54	C₇×D₅	—	13	2	no	—
55	C₇×D₅	—	39	2	no	—
56	C₅×C₁₃	C₄×#3862	1	1	yes	characteristic, complemented, Hall
57	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
58	#261	—	5	1	no	Hall
59	C₇²	C₃×#63122	1	1	yes	characteristic, complemented, Sylow
60	C₂×ΓL(1,𝔽₈)	—	455	2	no	—
61	C₃×C₁₃	—	49	1	no	Hall
62	C₅×C₇	—	3	2	no	—
63	C₅×C₇	ΓL(1,𝔽₈)×#261	1	2	yes	complemented
64	C₃×D₅	—	637	1	no	—
65	C₄×C₇	—	65	2	no	—
66	C₄×C₇	—	195	2	no	—
67	D₁₃	—	5	1	no	—
68	ΓL(1,𝔽₈)	—	7	2	no	—
69	AGL(1,𝔽₅)	—	13	1	no	Hall
70	C₃×C₅	—	49	1	no	Hall
71	C₂×C₇	—	65	2	no	—
72	C₂×C₇	—	195	2	no	—
73	C₁₃	AGL(1,𝔽₅)×#3862	1	1	yes	characteristic, complemented, Sylow
74	C₃×C₄	—	3185	1	no	Hall
75	D₅	—	13	1	no	—
76	C₇	—	3	2	no	—
77	C₇	ΓL(1,𝔽₈)×#63122	1	2	yes	complemented
78	C₂×C₃	—	3185	1	no	—
79	C₅	#261×#3862	1	1	yes	characteristic, complemented, Sylow
80	C₄	—	65	1	no	Sylow
81	C₃	—	49	1	no	Sylow
82	C₂	—	65	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	D₆₅×#3862	C₂
cosocle	C₇²×D₆₅	C₂×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇²×#63122	C₃	semidirect
3. C₅×C₁₃×#3862	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₁₃×#3862	AGL(1,𝔽₅)	semidirect
14. C₇×#63122	ΓL(1,𝔽₈)	semidirect
23. C₅×#3862	#261	semidirect
24. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
29. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
36. #63122	#3862	direct (N × H)
37. C₅×C₇²	C₃×#261	semidirect
44. #3862	#63122	direct (N × H)
51. C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
56. C₅×C₁₃	C₄×#3862	semidirect
59. C₇²	C₃×#63122	semidirect
63. C₅×C₇	ΓL(1,𝔽₈)×#261	semidirect
73. C₁₃	AGL(1,𝔽₅)×#3862	semidirect
77. C₇	ΓL(1,𝔽₈)×#63122	semidirect
79. C₅	#261×#3862	semidirect

Maximal subgroups

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

subgroup	index
D₆₅×#3862	2
C₇²×#63122	3
#261×#3862	5
ΓL(1,𝔽₈)×#63122	7
AGL(1,𝔽₅)×#3862	13