Subgroups — #3861×#63121 #50303896

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₅×#3861	C₂	1	1	yes	maximal, characteristic
2	C₇²×#63121	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	#261×#3861	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)×#63121	—	7	8	no	maximal
7	D₁₃×#3861	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	AGL(1,𝔽₅)×#3861	—	13	1	no	maximal, Hall
10	ΓL(1,𝔽₈)×D₆₅	—	7	8	no	—
11	C₇²×#261	—	5	1	no	Hall
12	C₁₃×#3861	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇×#63121	ΓL(1,𝔽₈)	1	8	yes	complemented
14	D₅×#3861	—	13	1	no	—
15	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	8	no	—
16	C₇²×D₁₃	—	5	1	no	—
17	ΓL(1,𝔽₈)×#261	—	35	8	no	—
18	C₇²×AGL(1,𝔽₅)	—	13	1	no	Hall
19	C₇×D₆₅	C₂×ΓL(1,𝔽₈)	1	8	yes	—
20	C₃×#63121	—	49	1	no	Hall
21	C₅×#3861	#261	1	1	yes	characteristic, complemented, Hall
22	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
23	C₄×#3861	—	65	1	no	Hall
24	ΓL(1,𝔽₈)×D₁₃	—	35	8	no	—
25	C₇²×D₅	—	13	1	no	—
26	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
27	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	91	8	no	—
28	C₃×D₆₅	—	49	1	no	—
29	C₇×#261	—	5	8	no	—
30	C₂×#3861	—	65	1	no	—
31	C₁₃×ΓL(1,𝔽₈)	—	7	8	no	—
32	#63121	#3861	1	1	yes	characteristic, direct product factor, Hall
33	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
34	D₅×ΓL(1,𝔽₈)	—	91	8	no	—
35	C₄×C₇²	—	65	1	no	Hall
36	C₃×C₅×C₁₃	—	49	1	no	Hall
37	C₇×D₁₃	—	5	8	no	—
38	C₃×#261	—	245	1	no	Hall
39	#3861	#63121	1	1	yes	characteristic, direct product factor, Hall
40	C₇×AGL(1,𝔽₅)	—	13	8	no	—
41	D₆₅	C₂×#3861	1	1	yes	characteristic
42	C₅×ΓL(1,𝔽₈)	—	7	8	no	—
43	C₂×C₇²	—	65	1	no	—
44	C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	8	yes	complemented
45	C₄×ΓL(1,𝔽₈)	—	455	8	no	—
46	C₃×D₁₃	—	245	1	no	—
47	C₇×D₅	—	13	8	no	—
48	C₅×C₁₃	C₄×#3861	1	1	yes	characteristic, complemented, Hall
49	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
50	#261	—	5	1	no	Hall
51	C₇²	C₃×#63121	1	1	yes	characteristic, complemented, Sylow
52	C₂×ΓL(1,𝔽₈)	—	455	8	no	—
53	C₃×C₁₃	—	49	1	no	Hall
54	C₅×C₇	ΓL(1,𝔽₈)×#261	1	8	yes	complemented
55	C₃×D₅	—	637	1	no	—
56	C₄×C₇	—	65	8	no	—
57	D₁₃	—	5	1	no	—
58	ΓL(1,𝔽₈)	—	7	8	no	—
59	AGL(1,𝔽₅)	—	13	1	no	Hall
60	C₃×C₅	—	49	1	no	Hall
61	C₂×C₇	—	65	8	no	—
62	C₁₃	AGL(1,𝔽₅)×#3861	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	3185	1	no	Hall
64	D₅	—	13	1	no	—
65	C₇	ΓL(1,𝔽₈)×#63121	1	8	yes	complemented
66	C₂×C₃	—	3185	1	no	—
67	C₅	#261×#3861	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	65	1	no	Sylow
69	C₃	—	49	1	no	Sylow
70	C₂	—	65	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	D₆₅×#3861	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×#63121	C₃	semidirect
3. C₅×C₁₃×#3861	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₁₃×#3861	AGL(1,𝔽₅)	semidirect
13. C₇×#63121	ΓL(1,𝔽₈)	semidirect
21. C₅×#3861	#261	semidirect
22. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
26. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
32. #63121	#3861	direct (N × H)
33. C₅×C₇²	C₃×#261	semidirect
39. #3861	#63121	direct (N × H)
44. C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
48. C₅×C₁₃	C₄×#3861	semidirect
51. C₇²	C₃×#63121	semidirect
54. C₅×C₇	ΓL(1,𝔽₈)×#261	semidirect
62. C₁₃	AGL(1,𝔽₅)×#3861	semidirect
65. C₇	ΓL(1,𝔽₈)×#63121	semidirect
67. C₅	#261×#3861	semidirect

Maximal subgroups

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

subgroup	index
D₆₅×#3861	2
C₇²×#63121	3
#261×#3861	5
ΓL(1,𝔽₈)×#63121	7
AGL(1,𝔽₅)×#3861	13