Subgroups — #50303828

Order 38220.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊#16838450	C₂	1	1	yes	maximal, characteristic
2	#27164944	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439682	—	5	1	no	maximal, Hall
5	C₅⋊(C₇⋊D₉₁)	C₂×C₃	1	1	yes	characteristic
6	#16838450	—	5	1	no	—
7	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	#11758266⋊C₃	—	13	1	no	maximal, Hall
9	C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	—	14	4	no	—
10	#14579330	—	5	1	no	Hall
11	C₁₃×#3861	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₅⋊#64424	—	13	1	no	—
13	C₅×C₁₃×ΓL(1,𝔽₈)	—	14	4	no	—
14	C₇⋊D₉₁	—	5	1	no	—
15	#11758266	—	13	1	no	Hall
16	D₄₅₅	—	14	4	no	—
17	C₃×#63121	—	49	1	no	maximal, Hall
18	C₅×#3861	#261	1	1	yes	characteristic, complemented, Hall
19	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
20	#5997⋊C₃	—	65	1	no	Hall
21	ΓL(1,𝔽₈)⋊D₁₃	—	70	4	no	—
22	C₅⋊(C₇⋊D₇)	—	13	1	no	—
23	C₅×C₇×C₁₃	—	2	4	no	—
24	C₃×D₆₅	—	49	1	no	—
25	#64424	—	65	1	no	—
26	C₁₃×ΓL(1,𝔽₈)	—	14	4	no	—
27	#63121	—	49	1	no	Hall
28	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
29	C₅⋊AGL(1,𝔽₇)	—	182	4	no	—
30	#5997	—	65	1	no	Hall
31	C₃×C₅×C₁₃	—	49	1	no	Hall
32	D₉₁	—	70	4	no	—
33	C₃×#261	—	245	1	no	Hall
34	#3861	#63121	1	1	yes	characteristic, complemented, Hall
35	D₆₅	—	49	1	no	—
36	C₅×ΓL(1,𝔽₈)	—	14	4	no	—
37	C₇⋊D₇	—	65	1	no	—
38	C₇×C₁₃	—	2	4	no	—
39	C₃×D₁₃	—	245	1	no	—
40	D₃₅	—	182	4	no	—
41	C₅×C₁₃	#5997⋊C₃	1	1	yes	characteristic, complemented, Hall
42	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
43	#261	—	245	1	no	Hall
44	C₇²	C₃×#63121	1	1	yes	characteristic, complemented, Sylow
45	AGL(1,𝔽₇)	—	910	4	no	—
46	C₃×C₁₃	—	49	1	no	Hall
47	C₅×C₇	—	2	4	no	—
48	C₃×D₅	—	637	1	no	—
49	D₁₃	—	245	1	no	—
50	ΓL(1,𝔽₈)	—	14	4	no	—
51	AGL(1,𝔽₅)	—	637	1	no	Hall
52	C₃×C₅	—	49	1	no	Hall
53	D₇	—	910	4	no	—
54	C₁₃	#11758266⋊C₃	1	1	yes	characteristic, complemented, Sylow
55	C₃×C₄	—	3185	1	no	Hall
56	D₅	—	637	1	no	—
57	C₇	—	2	4	no	—
58	C₂×C₃	—	3185	1	no	—
59	C₅	#21439682	1	1	yes	characteristic, complemented, Sylow
60	C₄	—	3185	1	no	Sylow
61	C₃	—	49	1	no	Sylow
62	C₂	—	3185	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	C₅⋊#16838450	C₂
cosocle	C₅⋊(C₇⋊D₉₁)	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. #27164944	C₃	semidirect
3. C₅×C₁₃×#3861	C₄	semidirect
7. C₅×C₇²×C₁₃	C₃×C₄	semidirect
11. C₁₃×#3861	AGL(1,𝔽₅)	semidirect
18. C₅×#3861	#261	semidirect
19. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
28. C₅×C₇²	C₃×#261	semidirect
34. #3861	#63121	semidirect
41. C₅×C₁₃	#5997⋊C₃	semidirect
44. C₇²	C₃×#63121	semidirect
54. C₁₃	#11758266⋊C₃	semidirect
59. C₅	#21439682	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊#16838450	2
#27164944	3
#21439682	5
#11758266⋊C₃	13
C₃×#63121	49