Subgroups — #50303808

Order 38220.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊((C₁₃⋊C₃)⋊(C₇⋊D₇))	C₂	1	1	yes	maximal, characteristic
2	#27164943	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇²×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439686	—	5	1	no	maximal, Hall
5	C₅⋊(C₇⋊D₉₁)	C₂×C₃	1	1	yes	characteristic
6	(C₁₃⋊C₃)⋊(C₇⋊D₇)	—	5	1	no	—
7	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₃×#11758266	—	13	1	no	maximal, Hall
9	C₅⋊((C₁₃⋊C₃)⋊D₇)	—	14	4	no	—
10	#14579330	—	5	1	no	Hall
11	C₇²×C₁₃⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅⋊(C₇⋊D₇)	—	13	1	no	—
13	C₅×C₇×C₁₃⋊C₃	—	2	4	no	—
14	C₇⋊D₉₁	—	5	1	no	—
15	#11758266	—	13	1	no	Hall
16	D₄₅₅	—	14	4	no	—
17	#63122⋊C₃	—	49	1	no	maximal, Hall
18	C₃×C₅×C₇²	—	13	1	no	Hall
19	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
20	C₃×#5997	—	65	1	no	Hall
21	(C₁₃⋊C₃)⋊D₇	—	70	4	no	—
22	C₅⋊(C₇⋊D₇)	—	13	1	no	—
23	C₅×C₇×C₁₃	—	2	4	no	—
24	C₅⋊(D₁₃⋊C₃)	—	49	1	no	—
25	C₃×C₇⋊D₇	—	65	1	no	—
26	C₇×C₁₃⋊C₃	—	2	4	no	—
27	#63122	—	49	1	no	Hall
28	C₅×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic, complemented, Hall
29	C₃×D₃₅	—	182	4	no	—
30	#5997	—	65	1	no	Hall
31	C₅×C₁₃⋊C₃	#5997	1	1	yes	characteristic, complemented, Hall
32	D₉₁	—	70	4	no	—
33	AGL(1,𝔽₁₃)	—	245	1	no	Hall
34	C₃×C₇²	—	13	1	no	Hall
35	D₆₅	—	49	1	no	—
36	C₃×C₅×C₇	—	26	4	no	—
37	C₇⋊D₇	—	65	1	no	—
38	C₇×C₁₃	—	2	4	no	—
39	D₁₃⋊C₃	—	245	1	no	—
40	D₃₅	—	182	4	no	—
41	C₅×C₁₃	C₃×#5997	1	1	yes	characteristic, complemented, Hall
42	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
43	#261	—	245	1	no	Hall
44	C₇²	#63122⋊C₃	1	1	yes	characteristic, complemented, Sylow
45	C₃×D₇	—	910	4	no	—
46	C₁₃⋊C₃	#11758266	1	1	yes	characteristic, complemented, Hall
47	C₅×C₇	—	2	4	no	—
48	C₃×D₅	—	637	1	no	—
49	D₁₃	—	245	1	no	—
50	C₃×C₇	—	26	4	no	—
51	AGL(1,𝔽₅)	—	637	1	no	Hall
52	C₃×C₅	—	13	1	no	Hall
53	D₇	—	910	4	no	—
54	C₁₃	C₃×#11758266	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	#21439686	1	1	yes	characteristic, complemented, Sylow
60	C₄	—	3185	1	no	Sylow
61	C₃	—	13	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⋊((C₁₃⋊C₃)⋊(C₇⋊D₇))	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. #27164943	C₃	semidirect
3. C₅×C₇²×C₁₃⋊C₃	C₄	semidirect
7. C₅×C₇²×C₁₃	C₃×C₄	semidirect
11. C₇²×C₁₃⋊C₃	AGL(1,𝔽₅)	semidirect
19. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
28. C₅×C₇²	AGL(1,𝔽₁₃)	semidirect
31. C₅×C₁₃⋊C₃	#5997	semidirect
41. C₅×C₁₃	C₃×#5997	semidirect
44. C₇²	#63122⋊C₃	semidirect
46. C₁₃⋊C₃	#11758266	semidirect
54. C₁₃	C₃×#11758266	semidirect
59. C₅	#21439686	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊((C₁₃⋊C₃)⋊(C₇⋊D₇))	2
#27164943	3
#21439686	5
C₃×#11758266	13
#63122⋊C₃	49