Subgroups — #44477649

Order 30420.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊#15690150	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₃×#63122	C₃	1	1	yes	maximal, characteristic
3	C₅×#13095200	C₄	1	1	yes	characteristic, complemented, Hall
4	#20258776	—	5	1	no	maximal, Hall
5	C₃×C₁₃×D₆₅	C₂×C₃	1	1	yes	characteristic
6	C₁₃×#63122	C₉	1	1	yes	characteristic, complemented, Hall
7	#15690150	—	5	1	no	—
8	C₃×C₅×C₁₃²	C₃×C₄	1	1	yes	characteristic
9	#63122⋊C₉	—	13	1	no	maximal
10	AGL(1,𝔽₅)×C₁₃⋊C₉	—	13	1	no	maximal
11	C₃×C₁₃×#261	—	5	1	no	—
12	C₁₃×D₆₅	C₂×C₉	1	1	yes	characteristic
13	#13095200	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
14	C₅⋊(D₁₃⋊C₉)	—	13	1	no	—
15	D₅×C₁₃⋊C₉	—	13	1	no	—
16	C₃×C₁₃×D₁₃	—	5	1	no	—
17	C₅×C₁₃²	C₄×C₉	1	1	yes	characteristic, complemented, Hall
18	C₃×#63122	C₁₃⋊C₃	1	1	yes	characteristic
19	C₃×C₁₃×AGL(1,𝔽₅)	—	39	1	no	—
20	C₁₃×#261	—	5	1	no	Hall
21	C₅×C₁₃⋊C₉	—	13	1	no	—
22	C₅×C₁₃⋊C₉	—	13	1	no	—
23	C₅×C₁₃⋊C₉	—	52	3	no	—
24	C₃×C₁₃²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
25	#261⋊C₉	—	65	1	no	—
26	C₄×C₁₃⋊C₉	—	65	1	no	—
27	C₃×D₆₅	C₂×C₁₃⋊C₃	1	1	yes	characteristic
28	C₃×D₅×C₁₃	—	39	1	no	—
29	C₁₃×D₁₃	—	5	1	no	—
30	#63122	C₁₃⋊C₉	1	1	yes	characteristic, complemented
31	C₁₃×AGL(1,𝔽₅)	—	39	1	no	—
32	C₂×C₁₃⋊C₉	—	65	1	no	—
33	D₁₃⋊C₉	—	65	1	no	—
34	C₃×C₅×C₁₃	—	4	3	no	—
35	C₃×C₅×C₁₃	C₄×C₁₃⋊C₃	1	1	yes	characteristic
36	C₃×C₅×C₁₃	AGL(1,𝔽₁₃)	1	1	yes	characteristic
37	C₉×AGL(1,𝔽₅)	—	169	1	no	Hall
38	C₁₃²	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
39	C₃×#261	—	5	1	no	—
40	C₃×C₄×C₁₃	—	195	1	no	—
41	D₆₅	C₂×C₁₃⋊C₉	1	1	yes	characteristic
42	D₅×C₁₃	—	39	1	no	—
43	C₁₃⋊C₉	—	13	1	no	—
44	C₁₃⋊C₉	—	13	1	no	—
45	C₁₃⋊C₉	—	52	3	no	—
46	C₉×D₅	—	169	1	no	—
47	C₂×C₃×C₁₃	—	195	1	no	—
48	C₃×D₁₃	—	5	1	no	—
49	C₅×C₁₃	—	4	3	no	—
50	C₅×C₁₃	C₄×C₁₃⋊C₉	1	1	yes	characteristic, complemented
51	C₅×C₁₃	#261⋊C₉	1	1	yes	characteristic, complemented
52	C₃×AGL(1,𝔽₅)	—	39	1	no	—
53	#261	—	5	1	no	—
54	C₄×C₁₃	—	195	1	no	—
55	C₅×C₉	—	169	1	no	Hall
56	C₃×C₁₃	—	4	3	no	—
57	C₃×C₁₃	AGL(1,𝔽₅)×C₁₃⋊C₃	1	1	yes	characteristic
58	C₃×C₁₃	#63122⋊C₃	1	1	yes	characteristic
59	C₄×C₉	—	845	1	no	Hall
60	C₃×D₅	—	39	1	no	—
61	C₂×C₁₃	—	195	1	no	—
62	D₁₃	—	5	1	no	—
63	AGL(1,𝔽₅)	—	39	1	no	Hall
64	C₂×C₉	—	845	1	no	—
65	C₃×C₅	#14523152	1	1	yes	characteristic
66	C₁₃	—	4	3	no	—
67	C₁₃	AGL(1,𝔽₅)×C₁₃⋊C₉	1	1	yes	characteristic, complemented
68	C₁₃	#63122⋊C₉	1	1	yes	characteristic, complemented
69	C₃×C₄	—	195	1	no	—
70	D₅	—	39	1	no	—
71	C₉	—	169	1	no	Sylow
72	C₂×C₃	—	195	1	no	—
73	C₅	#20258776	1	1	yes	characteristic, complemented, Sylow
74	C₄	—	195	1	no	Sylow
75	C₃	#23778467	1	1	yes	characteristic
76	C₂	—	195	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#23778467
commutator, hypocenter	C₅×C₁₃²	C₄×C₉
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical, cosocle	C₃×C₁₃×D₆₅	C₂×C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. C₅×#13095200	C₄	semidirect
6. C₁₃×#63122	C₉	semidirect
13. #13095200	AGL(1,𝔽₅)	semidirect
17. C₅×C₁₃²	C₄×C₉	semidirect
30. #63122	C₁₃⋊C₉	semidirect
38. C₁₃²	C₉×AGL(1,𝔽₅)	semidirect
50. C₅×C₁₃	C₄×C₁₃⋊C₉	semidirect
51. C₅×C₁₃	#261⋊C₉	semidirect
67. C₁₃	AGL(1,𝔽₅)×C₁₃⋊C₉	semidirect
68. C₁₃	#63122⋊C₉	semidirect
73. C₅	#20258776	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊#15690150	2
C₃×C₁₃×#63122	3
#20258776	5
#63122⋊C₉	13
AGL(1,𝔽₅)×C₁₃⋊C₉	13