Subgroups — #44477639

Order 30420.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊#15690148	C₂	1	1	yes	maximal, characteristic
2	C₃×#16810004	C₃	1	1	yes	maximal, characteristic
3	C₅×#13095200	C₄	1	1	yes	characteristic, complemented, Hall
4	#20258770	—	5	1	no	maximal, Hall
5	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃	1	1	yes	characteristic
6	#16810004	C₉	1	1	yes	characteristic, complemented, Hall
7	#15690148	—	5	1	no	—
8	C₃×C₅×C₁₃²	C₃×C₄	1	1	yes	characteristic
9	#63121⋊C₉	—	13	1	no	maximal
10	#63122⋊C₉	—	13	1	no	maximal
11	C₃×#10621729	—	5	1	no	—
12	C₅⋊(C₁₃⋊D₁₃)	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	C₅⋊(D₁₃⋊C₉)	—	13	1	no	—
16	C₅⋊(D₁₃⋊C₉)	—	26	6	no	—
17	C₃×C₁₃⋊D₁₃	—	5	1	no	—
18	C₅×C₁₃²	C₄×C₉	1	1	yes	characteristic, complemented, Hall
19	C₃×#63121	—	39	1	no	—
20	C₃×#63122	—	39	1	no	—
21	#10621729	—	5	1	no	Hall
22	C₅×C₁₃⋊C₉	—	13	1	no	—
23	C₅×C₁₃⋊C₉	—	13	1	no	—
24	C₅×C₁₃⋊C₉	—	26	6	no	—
25	C₃×C₁₃²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
26	#261⋊C₉	—	65	1	no	—
27	#261⋊C₉	—	65	1	no	—
28	C₃×D₆₅	—	39	1	no	—
29	C₃×D₆₅	—	39	1	no	—
30	C₃×D₆₅	—	78	6	no	—
31	C₁₃⋊D₁₃	—	5	1	no	—
32	#63122	—	39	1	no	—
33	#63121	—	39	1	no	—
34	D₁₃⋊C₉	—	65	1	no	—
35	D₁₃⋊C₉	—	65	1	no	—
36	D₁₃⋊C₉	—	130	6	no	—
37	C₃×C₅×C₁₃	—	2	6	no	—
38	C₃×C₅×C₁₃	AGL(1,𝔽₁₃)	1	1	yes	characteristic
39	C₃×C₅×C₁₃	AGL(1,𝔽₁₃)	1	1	yes	characteristic
40	C₉×AGL(1,𝔽₅)	—	169	1	no	Hall
41	C₁₃²	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
42	C₃×#261	—	65	1	no	—
43	C₃×#261	—	195	1	no	—
44	D₆₅	—	39	1	no	—
45	D₆₅	—	39	1	no	—
46	D₆₅	—	78	6	no	—
47	C₁₃⋊C₉	—	13	1	no	—
48	C₁₃⋊C₉	—	13	1	no	—
49	C₁₃⋊C₉	—	26	6	no	—
50	C₉×D₅	—	169	1	no	—
51	C₃×D₁₃	—	390	6	no	—
52	C₃×D₁₃	—	195	1	no	—
53	C₃×D₁₃	—	195	1	no	—
54	C₅×C₁₃	—	2	6	no	—
55	C₅×C₁₃	#261⋊C₉	1	1	yes	characteristic, complemented
56	C₅×C₁₃	#261⋊C₉	1	1	yes	characteristic, complemented
57	C₃×AGL(1,𝔽₅)	—	507	1	no	—
58	#261	—	65	1	no	—
59	#261	—	195	1	no	—
60	C₅×C₉	—	169	1	no	Hall
61	C₃×C₁₃	—	2	6	no	—
62	C₃×C₁₃	#63122⋊C₃	1	1	yes	characteristic
63	C₃×C₁₃	#63121⋊C₃	1	1	yes	characteristic
64	C₄×C₉	—	845	1	no	Hall
65	C₃×D₅	—	507	1	no	—
66	D₁₃	—	390	6	no	—
67	D₁₃	—	195	1	no	—
68	D₁₃	—	195	1	no	—
69	AGL(1,𝔽₅)	—	507	1	no	Hall
70	C₂×C₉	—	845	1	no	—
71	C₃×C₅	#14523146	1	1	yes	characteristic
72	C₁₃	—	2	6	no	—
73	C₁₃	#63122⋊C₉	1	1	yes	characteristic, complemented
74	C₁₃	#63121⋊C₉	1	1	yes	characteristic, complemented
75	C₃×C₄	—	2535	1	no	—
76	D₅	—	507	1	no	—
77	C₉	—	169	1	no	Sylow
78	C₂×C₃	—	2535	1	no	—
79	C₅	#20258770	1	1	yes	characteristic, complemented, Sylow
80	C₄	—	2535	1	no	Sylow
81	C₃	#23778457	1	1	yes	characteristic
82	C₂	—	2535	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#23778457
commutator, hypocenter	C₅×C₁₃²	C₄×C₉
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical, cosocle	C₃×C₅⋊(C₁₃⋊D₁₃)	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. #16810004	C₉	semidirect
13. #13095200	AGL(1,𝔽₅)	semidirect
18. C₅×C₁₃²	C₄×C₉	semidirect
41. C₁₃²	C₉×AGL(1,𝔽₅)	semidirect
55. C₅×C₁₃	#261⋊C₉	semidirect
56. C₅×C₁₃	#261⋊C₉	semidirect
73. C₁₃	#63122⋊C₉	semidirect
74. C₁₃	#63121⋊C₉	semidirect
79. C₅	#20258770	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊#15690148	2
C₃×#16810004	3
#20258770	5
#63121⋊C₉	13
#63122⋊C₉	13