Subgroups — #44477668

Order 30420.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊#15690149	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#16810004	—	3	1	no	maximal
3	C₅×#13095201	C₄	1	1	yes	characteristic, complemented, Hall
4	#20258785	—	5	1	no	maximal, Hall
5	C₃×C₅⋊(C₁₃⋊D₁₃)	D₃	1	1	yes	characteristic
6	#16810004	—	9	1	no	Hall
7	#15690149	—	5	1	no	—
8	C₃×C₅×C₁₃²	Dic₃	1	1	yes	characteristic
9	C₃⋊#10621729	—	15	1	no	—
10	C₅⋊(C₁₃⋊D₁₃)	D₉	1	1	yes	characteristic
11	#13095201	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₅⋊(D₁₃⋊C₉)	—	26	1	no	—
13	C₃×C₁₃⋊D₁₃	—	5	1	no	—
14	C₅×C₁₃²	Dic₉	1	1	yes	characteristic, complemented, Hall
15	C₃⋊#63122	—	39	1	no	—
16	C₃⋊#63121	—	39	1	no	—
17	#10621729	—	45	1	no	Hall
18	C₅×C₁₃⋊C₉	—	52	1	no	—
19	C₃×C₁₃²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
20	C₃×D₆₅	—	26	1	no	—
21	C₃×D₆₅	—	78	1	no	—
22	C₃×D₆₅	—	78	1	no	—
23	C₃×D₆₅	—	78	1	no	—
24	C₁₃⋊D₁₃	—	5	1	no	—
25	#63122	—	117	1	no	—
26	#63121	—	117	1	no	—
27	D₁₃⋊C₉	—	130	1	no	—
28	C₃×C₅×C₁₃	—	2	1	no	—
29	C₃×C₅×C₁₃	—	6	1	no	—
30	C₃×C₅×C₁₃	—	6	1	no	—
31	C₃×C₅×C₁₃	—	6	1	no	—
32	C₉⋊AGL(1,𝔽₅)	—	169	1	no	maximal, Hall
33	C₁₃²	C₉⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
34	C₃⋊#261	—	195	1	no	—
35	C₃⋊#261	—	195	1	no	—
36	D₆₅	—	26	1	no	—
37	D₆₅	—	39	1	no	—
38	D₆₅	—	78	1	no	—
39	D₆₅	—	78	1	no	—
40	C₁₃⋊C₉	—	52	1	no	—
41	C₉×D₅	—	169	1	no	—
42	C₃×D₁₃	—	130	1	no	—
43	C₃×D₁₃	—	390	1	no	—
44	C₃×D₁₃	—	390	1	no	—
45	C₃×D₁₃	—	390	1	no	—
46	C₅×C₁₃	—	2	1	no	—
47	C₅×C₁₃	—	3	1	no	—
48	C₅×C₁₃	—	6	1	no	—
49	C₅×C₁₃	—	6	1	no	—
50	ΓL(1,𝔽₁₆)	—	507	1	no	—
51	#261	—	585	1	no	—
52	#261	—	585	1	no	—
53	C₅×C₉	—	676	1	no	Hall
54	C₃×C₁₃	—	2	1	no	—
55	C₃×C₁₃	—	6	1	no	—
56	C₃×C₁₃	—	6	1	no	—
57	C₃×C₁₃	—	6	1	no	—
58	Dic₉	—	845	1	no	Hall
59	C₃×D₅	—	169	1	no	—
60	D₁₃	—	130	1	no	—
61	D₁₃	—	390	1	no	—
62	D₁₃	—	390	1	no	—
63	D₁₃	—	195	1	no	—
64	AGL(1,𝔽₅)	—	1521	1	no	Hall
65	C₂×C₉	—	845	1	no	—
66	C₃×C₅	#14523161	1	1	yes	characteristic
67	C₁₃	—	2	1	no	—
68	C₁₃	—	3	1	no	—
69	C₁₃	—	6	1	no	—
70	C₁₃	—	6	1	no	—
71	Dic₃	—	2535	1	no	—
72	D₅	—	169	1	no	—
73	C₉	—	676	1	no	Sylow
74	C₂×C₃	—	845	1	no	—
75	C₅	#20258785	1	1	yes	characteristic, complemented, Sylow
76	C₄	—	7605	1	no	Sylow
77	C₃	#23778486	1	1	yes	characteristic
78	C₂	—	845	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×#13095201	C₄
Frattini	C₃	#23778486
Fitting, socle	C₃×C₅×C₁₃²	Dic₃
prime radical	C₃×C₅⋊(C₁₃⋊D₁₃)	D₃
cosocle	C₅⋊#15690149	C₂
LCS	G ▷ C₅×#13095201
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅×#13095201	C₄	semidirect
11. #13095201	AGL(1,𝔽₅)	semidirect
14. C₅×C₁₃²	Dic₉	semidirect
33. C₁₃²	C₉⋊AGL(1,𝔽₅)	semidirect
75. C₅	#20258785	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊#15690149	2
C₃⋊#16810004	3
#20258785	5
C₉⋊AGL(1,𝔽₅)	169