Subgroups — #72276706

Order 3600.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(#1043⋊C₃)⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×AGL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#10597167	C₂	1	1	yes	maximal, characteristic, complemented
4	#11925748	—	3	1	no	maximal
5	AΓL(1,𝔽₂₅)	—	3	3	no	maximal
6	((C₅⋊D₅)⋊C₃)⋊D₃	C₄	1	1	yes	characteristic
7	C₃×#1043⋊C₃	C₂²	1	1	yes	characteristic
8	C₃⋊#64480	C₄	1	1	yes	characteristic
9	#10597182	—	3	1	no	—
10	C₃⋊#6053	—	3	1	no	—
11	C₃×#6053	D₃	1	1	yes	characteristic, complemented
12	#64480⋊C₂	—	3	3	no	—
13	AGL(1,𝔽₂₅)	D₃	1	3	yes	complemented
14	#10597167	—	3	3	no	—
15	C₃×(C₅⋊D₅)⋊C₃	C₂×C₄	1	1	yes	characteristic
16	C₃⋊#3874	—	2	1	no	—
17	#6053⋊C₂	—	9	1	no	Hall
18	D₁₅⋊D₅	—	3	1	no	—
19	C₃⋊#1044	—	3	1	no	—
20	C₃×#1043	C₂×D₃	1	1	yes	characteristic
21	#3874⋊C₂	—	3	3	no	—
22	#1043⋊C₃	C₂×D₃	1	3	yes	—
23	#64480	—	3	3	no	—
24	C₃×C₅²⋊C₃	M₁₆	1	1	yes	characteristic, complemented, Hall
25	D₅⋊AGL(1,𝔽₅)	—	9	1	no	—
26	#6053	—	9	1	no	—
27	#6053	C₃⋊D₃	1	1	yes	characteristic, complemented
28	C₅×D₁₅	—	6	1	no	—
29	C₃×C₅⋊D₅	C₄×D₃	1	1	yes	characteristic
30	(C₅⋊D₅)⋊C₃	C₄×D₃	1	3	yes	—
31	#3874	—	6	3	no	—
32	#3743	—	25	1	no	maximal, Hall
33	D₃×AGL(1,𝔽₅)	—	30	1	no	—
34	D₅²	—	9	1	no	—
35	#1044	—	9	1	no	—
36	#1043	C₂×C₃⋊D₃	1	1	yes	characteristic
37	C₃×C₅²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
38	C₅²⋊C₃	ΓL(1,𝔽₂₅)	1	3	yes	complemented
39	C₄×C₃⋊D₃	—	25	1	no	—
40	C₃²×C₈	—	25	1	no	—
41	C₃⋊(C₃⋊C₈)	—	25	1	no	—
42	D₃×D₅	—	30	1	no	—
43	ΓL(1,𝔽₁₆)	—	30	1	no	—
44	C₃×AGL(1,𝔽₅)	—	30	1	no	—
45	C₅⋊D₅	C₄×C₃⋊D₃	1	1	yes	characteristic
46	C₅×D₅	—	18	1	no	—
47	ΓL(1,𝔽₂₅)	—	75	3	no	—
48	ΓL(1,𝔽₂₅)	—	75	1	no	—
49	C₂×AGL(1,𝔽₅)	—	90	1	no	—
50	C₂×C₃⋊D₃	—	25	1	no	—
51	C₃²×C₄	—	25	1	no	—
52	C₃⋊Dic₃	—	25	1	no	—
53	D₁₅	—	6	1	no	—
54	C₃×D₅	—	30	1	no	—
55	C₅×D₃	—	30	1	no	—
56	C₅²	#3743	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₃	—	75	3	no	—
58	C₄×D₃	—	75	1	no	—
59	C₃×C₈	—	25	3	no	—
60	C₃×C₈	—	25	1	no	—
61	C₃⋊C₈	—	75	3	no	—
62	C₃⋊C₈	—	75	1	no	—
63	C₂×D₅	—	90	1	no	—
64	AGL(1,𝔽₅)	—	30	1	no	—
65	AGL(1,𝔽₅)	—	90	1	no	—
66	C₂×C₃²	—	25	1	no	—
67	C₃⋊D₃	—	50	1	no	—
68	M₁₆	—	225	1	no	Sylow
69	C₃×C₅	—	6	1	no	—
70	C₂×D₃	—	75	1	no	—
71	C₂×D₃	—	75	3	no	—
72	C₃×C₄	—	25	1	no	—
73	C₃×C₄	—	25	3	no	—
74	Dic₃	—	75	1	no	—
75	Dic₃	—	75	3	no	—
76	C₂×C₅	—	90	1	no	—
77	D₅	—	18	1	no	—
78	D₅	—	30	1	no	—
79	C₃²	—	25	1	no	Sylow
80	C₂×C₄	—	225	1	no	—
81	C₈	—	25	1	no	—
82	C₈	—	225	1	no	—
83	C₂×C₃	—	25	3	no	—
84	C₂×C₃	—	25	1	no	—
85	D₃	—	30	1	no	—
86	D₃	—	150	3	no	—
87	C₅	—	6	1	no	—
88	C₂²	—	225	1	no	—
89	C₄	—	25	1	no	—
90	C₄	—	225	1	no	—
91	C₃	—	25	3	no	—
92	C₃	AΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
93	C₂	—	25	1	no	—
94	C₂	—	90	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	C₃×(C₅⋊D₅)⋊C₃	C₂×C₄
Fitting, socle	C₃×C₅²	ΓL(1,𝔽₂₅)
prime radical	((C₅⋊D₅)⋊C₃)⋊D₃	C₄
cosocle	C₃×#1043⋊C₃	C₂²
hypocenter	C₃×C₅²⋊C₃	M₁₆
LCS	G ▷ C₃×(C₅⋊D₅)⋊C₃ ▷ C₃×C₅²⋊C₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃×AGL(1,𝔽₂₅)	C₂	semidirect
3. C₃⋊#10597167	C₂	semidirect
11. C₃×#6053	D₃	semidirect
13. AGL(1,𝔽₂₅)	D₃	semidirect
24. C₃×C₅²⋊C₃	M₁₆	semidirect
27. #6053	C₃⋊D₃	semidirect
37. C₃×C₅²	ΓL(1,𝔽₂₅)	semidirect
38. C₅²⋊C₃	ΓL(1,𝔽₂₅)	semidirect
56. C₅²	#3743	semidirect
92. C₃	AΓL(1,𝔽₂₅)	semidirect

Maximal subgroups

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

subgroup	index
(#1043⋊C₃)⋊D₃	2
C₃×AGL(1,𝔽₂₅)	2
C₃⋊#10597167	2
#11925748	3
AΓL(1,𝔽₂₅)	3
#3743	25