Subgroups — #11925152

Order 1200.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊#168	C₂	1	2	yes	maximal, complemented
3	#87769	—	3	1	no	maximal, Hall
4	C₂×C₃×C₅⋊D₅	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₃×#1039	C₄	1	1	yes	characteristic
7	#6649	—	5	2	no	maximal
8	C₄×C₅⋊D₅	D₃	1	1	yes	characteristic
9	C₅×#168	—	3	2	no	—
10	C₂×C₃×C₅²	C₂×C₄	1	1	yes	characteristic
11	C₃×C₅⋊D₅	—	2	1	no	—
12	C₃×C₄×D₅	—	5	2	no	—
13	C₃×C₄×D₅	—	10	2	no	—
14	C₃⋊#168	D₅	1	2	yes	complemented
15	C₅×C₃⋊C₈	—	5	2	no	—
16	C₂×C₅⋊D₅	Dic₃	1	1	yes	characteristic
17	C₄×C₅²	C₂×D₃	1	1	yes	characteristic
18	#1039	Dic₃	1	1	yes	characteristic
19	#676	—	15	2	no	—
20	C₃×C₅²	M₁₆	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×D₅	—	5	2	no	—
22	C₂×C₃×D₅	—	10	2	no	—
23	C₃×C₄×C₅	—	2	2	no	—
24	C₃×C₄×C₅	C₂×D₅	1	2	yes	—
25	C₃×Dic₅	—	5	2	no	—
26	C₃×Dic₅	—	10	2	no	—
27	C₂×C₅²	C₂×Dic₃	1	1	yes	characteristic
28	C₅⋊D₅	—	2	1	no	—
29	#208	—	25	1	no	Hall
30	C₄×D₅	—	5	2	no	—
31	C₄×D₅	—	10	2	no	—
32	C₅×C₈	—	15	2	no	—
33	#168	—	3	2	no	—
34	C₂×C₃×C₅	—	2	2	no	—
35	C₂×C₃×C₅	C₄×D₅	1	2	yes	—
36	C₃×D₅	—	10	4	no	—
37	C₃×D₅	—	10	2	no	—
38	C₅²	#208	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃×C₄	—	25	1	no	—
40	C₃⋊C₈	—	5	2	no	—
41	C₂×D₅	—	5	2	no	—
42	C₂×D₅	—	10	2	no	—
43	C₄×C₅	—	2	2	no	—
44	C₄×C₅	D₃×D₅	1	2	yes	—
45	Dic₅	—	5	2	no	—
46	Dic₅	—	10	2	no	—
47	M₁₆	—	75	1	no	Sylow
48	C₃×C₅	—	2	2	no	—
49	C₃×C₅	#676	1	2	yes	complemented
50	C₂²×C₃	—	25	1	no	—
51	C₃×C₄	—	25	1	no	—
52	C₃×C₄	D₅²	1	1	yes	characteristic
53	C₂×C₅	—	2	2	no	—
54	C₂×C₅	D₅×Dic₃	1	2	yes	—
55	D₅	—	10	4	no	—
56	D₅	—	10	2	no	—
57	C₂×C₄	—	25	1	no	—
58	C₈	—	15	2	no	—
59	C₂×C₃	—	50	1	no	—
60	C₂×C₃	#6036	1	1	yes	characteristic
61	C₅	—	2	2	no	—
62	C₅	#6649	1	2	yes	complemented
63	C₂²	—	25	1	no	—
64	C₄	—	25	1	no	—
65	C₄	D₁₅⋊D₅	1	1	yes	characteristic
66	C₃	#87769	1	1	yes	characteristic, complemented, Sylow
67	C₂	—	50	1	no	—
68	C₂	#10597126	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₅⋊D₅
commutator, socle	C₂×C₃×C₅²	C₂×C₄
Fitting, cosocle	C₃×C₄×C₅²	C₂²
prime radical	C₂×C₃×C₅⋊D₅	C₄
hypocenter	C₃×C₅²	M₁₆
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊#168	C₂	semidirect
14. C₃⋊#168	D₅	semidirect
20. C₃×C₅²	M₁₆	semidirect
38. C₅²	#208	semidirect
49. C₃×C₅	#676	semidirect
62. C₅	#6649	semidirect
66. C₃	#87769	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅⋊D₅	2
C₅×C₃⋊#168	2
#87769	3
#6649	5