Subgroups — #11924808

Order 1200.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₀₀	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₂×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6014	C₂	1	1	yes	maximal, characteristic, complemented
4	#87722	—	3	1	no	maximal, Hall
5	C₂×D₇₅	—	2	1	no	—
6	C₂×D₃×C₂₅	—	2	1	no	—
7	C₃×C₄×C₂₅	C₂²	1	1	yes	characteristic
8	#6654	—	5	1	no	maximal
9	D₄×C₂₅	—	3	1	no	—
10	D₁₀₀	—	3	1	no	—
11	#6014	D₃	1	1	yes	characteristic, complemented
12	C₂×C₃×C₂₅	D₄	1	1	yes	characteristic
13	D₇₅	—	4	1	no	—
14	D₃×C₂₅	—	4	1	no	—
15	D₆₀	—	5	1	no	—
16	C₅×D₁₂	D₅	1	1	yes	characteristic
17	C₃×#168	—	5	1	no	—
18	C₂²×C₂₅	—	6	1	no	—
19	C₂×D₂₅	—	6	1	no	—
20	C₄×C₂₅	C₂×D₃	1	1	yes	characteristic
21	#686	—	15	1	no	—
22	C₃×C₂₅	D₈	1	1	yes	characteristic, complemented, Hall
23	C₂×D₁₅	—	10	1	no	—
24	C₂×C₅×D₃	—	2	1	no	—
25	C₃×C₄×C₅	C₂×D₅	1	1	yes	characteristic
26	C₂×C₂₅	—	12	1	no	—
27	C₂×C₂₅	D₁₂	1	1	yes	characteristic
28	D₂₅	—	12	1	no	—
29	D₂₄	—	25	1	no	Hall
30	C₅×D₄	—	3	1	no	—
31	D₂₀	—	15	1	no	—
32	#168	—	5	1	no	—
33	C₂×C₃×C₅	C₂²⋊D₅	1	1	yes	characteristic
34	D₁₅	—	20	1	no	—
35	C₅×D₃	—	4	1	no	—
36	C₂₅	D₂₄	1	1	yes	characteristic, complemented, Sylow
37	D₁₂	—	25	1	no	—
38	D₁₂	D₂₅	1	1	yes	characteristic, complemented
39	C₃×C₈	—	25	1	no	—
40	C₂²×C₅	—	6	1	no	—
41	C₂×D₅	—	30	1	no	—
42	C₄×C₅	D₃×D₅	1	1	yes	characteristic
43	D₈	—	75	1	no	Sylow
44	C₃×C₅	#686	1	1	yes	characteristic
45	C₂×D₃	—	2	1	no	—
46	C₂×D₃	—	50	1	no	—
47	C₃×C₄	C₂×D₂₅	1	1	yes	characteristic
48	C₂×C₅	—	12	1	no	—
49	C₂×C₅	#1203	1	1	yes	characteristic
50	D₅	—	60	1	no	—
51	D₄	—	3	1	no	—
52	D₄	—	75	1	no	—
53	C₈	—	25	1	no	—
54	C₂×C₃	C₂²⋊D₂₅	1	1	yes	characteristic
55	D₃	—	4	1	no	—
56	D₃	—	100	1	no	—
57	C₅	#6654	1	1	yes	characteristic
58	C₂²	—	6	1	no	—
59	C₂²	—	150	1	no	—
60	C₄	D₃×D₂₅	1	1	yes	characteristic
61	C₃	#87722	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	12	1	no	—
63	C₂	—	300	1	no	—
64	C₂	#10597032	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10597032
commutator, Fitting, cosocle	C₃×C₄×C₂₅	C₂²
Frattini	C₄×C₅	D₃×D₅
socle	C₂×C₃×C₅	C₂²⋊D₅
hypercenter	C₄	D₃×D₂₅
hypocenter	C₃×C₂₅	D₈
LCS	G ▷ C₃×C₄×C₂₅ ▷ C₂×C₃×C₂₅ ▷ C₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. D₃₀₀	C₂	semidirect
2. D₁₂×C₂₅	C₂	semidirect
3. C₃×#6014	C₂	semidirect
11. #6014	D₃	semidirect
22. C₃×C₂₅	D₈	semidirect
36. C₂₅	D₂₄	semidirect
38. D₁₂	D₂₅	semidirect
61. C₃	#87722	semidirect

Maximal subgroups

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

subgroup	index
D₃₀₀	2
D₁₂×C₂₅	2
C₃×#6014	2
#87722	3
#6654	5