Subgroups — C₅²×#208 #11925213

Order 1200.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₅²×M₁₆	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅²	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅²	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₅×#208	C₅	1	6	yes	maximal, direct product factor
8	C₂×C₄×C₅²	D₃	1	1	yes	characteristic
9	C₅²×C₈	—	3	2	no	—
10	C₂×C₃×C₅²	—	2	1	no	—
11	C₂×C₃×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×C₅	C₂×C₅	1	6	yes	—
13	C₅×C₃⋊C₈	C₂×C₅	1	12	yes	complemented
14	C₂²×C₅²	Dic₃	1	1	yes	characteristic
15	C₄×C₅²	Dic₃	1	1	yes	characteristic
16	C₄×C₅²	C₂×D₃	1	1	yes	characteristic
17	C₅×M₁₆	—	3	6	no	—
18	C₃×C₅²	M₁₆	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₃×C₅	C₄×C₅	1	6	yes	—
20	C₃×C₄×C₅	C₄×C₅	1	6	yes	—
21	C₃×C₄×C₅	C₂²×C₅	1	6	yes	—
22	C₂×C₅²	—	2	1	no	—
23	C₂×C₅²	C₂×Dic₃	1	1	yes	characteristic
24	#208	C₅²	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₄×C₅	C₅×D₃	1	6	yes	—
26	C₅×C₈	—	3	12	no	—
27	C₂×C₃×C₅	—	2	6	no	—
28	C₂×C₃×C₅	C₂×C₄×C₅	1	6	yes	—
29	C₅²	#208	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃×C₄	C₂×C₅²	1	1	yes	characteristic
31	C₃⋊C₈	C₂×C₅²	1	2	yes	complemented
32	C₂²×C₅	C₅×Dic₃	1	6	yes	—
33	C₄×C₅	C₅×Dic₃	1	6	yes	—
34	C₄×C₅	C₂×C₅×D₃	1	6	yes	—
35	M₁₆	—	3	1	no	Sylow
36	C₃×C₅	C₅×M₁₆	1	6	yes	complemented
37	C₂²×C₃	C₄×C₅²	1	1	yes	characteristic
38	C₃×C₄	C₄×C₅²	1	1	yes	characteristic
39	C₃×C₄	C₂²×C₅²	1	1	yes	characteristic
40	C₂×C₅	—	2	6	no	—
41	C₂×C₅	C₂×C₅×Dic₃	1	6	yes	—
42	C₂×C₄	C₅²×D₃	1	1	yes	characteristic
43	C₈	—	3	2	no	—
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	C₂×C₄×C₅²	1	1	yes	characteristic
46	C₅	C₅×#208	1	6	yes	direct product factor
47	C₂²	C₅²×Dic₃	1	1	yes	characteristic
48	C₄	C₅²×Dic₃	1	1	yes	characteristic
49	C₄	C₂×C₅²×D₃	1	1	yes	characteristic
50	C₃	C₅²×M₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂	—	2	1	no	—
52	C₂	C₂×C₅²×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅²	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅²
Frattini	C₄	C₂×C₅²×D₃
Fitting	C₂×C₃×C₄×C₅²	C₂
socle	C₂×C₃×C₅²	C₂×C₄
prime radical	C₂²×C₃×C₅²	C₄
cosocle	C₃×C₄	C₂²×C₅²
hypercenter	C₂×C₄×C₅²	D₃
hypocenter	C₃	C₅²×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅² ◁ C₂×C₄×C₅²

Complements

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

N	H ≅ G/N	type
2. C₅²×C₃⋊C₈	C₂	semidirect
7. C₅×#208	C₅	direct (N × H)
13. C₅×C₃⋊C₈	C₂×C₅	semidirect
18. C₃×C₅²	M₁₆	semidirect
24. #208	C₅²	direct (N × H)
29. C₅²	#208	direct (N × H)
31. C₃⋊C₈	C₂×C₅²	semidirect
36. C₃×C₅	C₅×M₁₆	semidirect
46. C₅	C₅×#208	direct (N × H)
50. C₃	C₅²×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅²	2
C₅²×C₃⋊C₈	2
C₅²×M₁₆	3
C₅×#208	5

Subgroups — C52×#208 #11925213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#208 #11925213