Subgroups — #11717246

Order 800.

304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#1039	C₂	1	2	yes	maximal
3	C₄⋊#1039	C₂	1	2	yes	maximal, complemented
4	C₄×#1039	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅²	C₄	1	4	yes	complemented
7	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×#1039	C₂²	1	4	yes	—
10	#4072	—	5	6	no	maximal
11	C₂²×C₅²	—	2	2	no	—
12	C₂²×C₅²	C₂×C₄	1	2	yes	—
13	C₂²×C₅²	C₂³	1	1	yes	characteristic
14	C₄×C₅²	C₂×C₄	1	2	yes	—
15	C₄×C₅²	C₂×C₄	1	2	yes	—
16	#1039	—	2	4	no	—
17	C₂²×C₄×C₅	D₅	1	6	yes	—
18	C₂²⋊Dic₅	—	5	12	no	—
19	C₄⋊Dic₅	—	5	12	no	—
20	C₄×Dic₅	—	5	12	no	—
21	C₂×C₅²	—	2	2	no	—
22	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₅²	#41	1	2	yes	—
24	C₂³×C₅	C₂×D₅	1	6	yes	—
25	C₂×C₄×C₅	Dic₅	1	24	yes	complemented
26	C₂×C₄×C₅	C₂×D₅	1	6	yes	—
27	C₂×C₄×C₅	C₂×D₅	1	6	yes	—
28	C₂×Dic₅	—	5	24	no	—
29	#117	—	25	1	no	Sylow
30	C₅²	#117	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₅	—	2	12	no	—
32	C₂²×C₅	C₂×Dic₅	1	12	yes	—
33	C₂²×C₅	C₂²×D₅	1	6	yes	—
34	C₄×C₅	C₂×Dic₅	1	12	yes	—
35	C₄×C₅	C₂×Dic₅	1	12	yes	—
36	Dic₅	—	10	24	no	—
37	C₂²×C₄	C₅⋊D₅	1	1	yes	characteristic
38	C₄⋊C₄	—	25	2	no	—
39	C₂²⋊C₄	—	25	2	no	—
40	C₄²	—	25	2	no	—
41	C₂×C₅	—	2	12	no	—
42	C₂×C₅	#709	1	12	yes	—
43	C₂×C₅	C₂²×Dic₅	1	6	yes	—
44	C₂³	C₂×C₅⋊D₅	1	1	yes	characteristic
45	C₂×C₄	—	25	4	no	—
46	C₂×C₄	#1039	1	4	yes	complemented
47	C₂×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
48	C₂×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
49	C₅	#4072	1	6	yes	complemented
50	C₂²	—	2	2	no	—
51	C₂²	C₂×#1039	1	2	yes	—
52	C₂²	C₂²×C₅⋊D₅	1	1	yes	characteristic
53	C₄	—	50	4	no	—
54	C₄	C₂×#1039	1	2	yes	—
55	C₄	C₂×#1039	1	2	yes	—
56	C₂	—	2	2	no	—
57	C₂	#87900	1	2	yes	—
58	C₂	C₂²×#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₅⋊D₅
commutator	C₂×C₅²	C₂²×C₄
Frattini	C₂²	C₂²×C₅⋊D₅
Fitting	C₂²×C₄×C₅²	C₂
socle, cosocle	C₂²×C₅²	C₂³
prime radical	C₂³×C₅²	C₂²
hypercenter	C₂²×C₄	C₅⋊D₅
hypocenter	C₅²	#117
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊#1039	C₂	semidirect
4. C₄×#1039	C₂	semidirect
6. C₂×C₄×C₅²	C₄	semidirect
25. C₂×C₄×C₅	Dic₅	semidirect
30. C₅²	#117	semidirect
46. C₂×C₄	#1039	semidirect
49. C₅	#4072	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅²	2
C₂²⋊#1039	2
C₄⋊#1039	2
C₄×#1039	2
#4072	5