Subgroups — #11717439

Order 800.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#87902	C₂	1	1	yes	maximal, characteristic
2	#87860	C₂	1	2	yes	maximal, complemented
3	C₂×#6033	C₂	1	2	yes	maximal, complemented
4	#87856	C₂	1	1	yes	maximal, characteristic, complemented
5	#87855	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×D₄	C₄	1	1	yes	characteristic
7	C₂²⋊(C₅⋊D₅)	C₄	1	2	yes	—
8	C₂×#1039	C₂²	1	2	yes	—
9	C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
10	#6045	C₄	1	1	yes	characteristic
11	#6033	C₂²	1	1	yes	characteristic
12	#6033	C₂²	1	1	yes	characteristic
13	#6033	C₂²	1	2	yes	—
14	#4131	—	5	6	no	maximal
15	C₂²×C₅²	C₂×C₄	1	2	yes	—
16	C₂×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
17	C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	#1039	C₂×C₄	1	2	yes	—
19	#1039	C₂³	1	1	yes	characteristic
20	#711	—	5	6	no	—
21	#704	—	5	12	no	—
22	C₂×#170	—	5	12	no	—
23	#700	—	5	6	no	—
24	D₅⋊C₈	—	5	6	no	—
25	C₂×C₅²	—	2	2	no	—
26	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
27	C₅⋊D₅	—	2	1	no	—
28	C₅×D₄	AGL(1,𝔽₅)	1	6	yes	—
29	C₂²⋊D₅	—	5	12	no	—
30	C₂×Dic₅	—	5	12	no	—
31	C₄×D₅	—	5	6	no	—
32	Dic₁₀	—	5	6	no	—
33	#170	—	5	6	no	—
34	#170	—	5	6	no	—
35	#170	—	5	12	no	—
36	#131	—	25	1	no	Sylow
37	C₅²	#131	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₅	C₂×AGL(1,𝔽₅)	1	12	yes	—
39	C₂×D₅	—	5	6	no	—
40	C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
41	Dic₅	—	5	12	no	—
42	Dic₅	—	5	6	no	—
43	#41	—	25	1	no	—
44	M₁₆	—	25	1	no	—
45	M₁₆	—	25	2	no	—
46	C₂×C₈	—	25	1	no	—
47	C₂×C₈	—	25	2	no	—
48	C₂×C₅	—	2	12	no	—
49	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	6	yes	—
50	D₅	—	10	6	no	—
51	Q₈	—	25	1	no	—
52	D₄	—	25	2	no	—
53	D₄	#1043	1	1	yes	characteristic
54	C₂×C₄	—	25	2	no	—
55	C₂×C₄	—	25	1	no	—
56	C₈	—	25	1	no	—
57	C₈	—	25	1	no	—
58	C₈	—	25	2	no	—
59	C₅	#4131	1	6	yes	complemented
60	C₂²	—	25	1	no	—
61	C₂²	C₂×#1043	1	2	yes	—
62	C₄	—	25	1	no	—
63	C₄	—	25	2	no	—
64	C₄	C₂×#1043	1	1	yes	characteristic
65	C₂	—	2	2	no	—
66	C₂	—	50	1	no	—
67	C₂	C₂²×#1043	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#1043
commutator, socle	C₂×C₅²	C₂²×C₄
Fitting	C₅²×D₄	C₄
prime radical	#87902	C₂
cosocle	#1039	C₂³
hypercenter	D₄	#1043
hypocenter	C₅²	#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #87860	C₂	semidirect
3. C₂×#6033	C₂	semidirect
4. #87856	C₂	semidirect
5. #87855	C₂	semidirect
37. C₅²	#131	semidirect
59. C₅	#4131	semidirect

Maximal subgroups

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

subgroup	index
#87902	2
#87860	2
C₂×#6033	2
#87856	2
#87855	2
#4131	5