Subgroups — #11717269

Order 800.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	#87811	C₂	1	2	yes	maximal, complemented
4	#87809	C₂	1	2	yes	maximal, complemented
5	#87804	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅⋊D₅	—	2	1	no	—
7	C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
8	C₅²×D₄	—	2	1	no	—
9	C₅²×D₄	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₄×C₅²	—	2	1	no	—
11	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅⋊D₅)	—	2	1	no	—
13	C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
14	#6029	C₂²	1	2	yes	complemented
15	#4095	—	5	6	no	maximal
16	C₂²×C₅²	—	2	1	no	—
17	C₂²×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₅⋊D₅	—	2	2	no	—
19	C₂×C₅⋊D₅	—	4	2	no	—
20	C₄×C₅²	—	2	1	no	—
21	C₄×C₅²	D₄	1	1	yes	characteristic
22	C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₅×#41	D₅	1	6	yes	complemented
24	C₂×D₂₀	—	5	6	no	—
25	#688	—	5	12	no	—
26	#686	—	5	12	no	—
27	#681	—	5	6	no	—
28	C₂×C₅²	—	2	1	no	—
29	C₂×C₅²	—	4	1	no	—
30	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
31	C₅⋊D₅	—	4	2	no	—
32	C₂²×D₅	—	10	6	no	—
33	C₅×Q₈	C₂×D₅	1	6	yes	complemented
34	C₅×D₄	—	2	6	no	—
35	C₅×D₄	C₂×D₅	1	6	yes	complemented
36	C₂×C₄×C₅	—	2	6	no	—
37	C₂×C₄×C₅	C₂×D₅	1	6	yes	—
38	D₂₀	—	5	12	no	—
39	D₂₀	—	10	6	no	—
40	#168	—	5	12	no	—
41	AGL(1,ℤ/8)	—	25	1	no	Sylow
42	C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	6	no	—
44	C₂²×C₅	C₂²⋊D₅	1	6	yes	—
45	C₂×D₅	—	10	12	no	—
46	C₂×D₅	—	20	12	no	—
47	C₄×C₅	—	2	6	no	—
48	C₄×C₅	C₂²⋊D₅	1	6	yes	—
49	C₄×C₅	C₂²×D₅	1	6	yes	—
50	#41	C₅⋊D₅	1	1	yes	characteristic, complemented
51	C₂×D₄	—	25	1	no	—
52	SD₁₆	—	25	2	no	—
53	D₈	—	25	2	no	—
54	M₁₆	—	25	1	no	—
55	C₂×C₅	—	2	6	no	—
56	C₂×C₅	—	4	6	no	—
57	C₂×C₅	C₂×C₂²⋊D₅	1	6	yes	—
58	D₅	—	20	12	no	—
59	C₂³	—	50	1	no	—
60	Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
61	D₄	—	2	1	no	—
62	D₄	—	25	2	no	—
63	D₄	—	50	1	no	—
64	D₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
67	C₈	—	25	2	no	—
68	C₅	#4095	1	6	yes	complemented
69	C₂²	—	2	1	no	—
70	C₂²	—	50	2	no	—
71	C₂²	—	100	2	no	—
72	C₂²	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
75	C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	100	2	no	—
79	C₂	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₄×C₅²	C₂³
Frattini	C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#41	C₂
socle	C₂×C₅²	C₂×D₄
hypercenter	#41	C₅⋊D₅
hypocenter	C₅²	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₅²×#41	C₂	semidirect
2. C₂×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. #87811	C₂	semidirect
4. #87809	C₂	semidirect
5. #87804	C₂	semidirect
7. C₅²×Q₈	C₂²	semidirect
9. C₅²×D₄	C₂²	semidirect
14. #6029	C₂²	semidirect
23. C₅×#41	D₅	semidirect
33. C₅×Q₈	C₂×D₅	semidirect
35. C₅×D₄	C₂×D₅	semidirect
42. C₅²	AGL(1,ℤ/8)	semidirect
50. #41	C₅⋊D₅	semidirect
60. Q₈	C₂×C₅⋊D₅	semidirect
64. D₄	C₂×C₅⋊D₅	semidirect
68. C₅	#4095	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#41	2
C₂×C₄⋊(C₅⋊D₅)	2
#87811	2
#87809	2
#87804	2
#4095	5