Subgroups — #11717229

Order 800.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#87900	C₂	1	1	yes	maximal, characteristic
2	C₂×#6045	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#87802	C₂	1	2	yes	maximal, complemented
5	#87800	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
8	C₂×#1039	—	2	1	no	—
9	C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
10	C₄×C₅⋊D₅	—	2	1	no	—
11	#6045	—	2	1	no	—
12	#6045	C₂²	1	2	yes	—
13	#6045	C₂²	1	1	yes	characteristic
14	C₅²×C₈	C₂²	1	2	yes	—
15	#4055	—	5	6	no	maximal
16	C₂²×C₅²	D₄	1	1	yes	characteristic
17	C₂×C₅⋊D₅	—	2	1	no	—
18	C₄×C₅²	D₄	1	1	yes	characteristic
19	C₄×C₅²	C₂³	1	1	yes	characteristic
20	#1039	—	2	2	no	—
21	#1039	—	2	1	no	—
22	#709	—	5	6	no	—
23	C₂×Dic₁₀	—	5	6	no	—
24	C₅×M₁₆	D₅	1	6	yes	complemented
25	Dic₂₀	—	5	12	no	—
26	#677	—	5	12	no	—
27	C₂×C₅²	—	2	1	no	—
28	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
29	C₅⋊D₅	—	4	1	no	—
30	C₂×C₄×C₅	C₂×D₅	1	6	yes	—
31	C₂²⋊D₅	—	10	6	no	—
32	C₂×Dic₅	—	10	6	no	—
33	D₂₀	—	5	6	no	—
34	C₄×D₅	—	10	6	no	—
35	Dic₁₀	—	5	12	no	—
36	Dic₁₀	—	5	6	no	—
37	Dic₁₀	—	10	6	no	—
38	C₅×C₈	C₂×D₅	1	12	yes	—
39	#137	—	25	1	no	Sylow
40	C₅²	#137	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	D₂₀	1	6	yes	—
42	C₂×D₅	—	10	6	no	—
43	C₄×C₅	D₂₀	1	6	yes	—
44	C₄×C₅	C₂²×D₅	1	6	yes	—
45	Dic₅	—	10	12	no	—
46	Dic₅	—	10	6	no	—
47	#41	—	25	1	no	—
48	C₂×Q₈	—	25	1	no	—
49	Q₁₆	—	25	2	no	—
50	SD₁₆	—	25	2	no	—
51	M₁₆	C₅⋊D₅	1	1	yes	characteristic, complemented
52	C₂×C₅	—	2	6	no	—
53	C₂×C₅	C₂×D₂₀	1	6	yes	—
54	D₅	—	20	6	no	—
55	Q₈	—	25	2	no	—
56	Q₈	—	25	1	no	—
57	Q₈	—	50	1	no	—
58	D₄	—	25	1	no	—
59	D₄	—	50	1	no	—
60	C₂×C₄	—	50	1	no	—
61	C₂×C₄	—	50	1	no	—
62	C₂×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
63	C₈	C₂×C₅⋊D₅	1	2	yes	—
64	C₅	#4055	1	6	yes	complemented
65	C₂²	—	50	1	no	—
66	C₂²	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
67	C₄	—	50	1	no	—
68	C₄	—	50	2	no	—
69	C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
70	C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	100	1	no	—
73	C₂	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊(C₅⋊D₅)
commutator, cosocle	C₄×C₅²	C₂³
Frattini	C₄	C₂²×C₅⋊D₅
Fitting	C₅²×M₁₆	C₂
socle	C₂×C₅²	C₂×D₄
prime radical	#87900	C₂
hypercenter	M₁₆	C₅⋊D₅
hypocenter	C₅²	#137
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#6045	C₂	semidirect
3. C₅²×M₁₆	C₂	semidirect
4. #87802	C₂	semidirect
5. #87800	C₂	semidirect
24. C₅×M₁₆	D₅	semidirect
40. C₅²	#137	semidirect
51. M₁₆	C₅⋊D₅	semidirect
64. C₅	#4055	semidirect

Maximal subgroups

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

subgroup	index
#87900	2
C₂×#6045	2
C₅²×M₁₆	2
#87802	2
#87800	2
#4055	5