Subgroups — #11716923

Order 800.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#87873	C₂	1	1	yes	maximal, characteristic
2	C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#676	C₂	1	2	yes	maximal, complemented
4	#87769	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×#168	C₂	1	2	yes	maximal, complemented
6	C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×D₅	C₂²	1	2	yes	—
8	#6039	C₄	1	1	yes	characteristic
9	#6038	C₄	1	2	yes	—
10	#6037	C₄	1	1	yes	characteristic
11	C₅²×C₈	C₂²	1	1	yes	characteristic
12	#6029	C₂²	1	1	yes	characteristic
13	C₅×#168	C₂²	1	2	yes	—
14	#4053	—	5	2	no	maximal
15	C₂×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×D₅	C₂×C₄	1	2	yes	—
17	C₄×C₅²	C₂³	1	1	yes	characteristic
18	#1039	C₂×C₄	1	1	yes	characteristic
19	C₅×Dic₅	C₂×C₄	1	2	yes	—
20	#709	—	5	2	no	—
21	C₅×M₁₆	—	5	2	no	—
22	C₂×#168	—	5	2	no	—
23	#676	—	5	2	no	—
24	#676	D₅	1	2	yes	complemented
25	C₈×D₅	—	5	2	no	—
26	C₈×D₅	—	5	2	no	—
27	C₈×D₅	—	10	2	no	—
28	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₅⋊D₅	—	2	1	no	—
30	C₅×D₅	—	2	2	no	—
31	C₂×C₄×C₅	—	5	2	no	—
32	C₂²⋊D₅	—	5	2	no	—
33	C₂²⋊D₅	—	5	2	no	—
34	D₂₀	—	5	2	no	—
35	C₄×D₅	—	5	2	no	—
36	C₄×D₅	—	10	2	no	—
37	C₄×D₅	C₂×D₅	1	2	yes	—
38	Dic₁₀	—	5	2	no	—
39	C₅×C₈	—	2	2	no	—
40	C₅×C₈	—	5	2	no	—
41	C₅×C₈	C₂×D₅	1	2	yes	—
42	#168	—	5	2	no	—
43	#168	—	10	2	no	—
44	#168	C₂×D₅	1	2	yes	—
45	#131	—	25	1	no	Sylow
46	C₅²	#131	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	—	5	2	no	—
48	C₂×D₅	—	5	2	no	—
49	C₂×D₅	—	10	2	no	—
50	C₂×D₅	C₄×D₅	1	2	yes	—
51	C₄×C₅	—	2	2	no	—
52	C₄×C₅	—	5	2	no	—
53	C₄×C₅	C₂²×D₅	1	2	yes	—
54	Dic₅	—	5	2	no	—
55	Dic₅	—	10	2	no	—
56	Dic₅	C₄×D₅	1	2	yes	—
57	#41	—	25	1	no	—
58	M₁₆	—	5	2	no	—
59	M₁₆	—	25	1	no	—
60	C₂×C₈	—	25	2	no	—
61	C₂×C₈	—	25	1	no	—
62	C₂×C₅	—	2	2	no	—
63	C₂×C₅	—	10	2	no	—
64	C₂×C₅	C₂×C₄×D₅	1	2	yes	—
65	D₅	—	2	2	no	—
66	D₅	—	10	4	no	—
67	D₅	—	10	2	no	—
68	Q₈	—	25	1	no	—
69	D₄	—	25	2	no	—
70	D₄	—	25	1	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	25	1	no	—
73	C₈	—	5	2	no	—
74	C₈	—	25	1	no	—
75	C₈	D₅²	1	1	yes	characteristic
76	C₅	—	2	2	no	—
77	C₅	#4053	1	2	yes	complemented
78	C₂²	—	5	2	no	—
79	C₂²	—	25	1	no	—
80	C₄	—	5	2	no	—
81	C₄	—	25	1	no	—
82	C₄	C₂×D₅²	1	1	yes	characteristic
83	C₂	—	10	2	no	—
84	C₂	—	50	1	no	—
85	C₂	C₄×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₅²
commutator, socle	C₂×C₅²	C₂²×C₄
Fitting	C₅²×C₈	C₂²
prime radical	#87873	C₂
cosocle	C₄×C₅²	C₂³
hypercenter	C₈	D₅²
hypocenter	C₅²	#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₈×C₅⋊D₅	C₂	semidirect
3. C₅×#676	C₂	semidirect
4. #87769	C₂	semidirect
5. D₅×#168	C₂	semidirect
24. #676	D₅	semidirect
46. C₅²	#131	semidirect
77. C₅	#4053	semidirect

Maximal subgroups

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

subgroup	index
#87873	2
C₈×C₅⋊D₅	2
C₅×#676	2
#87769	2
D₅×#168	2
#4053	5