Subgroups — #11716607

Order 800.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#1039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
5	C₂×#1039	—	2	1	no	—
6	C₂×C₅×Dic₅	—	2	1	no	—
7	C₅×Dic₁₀	—	2	1	no	—
8	C₅×Dic₁₀	C₄	1	2	yes	complemented
9	C₅×#168	—	2	1	no	—
10	#3967	—	5	1	no	maximal
11	#3948	—	5	1	no	maximal
12	C₂²×C₅²	D₄	1	1	yes	characteristic
13	C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₄×C₅²	D₄	1	1	yes	characteristic
15	#1039	—	4	1	no	—
16	C₅×Dic₅	—	2	2	no	—
17	C₂×C₅×Q₈	—	5	1	no	—
18	C₂×Dic₁₀	D₅	1	1	yes	characteristic
19	C₂×C₅×C₈	—	5	1	no	—
20	C₄⋊Dic₅	—	5	1	no	—
21	C₄⋊Dic₅	—	5	1	no	—
22	C₄⋊Dic₅	—	10	2	no	—
23	C₂×#168	D₅	1	1	yes	characteristic
24	C₂×C₅²	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₅²	SD₁₆	1	1	yes	characteristic
26	C₂×C₅²	Q₁₆	1	1	yes	characteristic
27	C₅×Q₈	—	5	2	no	—
28	C₅×Q₈	—	10	1	no	—
29	C₂×C₄×C₅	—	2	2	no	—
30	C₂×C₄×C₅	—	10	1	no	—
31	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
33	C₂×Dic₅	—	2	1	no	—
34	C₂×Dic₅	—	10	4	no	—
35	C₂×Dic₅	—	10	1	no	—
36	C₂×Dic₅	—	10	1	no	—
37	Dic₁₀	—	2	1	no	—
38	Dic₁₀	Dic₅	1	2	yes	complemented
39	C₅×C₈	—	10	1	no	—
40	#168	—	2	1	no	—
41	#103	—	25	1	no	Sylow
42	C₅²	#103	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	2	no	—
44	C₂²×C₅	D₂₀	1	1	yes	characteristic
45	C₂²×C₅	C₂²⋊D₅	1	1	yes	characteristic
46	C₄×C₅	—	2	2	no	—
47	C₄×C₅	—	2	2	no	—
48	C₄×C₅	—	10	2	no	—
49	C₄×C₅	C₄×D₅	1	1	yes	characteristic
50	C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
51	C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
52	C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
53	Dic₅	—	2	2	no	—
54	Dic₅	—	20	4	no	—
55	Dic₅	—	20	1	no	—
56	Dic₅	—	20	1	no	—
57	C₂×Q₈	—	5	1	no	—
58	C₂×C₈	—	5	1	no	—
59	C₄⋊C₄	—	25	1	no	—
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	—	2	2	no	—
62	C₂×C₅	—	2	2	no	—
63	C₂×C₅	#677	1	1	yes	characteristic
64	C₂×C₅	Dic₂₀	1	1	yes	characteristic
65	C₂×C₅	#685	1	1	yes	characteristic
66	C₂×C₅	#688	1	1	yes	characteristic
67	C₂×C₅	#689	1	1	yes	characteristic
68	C₂×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
69	Q₈	—	5	2	no	—
70	Q₈	—	10	1	no	—
71	C₂×C₄	—	10	1	no	—
72	C₂×C₄	—	50	1	no	—
73	C₂×C₄	D₅²	1	1	yes	characteristic
74	C₈	—	10	1	no	—
75	C₅	—	2	2	no	—
76	C₅	#3948	1	1	yes	characteristic, complemented
77	C₅	#3967	1	1	yes	characteristic, complemented
78	C₂²	#6038	1	1	yes	characteristic
79	C₄	—	10	2	no	—
80	C₄	—	100	1	no	—
81	C₄	D₅×Dic₅	1	1	yes	characteristic
82	C₄	#6037	1	1	yes	characteristic
83	C₂	#87774	1	1	yes	characteristic
84	C₂	#87776	1	1	yes	characteristic
85	C₂	#87778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6038
commutator	C₄×C₅²	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₅²
Fitting, cosocle	C₂×C₄×C₅²	C₂²
socle, prime radical	C₂²×C₅²	D₄
hypocenter	C₅²	#103
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. C₅×Dic₁₀	C₄	semidirect
38. Dic₁₀	Dic₅	semidirect
42. C₅²	#103	semidirect
76. C₅	#3948	semidirect
77. C₅	#3967	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₁₀	2
C₄⋊#1039	2
C₂×C₅×#168	2
#3967	5
#3948	5