Subgroups — C₅²×#683 #14522614

Order 2000.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅³	C₂²	1	1	yes	characteristic
5	C₄×C₅³	—	2	1	no	—
6	C₅²×Dic₅	—	2	1	no	—
7	C₅²×Dic₅	C₄	1	2	yes	complemented
8	C₅²×C₄⋊C₄	—	5	1	no	maximal
9	C₅×#683	C₅	1	6	yes	maximal, direct product factor
10	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅³	D₄	1	1	yes	characteristic
12	C₂×C₅³	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	—	2	12	no	—
14	C₂×C₄×C₅²	—	5	1	no	—
15	C₂×C₄×C₅²	—	5	1	no	—
16	C₂×C₄×C₅²	D₅	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₅	1	6	yes	—
18	C₂×C₅×Dic₅	C₂×C₅	1	6	yes	—
19	C₂×C₅×Dic₅	C₂×C₅	1	6	yes	—
20	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅²	—	2	12	no	—
22	C₂²×C₅²	C₂×D₅	1	1	yes	characteristic
23	C₂²×C₅²	C₂²×C₅	1	6	yes	—
24	C₄×C₅²	—	2	24	no	—
25	C₄×C₅²	—	2	1	no	—
26	C₄×C₅²	—	2	6	no	—
27	C₄×C₅²	—	5	2	no	—
28	C₄×C₅²	—	10	1	no	—
29	C₅×Dic₅	—	2	6	no	—
30	C₅×Dic₅	C₄×C₅	1	12	yes	complemented
31	C₅×C₄⋊C₄	—	5	6	no	—
32	#683	C₅²	1	1	yes	characteristic, direct product factor
33	C₂×C₅²	—	2	12	no	—
34	C₂×C₅²	—	2	12	no	—
35	C₂×C₅²	—	2	12	no	—
36	C₂×C₅²	Dic₁₀	1	1	yes	characteristic
37	C₂×C₅²	C₄×D₅	1	1	yes	characteristic
38	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
39	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
40	C₂×C₅²	C₅×D₄	1	6	yes	—
41	C₂×C₅²	C₅×Q₈	1	6	yes	—
42	C₂×C₄×C₅	—	2	12	no	—
43	C₂×C₄×C₅	—	5	6	no	—
44	C₂×C₄×C₅	—	5	6	no	—
45	C₂×C₄×C₅	C₅×D₅	1	6	yes	—
46	C₂×C₄×C₅	C₂×C₅²	1	1	yes	characteristic
47	C₂×Dic₅	C₂×C₅²	1	1	yes	characteristic
48	C₂×Dic₅	C₂×C₅²	1	1	yes	characteristic
49	C₅²	—	2	12	no	—
50	C₅²	#683	1	1	yes	characteristic, direct product factor
51	C₅²	C₅×C₄⋊C₄	1	6	yes	complemented
52	C₂²×C₅	—	2	12	no	—
53	C₂²×C₅	C₂×C₅×D₅	1	6	yes	—
54	C₂²×C₅	C₂²×C₅²	1	1	yes	characteristic
55	C₄×C₅	—	2	24	no	—
56	C₄×C₅	—	2	6	no	—
57	C₄×C₅	—	2	1	no	—
58	C₄×C₅	—	5	12	no	—
59	C₄×C₅	—	10	6	no	—
60	Dic₅	—	2	1	no	—
61	Dic₅	C₄×C₅²	1	2	yes	complemented
62	C₄⋊C₄	—	5	1	no	Sylow
63	C₂×C₅	—	2	12	no	—
64	C₂×C₅	—	2	12	no	—
65	C₂×C₅	—	2	12	no	—
66	C₂×C₅	C₅×Dic₁₀	1	6	yes	—
67	C₂×C₅	C₄×C₅×D₅	1	6	yes	—
68	C₂×C₅	C₅×C₂²⋊D₅	1	6	yes	—
69	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
70	C₂×C₅	C₅²×D₄	1	1	yes	characteristic
71	C₂×C₅	C₅²×Q₈	1	1	yes	characteristic
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	C₅²×D₅	1	1	yes	characteristic
75	C₅	—	2	12	no	—
76	C₅	C₅×#683	1	6	yes	direct product factor
77	C₅	C₅²×C₄⋊C₄	1	1	yes	characteristic, complemented
78	C₂²	C₂×C₅²×D₅	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	5	2	no	—
81	C₄	—	10	1	no	—
82	C₂	C₅²×Dic₁₀	1	1	yes	characteristic
83	C₂	C₄×C₅²×D₅	1	1	yes	characteristic
84	C₂	C₅²×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅²	C₂×D₅
commutator	C₂×C₅	C₂×C₄×C₅²
Frattini	C₂²	C₂×C₅²×D₅
Fitting	C₂×C₄×C₅³	C₂
socle, prime radical	C₂²×C₅³	C₂²
cosocle	C₂²×C₅	C₂²×C₅²
hypercenter	C₂×C₄×C₅²	D₅
hypocenter	C₅	C₅²×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅² ◁ C₂×C₄×C₅²

Complements

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

N	H ≅ G/N	type
7. C₅²×Dic₅	C₄	semidirect
9. C₅×#683	C₅	direct (N × H)
20. C₅³	C₄⋊C₄	semidirect
30. C₅×Dic₅	C₄×C₅	semidirect
32. #683	C₅²	direct (N × H)
50. C₅²	#683	direct (N × H)
51. C₅²	C₅×C₄⋊C₄	semidirect
61. Dic₅	C₄×C₅²	semidirect
76. C₅	C₅×#683	direct (N × H)
77. C₅	C₅²×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅³	2
C₂×C₅²×Dic₅	2
C₅²×C₄⋊C₄	5
C₅×#683	5

Subgroups — C52×#683 #14522614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#683 #14522614