Subgroups — #13102556

Order 1600.

568 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87800	C₂	1	4	yes	maximal, complemented
2	#11717194	C₂	1	1	yes	maximal, characteristic
3	#11717189	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
7	C₂×#6045	C₂²	1	2	yes	—
8	C₂×C₅²×C₈	C₂²	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	—	2	2	no	—
11	#87800	—	2	8	no	—
12	#64982	—	5	6	no	maximal
13	C₂²×C₅⋊D₅	—	2	2	no	—
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
16	C₂×#1039	—	2	2	no	—
17	C₄⋊(C₅⋊D₅)	—	2	4	no	—
18	C₄⋊(C₅⋊D₅)	—	4	4	no	—
19	#6045	—	2	4	no	—
20	C₅²×C₈	D₄	1	4	yes	complemented
21	C₂×#677	—	5	24	no	—
22	#4020	—	5	6	no	—
23	#4015	—	5	6	no	—
24	C₄×C₅×C₈	D₅	1	6	yes	complemented
25	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
26	C₂×C₅⋊D₅	—	4	2	no	—
27	C₂×C₅⋊D₅	—	4	4	no	—
28	C₄×C₅²	SD₁₆	1	4	yes	complemented
29	C₄×C₅²	C₂×D₄	1	2	yes	—
30	#1039	—	4	2	no	—
31	C₂×D₂₀	—	5	12	no	—
32	C₂×D₂₀	—	10	12	no	—
33	C₂×Dic₁₀	—	5	12	no	—
34	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
35	C₄²×C₅	C₂×D₅	1	6	yes	—
36	C₄⋊Dic₅	—	10	12	no	—
37	#677	—	10	48	no	—
38	#492	—	25	1	no	Sylow
39	C₂×C₅²	#127	1	1	yes	characteristic
40	C₂×C₅²	C₂×SD₁₆	1	2	yes	—
41	C₅⋊D₅	—	8	2	no	—
42	C₂²×D₅	—	10	12	no	—
43	C₂×C₄×C₅	D₂₀	1	12	yes	—
44	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
45	C₂×Dic₅	—	10	12	no	—
46	D₂₀	—	10	24	no	—
47	D₂₀	—	20	24	no	—
48	Dic₁₀	—	10	24	no	—
49	C₅×C₈	D₂₀	1	24	yes	complemented
50	C₂×SD₁₆	—	25	4	no	—
51	C₄⋊Q₈	—	25	1	no	—
52	#127	—	25	1	no	—
53	C₄×C₈	C₅⋊D₅	1	1	yes	characteristic, complemented
54	C₅²	#492	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	C₂×D₂₀	1	6	yes	—
56	C₂×D₅	—	20	12	no	—
57	C₂×D₅	—	20	24	no	—
58	C₄×C₅	#677	1	24	yes	complemented
59	C₄×C₅	C₂×D₂₀	1	12	yes	—
60	Dic₅	—	20	12	no	—
61	C₂×Q₈	—	25	2	no	—
62	C₂×D₄	—	25	2	no	—
63	C₂×D₄	—	50	2	no	—
64	SD₁₆	—	50	8	no	—
65	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
66	C₄⋊C₄	—	50	2	no	—
67	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
68	C₂×C₅	#4020	1	6	yes	—
69	C₂×C₅	C₂×#677	1	12	yes	—
70	D₅	—	40	12	no	—
71	C₂³	—	50	2	no	—
72	Q₈	—	50	4	no	—
73	D₄	—	50	4	no	—
74	D₄	—	100	4	no	—
75	C₂×C₄	—	50	2	no	—
76	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
77	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
78	C₈	C₄⋊(C₅⋊D₅)	1	4	yes	complemented
79	C₅	#64982	1	6	yes	complemented
80	C₂²	—	100	2	no	—
81	C₂²	—	100	4	no	—
82	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
83	C₄	—	100	2	no	—
84	C₄	#87800	1	4	yes	complemented
85	C₄	C₂×C₄⋊(C₅⋊D₅)	1	2	yes	—
86	C₂	—	200	2	no	—
87	C₂	#11717194	1	1	yes	characteristic
88	C₂	C₂×#87800	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₄×C₅²×C₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	#11717194	C₂
hypercenter	C₄×C₈	C₅⋊D₅
hypocenter	C₅²	#492
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#87800	C₂	semidirect
3. #11717189	C₂	semidirect
4. C₄×C₅²×C₈	C₂	semidirect
20. C₅²×C₈	D₄	semidirect
24. C₄×C₅×C₈	D₅	semidirect
28. C₄×C₅²	SD₁₆	semidirect
49. C₅×C₈	D₂₀	semidirect
53. C₄×C₈	C₅⋊D₅	semidirect
54. C₅²	#492	semidirect
58. C₄×C₅	#677	semidirect
78. C₈	C₄⋊(C₅⋊D₅)	semidirect
79. C₅	#64982	semidirect
84. C₄	#87800	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87800	2
#11717194	2
#11717189	2
C₄×C₅²×C₈	2
#64982	5