Subgroups — #72221162

Order 3200.

856 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13102556	C₂	1	3	yes	maximal
2	#13102543	C₂	1	3	yes	maximal, complemented
3	C₅²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87800	—	2	6	no	—
5	#11717194	C₂²	1	1	yes	characteristic
6	#11717189	C₂²	1	3	yes	—
7	C₄×C₅²×C₈	C₂²	1	3	yes	—
8	#11716684	—	2	6	no	—
9	#10598850	—	5	6	no	maximal
10	C₂×C₄⋊(C₅⋊D₅)	—	2	3	no	—
11	C₂×#6045	—	2	3	no	—
12	C₂×C₅²×C₈	D₄	1	6	yes	—
13	C₄²×C₅²	C₂³	1	1	yes	characteristic
14	C₄⋊#1039	—	2	6	no	—
15	#87800	—	4	12	no	—
16	#64982	—	5	18	no	—
17	#64969	—	5	18	no	—
18	C₅×C₈²	D₅	1	6	yes	complemented
19	C₂²×C₅⋊D₅	—	4	1	no	—
20	C₂×C₄×C₅²	C₂×D₄	1	3	yes	—
21	C₂×#1039	—	4	3	no	—
22	C₄⋊(C₅⋊D₅)	—	4	6	no	—
23	#6045	—	4	6	no	—
24	C₅²×C₈	SD₁₆	1	12	yes	complemented
25	C₂×#677	—	10	36	no	—
26	#4020	—	5	6	no	—
27	#4015	—	5	18	no	—
28	C₄×C₅×C₈	C₂×D₅	1	18	yes	—
29	#3949	—	10	36	no	—
30	#1705	—	25	1	no	Sylow
31	C₂²×C₅²	#127	1	1	yes	characteristic
32	C₂×C₅⋊D₅	—	8	3	no	—
33	C₄×C₅²	C₂×SD₁₆	1	6	yes	—
34	#1039	—	8	3	no	—
35	C₂×D₂₀	—	10	18	no	—
36	C₂×Dic₁₀	—	10	18	no	—
37	C₂×C₅×C₈	D₂₀	1	36	yes	—
38	C₄²×C₅	C₂²×D₅	1	6	yes	—
39	C₄⋊Dic₅	—	10	36	no	—
40	#677	—	20	72	no	—
41	#498	—	25	3	no	—
42	#492	—	25	3	no	—
43	C₈²	C₅⋊D₅	1	1	yes	characteristic, complemented
44	C₂×C₅²	#492	1	3	yes	—
45	C₅⋊D₅	—	16	1	no	—
46	C₂²×D₅	—	20	6	no	—
47	C₂×C₄×C₅	C₂×D₂₀	1	18	yes	—
48	C₂×Dic₅	—	20	18	no	—
49	D₂₀	—	20	36	no	—
50	Dic₁₀	—	20	36	no	—
51	C₅×C₈	#677	1	72	yes	complemented
52	C₂×SD₁₆	—	50	6	no	—
53	C₄⋊Q₈	—	25	3	no	—
54	#127	—	25	1	no	—
55	#106	—	50	6	no	—
56	C₄×C₈	C₂×C₅⋊D₅	1	3	yes	—
57	C₅²	#1705	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₅	#4020	1	6	yes	—
59	C₂×D₅	—	40	18	no	—
60	C₄×C₅	C₂×#677	1	36	yes	—
61	Dic₅	—	40	18	no	—
62	C₂×Q₈	—	50	3	no	—
63	C₂×D₄	—	50	3	no	—
64	SD₁₆	—	100	12	no	—
65	C₂×C₈	C₄⋊(C₅⋊D₅)	1	6	yes	—
66	C₄⋊C₄	—	50	6	no	—
67	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
68	C₂×C₅	#64982	1	18	yes	—
69	D₅	—	80	6	no	—
70	C₂³	—	100	1	no	—
71	Q₈	—	100	6	no	—
72	D₄	—	100	6	no	—
73	C₂×C₄	—	100	3	no	—
74	C₂×C₄	C₂×C₄⋊(C₅⋊D₅)	1	3	yes	—
75	C₈	#87800	1	12	yes	complemented
76	C₅	#10598850	1	6	yes	complemented
77	C₂²	—	200	3	no	—
78	C₂²	#11717194	1	1	yes	characteristic
79	C₄	—	200	3	no	—
80	C₄	C₂×#87800	1	6	yes	—
81	C₂	—	400	1	no	—
82	C₂	#13102556	1	3	yes	—

	subgroup	G/H
center	C₂²	#11717194
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×C₈²	C₂
socle	C₂²×C₅²	#127
prime radical	#11717194	C₂²
hypercenter	C₈²	C₅⋊D₅
hypocenter	C₅²	#1705
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #13102543	C₂	semidirect
3. C₅²×C₈²	C₂	semidirect
18. C₅×C₈²	D₅	semidirect
24. C₅²×C₈	SD₁₆	semidirect
43. C₈²	C₅⋊D₅	semidirect
51. C₅×C₈	#677	semidirect
57. C₅²	#1705	semidirect
75. C₈	#87800	semidirect
76. C₅	#10598850	semidirect

Maximal subgroups

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

subgroup	index
#13102556	2
#13102543	2
C₅²×C₈²	2
#10598850	5