Subgroups — #44567098

Order 32192.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632911	C₂	1	2	yes	maximal, complemented
2	#30632897	C₂	1	6	yes	maximal, complemented
3	#30632886	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₂₀₁₂	C₂	1	3	yes	maximal, complemented
5	#131×C₅₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22548883	C₂²	1	6	yes	—
7	#22548882	—	2	6	no	—
8	C₂×Dic₁₀₀₆	C₂²	1	6	yes	—
9	Q₈×D₅₀₃	—	2	2	no	—
10	Dic₂₀₁₂	C₂²	1	3	yes	—
11	Dic₂₀₁₂	C₂²	1	6	yes	—
12	#22548872	C₂²	1	6	yes	—
13	D₄₀₂₄	C₂²	1	1	yes	characteristic
14	#41×C₅₀₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₀₃	C₂²	1	3	yes	—
16	C₂×C₈×C₅₀₃	C₂²	1	3	yes	—
17	Dic₁₀₀₆	—	2	6	no	—
18	Dic₁₀₀₆	C₂³	1	6	yes	—
19	D₂₀₁₂	C₂³	1	2	yes	—
20	C₂²⋊D₅₀₃	—	2	6	no	—
21	C₂×Dic₅₀₃	—	2	6	no	—
22	C₄×D₅₀₃	—	2	6	no	—
23	Q₈×C₅₀₃	D₄	1	1	yes	characteristic
24	D₄×C₅₀₃	D₄	1	3	yes	—
25	C₂×C₄×C₅₀₃	C₂³	1	3	yes	—
26	C₈×C₅₀₃	C₂³	1	1	yes	characteristic
27	C₈×C₅₀₃	C₂³	1	3	yes	—
28	C₂²×C₅₀₃	C₂×D₄	1	3	yes	—
29	C₂×D₅₀₃	—	2	2	no	—
30	C₄×C₅₀₃	C₂×D₄	1	3	yes	—
31	C₄×C₅₀₃	C₂⁴	1	1	yes	characteristic
32	Dic₅₀₃	—	2	6	no	—
33	C₂×C₅₀₃	—	2	3	no	—
34	C₂×C₅₀₃	C₂²×D₄	1	1	yes	characteristic
35	D₅₀₃	—	4	2	no	—
36	C₅₀₃	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	503	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	503	2	no	—
39	#137	—	503	6	no	—
40	#135	—	503	3	no	—
41	C₂×Q₁₆	—	503	3	no	—
42	#131	D₅₀₃	1	1	yes	characteristic, complemented
43	#41	—	1006	6	no	—
44	#41	—	503	6	no	—
45	#41	C₂×D₅₀₃	1	1	yes	characteristic
46	C₂×Q₈	—	1006	2	no	—
47	C₂×Q₈	—	503	6	no	—
48	Q₁₆	—	503	3	no	—
49	Q₁₆	—	503	6	no	—
50	SD₁₆	—	503	6	no	—
51	D₈	—	503	1	no	—
52	M₁₆	C₂×D₅₀₃	1	3	yes	—
53	C₂×C₈	C₂×D₅₀₃	1	3	yes	—
54	Q₈	—	1006	6	no	—
55	Q₈	—	503	6	no	—
56	Q₈	D₂₀₁₂	1	1	yes	characteristic
57	D₄	—	1006	6	no	—
58	D₄	—	503	2	no	—
59	D₄	D₂₀₁₂	1	3	yes	—
60	C₂×C₄	—	1006	6	no	—
61	C₂×C₄	—	1006	6	no	—
62	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
63	C₈	C₂²×D₅₀₃	1	1	yes	characteristic
64	C₈	C₂²×D₅₀₃	1	3	yes	—
65	C₂²	—	1006	2	no	—
66	C₂²	C₂×D₂₀₁₂	1	3	yes	—
67	C₄	—	1006	6	no	—
68	C₄	C₂×D₂₀₁₂	1	3	yes	—
69	C₄	C₂³×D₅₀₃	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	2012	2	no	—
72	C₂	C₂²×D₂₀₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₀₁₂
commutator, cosocle	C₄×C₅₀₃	C₂⁴
Frattini	C₄	C₂³×D₅₀₃
Fitting	#131×C₅₀₃	C₂
socle	C₂×C₅₀₃	C₂²×D₄
hypercenter	#131	D₅₀₃
hypocenter	C₅₀₃	#578
LCS	G ▷ C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #30632911	C₂	semidirect
2. #30632897	C₂	semidirect
3. #30632886	C₂	semidirect
4. C₂×Dic₂₀₁₂	C₂	semidirect
5. #131×C₅₀₃	C₂	semidirect
36. C₅₀₃	#578	semidirect
42. #131	D₅₀₃	semidirect

Maximal subgroups

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

subgroup	index
#30632911	2
#30632897	2
#30632886	2
C₂×Dic₂₀₁₂	2
#131×C₅₀₃	2
#578	503