Subgroups — #45688249

Order 33344.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30659755	C₂	1	2	yes	maximal, complemented
2	#30659741	C₂	1	6	yes	maximal, complemented
3	#30659730	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₂₀₈₄	C₂	1	3	yes	maximal, complemented
5	#131×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22552364	C₂²	1	6	yes	—
7	#22552363	—	2	6	no	—
8	C₂×Dic₁₀₄₂	C₂²	1	6	yes	—
9	Q₈×D₅₂₁	—	2	2	no	—
10	Dic₂₀₈₄	C₂²	1	3	yes	—
11	Dic₂₀₈₄	C₂²	1	6	yes	—
12	#22552353	C₂²	1	6	yes	—
13	D₄₁₆₈	C₂²	1	1	yes	characteristic
14	#41×C₅₂₁	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₂₁	C₂²	1	3	yes	—
16	C₂×C₈×C₅₂₁	C₂²	1	3	yes	—
17	Dic₁₀₄₂	—	2	6	no	—
18	Dic₁₀₄₂	C₂³	1	6	yes	—
19	D₂₀₈₄	C₂³	1	2	yes	—
20	C₂²⋊D₅₂₁	—	2	6	no	—
21	C₂×Dic₅₂₁	—	2	6	no	—
22	C₄×D₅₂₁	—	2	6	no	—
23	Q₈×C₅₂₁	D₄	1	1	yes	characteristic
24	D₄×C₅₂₁	D₄	1	3	yes	—
25	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
26	C₈×C₅₂₁	C₂³	1	1	yes	characteristic
27	C₈×C₅₂₁	C₂³	1	3	yes	—
28	C₂²×C₅₂₁	C₂×D₄	1	3	yes	—
29	C₂×D₅₂₁	—	2	2	no	—
30	C₄×C₅₂₁	C₂×D₄	1	3	yes	—
31	C₄×C₅₂₁	C₂⁴	1	1	yes	characteristic
32	Dic₅₂₁	—	2	6	no	—
33	C₂×C₅₂₁	—	2	3	no	—
34	C₂×C₅₂₁	C₂²×D₄	1	1	yes	characteristic
35	D₅₂₁	—	4	2	no	—
36	C₅₂₁	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	521	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	521	2	no	—
39	#137	—	521	6	no	—
40	#135	—	521	3	no	—
41	C₂×Q₁₆	—	521	3	no	—
42	#131	D₅₂₁	1	1	yes	characteristic, complemented
43	#41	—	521	6	no	—
44	#41	—	1042	6	no	—
45	#41	C₂×D₅₂₁	1	1	yes	characteristic
46	C₂×Q₈	—	521	6	no	—
47	C₂×Q₈	—	1042	2	no	—
48	Q₁₆	—	521	3	no	—
49	Q₁₆	—	521	6	no	—
50	SD₁₆	—	521	6	no	—
51	D₈	—	521	1	no	—
52	M₁₆	C₂×D₅₂₁	1	3	yes	—
53	C₂×C₈	C₂×D₅₂₁	1	3	yes	—
54	Q₈	—	521	6	no	—
55	Q₈	—	1042	6	no	—
56	Q₈	D₂₀₈₄	1	1	yes	characteristic
57	D₄	—	521	2	no	—
58	D₄	—	1042	6	no	—
59	D₄	D₂₀₈₄	1	3	yes	—
60	C₂×C₄	—	1042	6	no	—
61	C₂×C₄	—	1042	6	no	—
62	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
63	C₈	C₂²×D₅₂₁	1	1	yes	characteristic
64	C₈	C₂²×D₅₂₁	1	3	yes	—
65	C₂²	—	1042	2	no	—
66	C₂²	C₂×D₂₀₈₄	1	3	yes	—
67	C₄	—	1042	6	no	—
68	C₄	C₂×D₂₀₈₄	1	3	yes	—
69	C₄	C₂³×D₅₂₁	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	2084	2	no	—
72	C₂	C₂²×D₂₀₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₀₈₄
commutator, cosocle	C₄×C₅₂₁	C₂⁴
Frattini	C₄	C₂³×D₅₂₁
Fitting	#131×C₅₂₁	C₂
socle	C₂×C₅₂₁	C₂²×D₄
hypercenter	#131	D₅₂₁
hypocenter	C₅₂₁	#578
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #30659755	C₂	semidirect
2. #30659741	C₂	semidirect
3. #30659730	C₂	semidirect
4. C₂×Dic₂₀₈₄	C₂	semidirect
5. #131×C₅₂₁	C₂	semidirect
36. C₅₂₁	#578	semidirect
42. #131	D₅₂₁	semidirect

Maximal subgroups

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

subgroup	index
#30659755	2
#30659741	2
#30659730	2
C₂×Dic₂₀₈₄	2
#131×C₅₂₁	2
#578	521