Subgroups — #45687538

Order 33344.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₈₄	C₂	1	1	yes	maximal, characteristic
2	#30659673	C₂	1	2	yes	maximal
3	#30659631	C₂	1	2	yes	maximal
4	#30659618	C₂	1	1	yes	maximal, characteristic
5	#106×C₅₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄₂	C₂²	1	2	yes	—
7	Dic₂₀₈₄	C₄	1	4	yes	complemented
8	C₂×#16861960	C₂²	1	1	yes	characteristic
9	#22552343	—	2	2	no	—
10	C₄×Dic₅₂₁	—	2	2	no	—
11	C₄×Dic₅₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₅₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₅₂₁	C₂²	1	2	yes	—
14	Dic₁₀₄₂	—	2	2	no	—
15	Dic₁₀₄₂	C₂×C₄	1	4	yes	—
16	C₂×Dic₅₂₁	—	2	2	no	—
17	C₂×Dic₅₂₁	D₄	1	2	yes	—
18	#16861960	—	2	1	no	—
19	C₂×C₄×C₅₂₁	—	2	2	no	—
20	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
21	C₈×C₅₂₁	C₂×C₄	1	2	yes	—
22	C₂²×C₅₂₁	C₂×D₄	1	1	yes	characteristic
23	C₄×C₅₂₁	—	4	2	no	—
24	C₄×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₅₂₁	#41	1	1	yes	characteristic
26	Dic₅₂₁	—	2	4	no	—
27	Dic₅₂₁	—	2	2	no	—
28	C₂×C₅₂₁	C₄×D₄	1	1	yes	characteristic
29	C₂×C₅₂₁	#137	1	2	yes	—
30	C₅₂₁	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	521	1	no	maximal, Sylow
32	C₂×Q₁₆	—	521	1	no	—
33	C₄×Q₈	—	521	2	no	—
34	#106	D₅₂₁	1	1	yes	characteristic
35	#103	—	521	2	no	—
36	#97	—	521	1	no	—
37	C₂×Q₈	—	521	2	no	—
38	Q₁₆	—	521	4	no	—
39	C₂×C₈	—	521	1	no	—
40	C₂×C₈	C₂×D₅₂₁	1	1	yes	characteristic
41	C₄⋊C₄	—	1042	2	no	—
42	C₄⋊C₄	C₂×D₅₂₁	1	2	yes	—
43	C₄²	—	521	1	no	—
44	C₄²	—	1042	2	no	—
45	Q₈	—	521	4	no	—
46	Q₈	—	1042	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	521	2	no	—
49	C₂×C₄	—	1042	2	no	—
50	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
51	C₈	—	1042	1	no	—
52	C₈	C₄×D₅₂₁	1	2	yes	—
53	C₂²	D₄×D₅₂₁	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	1042	4	no	—
56	C₄	—	1042	2	no	—
57	C₄	C₂×C₄×D₅₂₁	1	1	yes	characteristic
58	C₄	#22552365	1	1	yes	characteristic
59	C₂	#30659670	1	1	yes	characteristic
60	C₂	#30659737	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₂₁
commutator	C₄×C₅₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₂₁
Fitting	#106×C₅₂₁	C₂
socle, prime radical	C₂²×C₅₂₁	C₂×D₄
cosocle	C₂×C₄×C₅₂₁	C₂³
hypercenter	#106	D₅₂₁
hypocenter	C₅₂₁	#441
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
7. Dic₂₀₈₄	C₄	semidirect
30. C₅₂₁	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₈₄	2
#30659673	2
#30659631	2
#30659618	2
#106×C₅₂₁	2
#441	521