Subgroups — #45687543

Order 33344.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₁₆₈	C₂	1	1	yes	maximal, characteristic
2	#30659670	C₂	1	2	yes	maximal, complemented
3	#30659628	C₂	1	2	yes	maximal, complemented
4	#30659618	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₈₄	C₂²	1	2	yes	—
7	C₂×C₄×D₅₂₁	—	2	2	no	—
8	D₄₁₆₈	C₄	1	4	yes	complemented
9	C₂×#16861960	C₂²	1	1	yes	characteristic
10	#22552341	—	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	C₂²×D₅₂₁	—	2	2	no	—
15	D₂₀₈₄	—	2	2	no	—
16	D₂₀₈₄	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₅₂₁	D₄	1	2	yes	—
18	C₄×D₅₂₁	—	4	4	no	—
19	#16861960	—	2	1	no	—
20	C₂×C₄×C₅₂₁	—	2	2	no	—
21	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
22	C₈×C₅₂₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₅₂₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₂₁	—	2	4	no	—
25	C₂×D₅₂₁	—	4	2	no	—
26	C₂×D₅₂₁	—	4	2	no	—
27	C₄×C₅₂₁	—	4	2	no	—
28	C₄×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅₂₁	#41	1	1	yes	characteristic
30	Dic₅₂₁	—	2	2	no	—
31	C₂×C₅₂₁	C₄×D₄	1	1	yes	characteristic
32	C₂×C₅₂₁	AGL(1,ℤ/8)	1	2	yes	—
33	D₅₂₁	—	4	4	no	—
34	C₅₂₁	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	521	1	no	maximal, Sylow
36	C₂×D₈	—	521	1	no	—
37	C₄×D₄	—	521	2	no	—
38	#106	D₅₂₁	1	1	yes	characteristic, complemented
39	#102	—	521	2	no	—
40	#97	—	521	1	no	—
41	C₂×D₄	—	521	2	no	—
42	C₂²×C₄	—	1042	2	no	—
43	D₈	—	521	4	no	—
44	C₂×C₈	—	521	1	no	—
45	C₂×C₈	C₂×D₅₂₁	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₅₂₁	1	2	yes	—
47	C₂²⋊C₄	—	1042	2	no	—
48	C₄²	—	521	1	no	—
49	C₂³	—	1042	2	no	—
50	D₄	—	521	4	no	—
51	D₄	—	1042	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	521	2	no	—
54	C₂×C₄	—	2084	4	no	—
55	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
56	C₈	—	1042	1	no	—
57	C₈	C₄×D₅₂₁	1	2	yes	complemented
58	C₂²	—	1042	4	no	—
59	C₂²	—	2084	2	no	—
60	C₂²	—	2084	2	no	—
61	C₂²	D₄×D₅₂₁	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	1042	2	no	—
64	C₄	C₂×C₄×D₅₂₁	1	1	yes	characteristic
65	C₄	#22552365	1	1	yes	characteristic
66	C₂	—	2084	4	no	—
67	C₂	#30659670	1	1	yes	characteristic
68	C₂	#30659735	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	C₂²×C₅₂₁	C₂×D₄
prime radical	C₂×D₄₁₆₈	C₂
cosocle	C₂×C₄×C₅₂₁	C₂³
hypercenter	#106	D₅₂₁
hypocenter	C₅₂₁	#442
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
2. #30659670	C₂	semidirect
3. #30659628	C₂	semidirect
4. #30659618	C₂	semidirect
5. #106×C₅₂₁	C₂	semidirect
8. D₄₁₆₈	C₄	semidirect
16. D₂₀₈₄	C₂×C₄	semidirect
22. C₈×C₅₂₁	C₂×C₄	semidirect
34. C₅₂₁	#442	semidirect
38. #106	D₅₂₁	semidirect
57. C₈	C₄×D₅₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₁₆₈	2
#30659670	2
#30659628	2
#30659618	2
#106×C₅₂₁	2
#442	521