Subgroups — #45688335

Order 33344.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30659768	C₂	1	1	yes	maximal, characteristic
2	#30659766	C₂	1	1	yes	maximal, characteristic
3	#30659692	C₂	1	1	yes	maximal, characteristic
4	#22552373	—	2	1	no	—
5	C₂²⋊#14523438	—	2	1	no	—
6	C₂×C₂²⋊D₅₂₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅₂₁	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₅₂₁	—	2	1	no	—
9	#22552341	—	2	1	no	—
10	C₂²⋊C₄×C₅₂₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#14523438	—	4	1	no	—
12	#16861967	—	4	1	no	—
13	C₂²×D₅₂₁	D₄	1	1	yes	characteristic
14	C₂²⋊D₅₂₁	—	4	1	no	—
15	C₂×Dic₅₂₁	—	2	1	no	—
16	C₂×Dic₅₂₁	—	4	1	no	—
17	C₂×Dic₅₂₁	D₄	1	1	yes	characteristic
18	C₂³×C₅₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₂₁	—	2	1	no	—
20	C₂²×C₅₂₁	—	2	1	no	—
21	C₂²×C₅₂₁	—	4	1	no	—
22	C₂²×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₅₂₁	—	2	1	no	—
24	C₂×D₅₂₁	—	4	1	no	—
25	#14523438	—	8	1	no	—
26	C₄×C₅₂₁	—	4	1	no	—
27	Dic₅₂₁	—	2	1	no	—
28	Dic₅₂₁	—	4	1	no	—
29	C₂×C₅₂₁	—	2	1	no	—
30	C₂×C₅₂₁	—	4	1	no	—
31	C₂×C₅₂₁	#99	1	1	yes	characteristic
32	D₅₂₁	—	4	1	no	—
33	C₅₂₁	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	521	1	no	maximal, Sylow
35	#123	—	521	1	no	—
36	#100	—	521	1	no	—
37	#99	—	521	1	no	—
38	C₂×D₄	—	521	1	no	—
39	C₂²×C₄	—	521	1	no	—
40	M₁₆	—	1042	1	no	—
41	C₄⋊C₄	—	1042	1	no	—
42	C₂²⋊C₄	—	1042	1	no	—
43	C₂²⋊C₄	—	1042	1	no	—
44	C₂²⋊C₄	#14523438	1	1	yes	characteristic, complemented
45	C₂³	—	521	1	no	—
46	C₂³	C₂×#14523438	1	1	yes	characteristic
47	D₄	—	2084	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	521	1	no	—
50	C₂×C₄	—	1042	1	no	—
51	C₂×C₄	—	2084	1	no	—
52	C₂×C₄	—	2084	1	no	—
53	C₈	—	2084	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	1042	1	no	—
57	C₂²	—	2084	1	no	—
58	C₂²	C₂²⋊#14523438	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	1042	1	no	—
61	C₄	—	2084	1	no	—
62	C₄	—	4168	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	2084	1	no	—
66	C₂	#30659767	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30659767
commutator	C₂³×C₅₂₁	C₂×C₄
Frattini	C₂³	C₂×#14523438
Fitting	C₂²⋊C₄×C₅₂₁	C₄
socle	C₂×C₅₂₁	#99
prime radical, cosocle	C₂×C₂²⋊D₅₂₁	C₂²
hypercenter	C₂²⋊C₄	#14523438
hypocenter	C₅₂₁	#352
LCS	G ▷ C₂³×C₅₂₁ ▷ C₂²×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂²×Dic₅₂₁	C₄	semidirect
10. C₂²⋊C₄×C₅₂₁	C₄	semidirect
33. C₅₂₁	#352	semidirect
44. C₂²⋊C₄	#14523438	semidirect

Maximal subgroups

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

subgroup	index
#30659768	2
#30659766	2
#30659692	2
#352	521