Subgroups — #127×D₅₂₁ #45687970

Order 33344.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₅₂₁	C₂	1	6	yes	maximal, complemented
2	#30659705	C₂	1	1	yes	maximal, characteristic, complemented
3	#30659704	C₂	1	6	yes	maximal, complemented
4	C₄²×D₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₅₂₁	C₂²	1	4	yes	—
7	C₂×D₂₀₈₄	C₂²	1	6	yes	complemented
8	C₂×C₂²⋊D₅₂₁	C₂²	1	12	yes	complemented
9	D₄×D₅₂₁	—	2	24	no	—
10	C₂×C₄×D₅₂₁	C₂²	1	3	yes	—
11	C₄×Dic₅₂₁	C₂²	1	3	yes	complemented
12	C₂×D₄×C₅₂₁	C₂²	1	6	yes	complemented
13	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₅₂₁	—	2	24	no	—
15	C₂²×D₅₂₁	C₂³	1	4	yes	—
16	C₂²×D₅₂₁	C₂³	1	1	yes	characteristic
17	D₂₀₈₄	—	2	12	no	—
18	C₂²⋊D₅₂₁	—	2	24	no	—
19	C₂×Dic₅₂₁	C₂³	1	3	yes	—
20	C₄×D₅₂₁	D₄	1	12	yes	complemented
21	C₂³×C₅₂₁	C₂³	1	4	yes	—
22	D₄×C₅₂₁	—	2	12	no	—
23	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
24	C₂²×C₅₂₁	—	2	12	no	—
25	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₅₂₁	—	2	12	no	—
27	C₂×D₅₂₁	—	4	16	no	—
28	C₂×D₅₂₁	C₂×D₄	1	6	yes	—
29	C₄×C₅₂₁	C₂×D₄	1	6	yes	complemented
30	Dic₅₂₁	C₂×D₄	1	6	yes	complemented
31	C₂×C₅₂₁	—	4	4	no	—
32	C₂×C₅₂₁	C₂²×D₄	1	3	yes	—
33	D₅₂₁	—	4	4	no	—
34	D₅₂₁	#127	1	4	yes	direct product factor
35	C₅₂₁	C₂×#127	1	1	yes	characteristic, complemented, Sylow
36	C₂×#127	—	521	1	no	maximal, Sylow
37	C₂²×D₄	—	521	6	no	—
38	#127	—	521	6	no	—
39	#127	—	521	1	no	—
40	#127	D₅₂₁	1	1	yes	characteristic, direct product factor
41	C₂×C₄²	—	521	1	no	—
42	C₂⁴	—	521	4	no	—
43	C₂×D₄	—	521	12	no	—
44	C₂×D₄	—	521	6	no	—
45	C₂×D₄	—	1042	24	no	—
46	C₂×D₄	C₂×D₅₂₁	1	6	yes	complemented
47	C₂²×C₄	—	521	3	no	—
48	C₄²	—	521	3	no	—
49	C₄²	C₂×D₅₂₁	1	1	yes	characteristic, complemented
50	C₂³	—	521	1	no	—
51	C₂³	—	521	4	no	—
52	C₂³	—	1042	24	no	—
53	C₂³	C₂²×D₅₂₁	1	4	yes	—
54	D₄	—	2	12	no	—
55	D₄	—	1042	24	no	—
56	D₄	—	1042	12	no	—
57	C₂×C₄	—	521	3	no	—
58	C₂×C₄	—	521	12	no	—
59	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
60	C₂²	—	2	12	no	—
61	C₂²	—	521	6	no	—
62	C₂²	—	1042	12	no	—
63	C₂²	—	2084	16	no	—
64	C₂²	C₂³×D₅₂₁	1	1	yes	characteristic
65	C₄	—	521	6	no	—
66	C₄	D₄×D₅₂₁	1	6	yes	complemented
67	C₂	—	4	4	no	—
68	C₂	—	521	4	no	—
69	C₂	—	2084	4	no	—
70	C₂	C₂×D₄×D₅₂₁	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₂₁
commutator, socle, cosocle	C₂²×C₅₂₁	C₂⁴
Fitting	#127×C₅₂₁	C₂
hypercenter	#127	D₅₂₁
hypocenter	C₅₂₁	C₂×#127
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₅₂₁	C₂	semidirect
2. #30659705	C₂	semidirect
3. #30659704	C₂	semidirect
4. C₄²×D₅₂₁	C₂	semidirect
5. #127×C₅₂₁	C₂	semidirect
7. C₂×D₂₀₈₄	C₂²	semidirect
8. C₂×C₂²⋊D₅₂₁	C₂²	semidirect
11. C₄×Dic₅₂₁	C₂²	semidirect
12. C₂×D₄×C₅₂₁	C₂²	semidirect
13. C₄²×C₅₂₁	C₂²	semidirect
20. C₄×D₅₂₁	D₄	semidirect
29. C₄×C₅₂₁	C₂×D₄	semidirect
30. Dic₅₂₁	C₂×D₄	semidirect
34. D₅₂₁	#127	direct (N × H)
35. C₅₂₁	C₂×#127	semidirect
40. #127	D₅₂₁	direct (N × H)
46. C₂×D₄	C₂×D₅₂₁	semidirect
49. C₄²	C₂×D₅₂₁	semidirect
66. C₄	D₄×D₅₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₅₂₁	2
#30659705	2
#30659704	2
C₄²×D₅₂₁	2
#127×C₅₂₁	2
C₂×#127	521

Subgroups — #127×D521 #45687970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #127×D₅₂₁ #45687970