Subgroups — #44566916

Order 32192.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₅₀₃	C₂	1	4	yes	maximal, complemented
2	#30632835	C₂	1	4	yes	maximal, complemented
3	#30632832	C₂	1	4	yes	maximal, complemented
4	C₄×D₂₀₁₂	C₂	1	2	yes	maximal, complemented
5	#127×C₅₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₅₀₃	C₂²	1	4	yes	complemented
7	C₂×D₂₀₁₂	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₅₀₃	C₂²	1	8	yes	complemented
9	D₄×D₅₀₃	—	2	16	no	—
10	C₂×C₄×D₅₀₃	C₂²	1	4	yes	complemented
11	C₄⋊Dic₅₀₃	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₅₀₃	C₂²	1	4	yes	complemented
13	(C₂×D₅₀₃)⋊C₄	C₂²	1	4	yes	complemented
14	C₂×D₄×C₅₀₃	C₂²	1	4	yes	complemented
15	C₂×D₄×C₅₀₃	C₂²	1	2	yes	complemented
16	C₄²×C₅₀₃	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₅₀₃	—	2	16	no	—
18	C₂²×D₅₀₃	—	4	4	no	—
19	C₂²×D₅₀₃	C₂³	1	4	yes	—
20	D₂₀₁₂	D₄	1	8	yes	complemented
21	C₂²⋊D₅₀₃	—	2	16	no	—
22	C₂×Dic₅₀₃	C₂³	1	4	yes	—
23	C₄×D₅₀₃	—	2	8	no	—
24	C₂³×C₅₀₃	C₂³	1	4	yes	—
25	D₄×C₅₀₃	—	2	8	no	—
26	D₄×C₅₀₃	—	4	2	no	—
27	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅₀₃	C₂³	1	2	yes	—
29	C₂²×C₅₀₃	—	2	8	no	—
30	C₂²×C₅₀₃	—	2	4	no	—
31	C₂²×C₅₀₃	C₂⁴	1	1	yes	characteristic
32	C₂×D₅₀₃	—	2	4	no	—
33	C₂×D₅₀₃	—	2	4	no	—
34	C₂×D₅₀₃	—	4	16	no	—
35	C₂×D₅₀₃	C₂×D₄	1	8	yes	complemented
36	C₄×C₅₀₃	—	2	1	no	—
37	C₄×C₅₀₃	C₂×D₄	1	4	yes	complemented
38	Dic₅₀₃	—	2	4	no	—
39	C₂×C₅₀₃	—	4	4	no	—
40	C₂×C₅₀₃	C₂²×D₄	1	2	yes	—
41	C₂×C₅₀₃	H(5,𝔽₂)	1	1	yes	characteristic
42	D₅₀₃	—	2	8	no	—
43	C₅₀₃	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	503	1	no	maximal, Sylow
45	C₂²×D₄	—	503	4	no	—
46	#127	D₅₀₃	1	1	yes	characteristic, complemented
47	#121	—	503	4	no	—
48	#120	—	503	4	no	—
49	C₄×D₄	—	503	2	no	—
50	C₂⁴	—	503	4	no	—
51	C₂×D₄	—	1006	16	no	—
52	C₂×D₄	—	503	8	no	—
53	C₂×D₄	—	503	2	no	—
54	C₂×D₄	C₂×D₅₀₃	1	4	yes	complemented
55	C₂×D₄	C₂×D₅₀₃	1	2	yes	complemented
56	C₂²×C₄	—	503	4	no	—
57	C₄⋊C₄	—	503	2	no	—
58	C₂²⋊C₄	—	503	4	no	—
59	C₂²⋊C₄	—	503	4	no	—
60	C₄²	C₂×D₅₀₃	1	1	yes	characteristic, complemented
61	C₂³	—	2012	4	no	—
62	C₂³	—	1006	16	no	—
63	C₂³	—	503	4	no	—
64	C₂³	C₂²×D₅₀₃	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	1006	16	no	—
68	D₄	—	503	8	no	—
69	C₂×C₄	—	1006	8	no	—
70	C₂×C₄	—	503	4	no	—
71	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₅₀₃	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2012	16	no	—
76	C₂²	—	1006	4	no	—
77	C₂²	—	1006	4	no	—
78	C₂²	—	503	8	no	—
79	C₂²	C₂³×D₅₀₃	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	1006	4	no	—
82	C₄	D₄×D₅₀₃	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	1006	8	no	—
85	C₂	C₂×D₄×D₅₀₃	1	2	yes	—
86	C₂	#30632909	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₃
commutator, socle, cosocle	C₂²×C₅₀₃	C₂⁴
Fitting	#127×C₅₀₃	C₂
hypercenter	#127	D₅₀₃
hypocenter	C₅₀₃	D₄²
LCS	G ▷ C₂²×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₅₀₃	C₂	semidirect
2. #30632835	C₂	semidirect
3. #30632832	C₂	semidirect
4. C₄×D₂₀₁₂	C₂	semidirect
5. #127×C₅₀₃	C₂	semidirect
6. C₂³×D₅₀₃	C₂²	semidirect
8. C₂×C₂²⋊D₅₀₃	C₂²	semidirect
10. C₂×C₄×D₅₀₃	C₂²	semidirect
11. C₄⋊Dic₅₀₃	C₂²	semidirect
12. C₂²⋊Dic₅₀₃	C₂²	semidirect
13. (C₂×D₅₀₃)⋊C₄	C₂²	semidirect
14. C₂×D₄×C₅₀₃	C₂²	semidirect
15. C₂×D₄×C₅₀₃	C₂²	semidirect
16. C₄²×C₅₀₃	C₂²	semidirect
20. D₂₀₁₂	D₄	semidirect
35. C₂×D₅₀₃	C₂×D₄	semidirect
37. C₄×C₅₀₃	C₂×D₄	semidirect
43. C₅₀₃	D₄²	semidirect
46. #127	D₅₀₃	semidirect
54. C₂×D₄	C₂×D₅₀₃	semidirect
55. C₂×D₄	C₂×D₅₀₃	semidirect
60. C₄²	C₂×D₅₀₃	semidirect
82. C₄	D₄×D₅₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₅₀₃	2
#30632835	2
#30632832	2
C₄×D₂₀₁₂	2
#127×C₅₀₃	2
D₄²	503