Subgroups — #44567915

Order 32288.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30633333	C₂	1	2	yes	maximal, complemented
2	#30633332	C₂	1	3	yes	maximal, complemented
3	C₂×D₄₀₃₆	C₂	1	3	yes	maximal, complemented
4	D₄×D₁₀₀₉	C₂	1	6	yes	maximal, complemented
5	#41×C₁₀₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₀₉	C₂²	1	6	yes	complemented
7	Dic₂₀₁₈	C₂²	1	1	yes	characteristic, complemented
8	D₄₀₃₆	C₂²	1	6	yes	complemented
9	D₄₀₃₆	C₂²	1	3	yes	complemented
10	C₂²⋊D₁₀₀₉	C₂²	1	6	yes	complemented
11	C₄×D₁₀₀₉	C₂²	1	6	yes	complemented
12	Q₈×C₁₀₀₉	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₁₀₀₉	C₂²	1	3	yes	complemented
14	C₂×C₄×C₁₀₀₉	C₂²	1	3	yes	complemented
15	C₂²×C₁₀₀₉	C₂³	1	3	yes	—
16	C₂×D₁₀₀₉	—	4	6	no	—
17	C₂×D₁₀₀₉	C₂³	1	6	yes	—
18	C₄×C₁₀₀₉	C₂³	1	1	yes	characteristic
19	C₄×C₁₀₀₉	C₂³	1	3	yes	—
20	Dic₁₀₀₉	C₂³	1	2	yes	—
21	C₂×C₁₀₀₉	—	2	3	no	—
22	C₂×C₁₀₀₉	C₂⁴	1	1	yes	characteristic
23	D₁₀₀₉	—	2	6	no	—
24	C₁₀₀₉	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	1009	1	no	maximal, Sylow
26	#41	—	1009	2	no	—
27	#41	—	1009	3	no	—
28	#41	D₁₀₀₉	1	1	yes	characteristic, complemented
29	C₂×D₄	—	1009	6	no	—
30	C₂×D₄	—	1009	3	no	—
31	C₂³	—	1009	6	no	—
32	Q₈	—	1009	1	no	—
33	Q₈	C₂×D₁₀₀₉	1	1	yes	characteristic, complemented
34	D₄	—	1009	6	no	—
35	D₄	—	1009	6	no	—
36	D₄	—	1009	3	no	—
37	D₄	C₂×D₁₀₀₉	1	3	yes	complemented
38	C₂×C₄	—	1009	6	no	—
39	C₂×C₄	C₂×D₁₀₀₉	1	3	yes	complemented
40	C₂²	—	4036	6	no	—
41	C₂²	—	1009	6	no	—
42	C₂²	C₂²×D₁₀₀₉	1	3	yes	—
43	C₄	—	1009	2	no	—
44	C₄	C₂²×D₁₀₀₉	1	3	yes	—
45	C₄	C₂²×D₁₀₀₉	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	2018	6	no	—
48	C₂	C₂³×D₁₀₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₀₀₉
commutator, socle, cosocle	C₂×C₁₀₀₉	C₂⁴
Fitting	#41×C₁₀₀₉	C₂
hypercenter	#41	D₁₀₀₉
hypocenter	C₁₀₀₉	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₀₀₉ ▷ C₁₀₀₉
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #30633333	C₂	semidirect
2. #30633332	C₂	semidirect
3. C₂×D₄₀₃₆	C₂	semidirect
4. D₄×D₁₀₀₉	C₂	semidirect
5. #41×C₁₀₀₉	C₂	semidirect
6. C₂²×D₁₀₀₉	C₂²	semidirect
7. Dic₂₀₁₈	C₂²	semidirect
8. D₄₀₃₆	C₂²	semidirect
9. D₄₀₃₆	C₂²	semidirect
10. C₂²⋊D₁₀₀₉	C₂²	semidirect
11. C₄×D₁₀₀₉	C₂²	semidirect
12. Q₈×C₁₀₀₉	C₂²	semidirect
13. D₄×C₁₀₀₉	C₂²	semidirect
14. C₂×C₄×C₁₀₀₉	C₂²	semidirect
24. C₁₀₀₉	H(5,𝔽₂)	semidirect
28. #41	D₁₀₀₉	semidirect
33. Q₈	C₂×D₁₀₀₉	semidirect
37. D₄	C₂×D₁₀₀₉	semidirect
39. C₂×C₄	C₂×D₁₀₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30633333	2
#30633332	2
C₂×D₄₀₃₆	2
D₄×D₁₀₀₉	2
#41×C₁₀₀₉	2
H(5,𝔽₂)	1009