Subgroups — #31772674

Order 17312.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575837	C₂	1	2	yes	maximal, complemented
2	#22575836	C₂	1	3	yes	maximal, complemented
3	C₂×D₂₁₆₄	C₂	1	3	yes	maximal, complemented
4	D₄×D₅₄₁	C₂	1	6	yes	maximal, complemented
5	#41×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₄₁	C₂²	1	6	yes	complemented
7	Dic₁₀₈₂	C₂²	1	1	yes	characteristic, complemented
8	D₂₁₆₄	C₂²	1	6	yes	complemented
9	D₂₁₆₄	C₂²	1	3	yes	complemented
10	C₂²⋊D₅₄₁	C₂²	1	6	yes	complemented
11	C₄×D₅₄₁	C₂²	1	6	yes	complemented
12	Q₈×C₅₄₁	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₅₄₁	C₂²	1	3	yes	complemented
14	C₂×C₄×C₅₄₁	C₂²	1	3	yes	complemented
15	C₂²×C₅₄₁	C₂³	1	3	yes	—
16	C₂×D₅₄₁	—	4	6	no	—
17	C₂×D₅₄₁	C₂³	1	6	yes	—
18	C₄×C₅₄₁	C₂³	1	3	yes	—
19	C₄×C₅₄₁	C₂³	1	1	yes	characteristic
20	Dic₅₄₁	C₂³	1	2	yes	—
21	C₂×C₅₄₁	—	2	3	no	—
22	C₂×C₅₄₁	C₂⁴	1	1	yes	characteristic
23	D₅₄₁	—	2	6	no	—
24	C₅₄₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	541	1	no	maximal, Sylow
26	#41	—	541	2	no	—
27	#41	—	541	3	no	—
28	#41	D₅₄₁	1	1	yes	characteristic, complemented
29	C₂×D₄	—	541	6	no	—
30	C₂×D₄	—	541	3	no	—
31	C₂³	—	541	6	no	—
32	Q₈	—	541	1	no	—
33	Q₈	C₂×D₅₄₁	1	1	yes	characteristic, complemented
34	D₄	—	541	6	no	—
35	D₄	—	541	6	no	—
36	D₄	—	541	3	no	—
37	D₄	C₂×D₅₄₁	1	3	yes	complemented
38	C₂×C₄	—	541	6	no	—
39	C₂×C₄	C₂×D₅₄₁	1	3	yes	complemented
40	C₂²	—	541	6	no	—
41	C₂²	—	2164	6	no	—
42	C₂²	C₂²×D₅₄₁	1	3	yes	—
43	C₄	—	541	2	no	—
44	C₄	C₂²×D₅₄₁	1	3	yes	—
45	C₄	C₂²×D₅₄₁	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	1082	6	no	—
48	C₂	C₂³×D₅₄₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅₄₁
commutator, socle, cosocle	C₂×C₅₄₁	C₂⁴
Fitting	#41×C₅₄₁	C₂
hypercenter	#41	D₅₄₁
hypocenter	C₅₄₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #22575837	C₂	semidirect
2. #22575836	C₂	semidirect
3. C₂×D₂₁₆₄	C₂	semidirect
4. D₄×D₅₄₁	C₂	semidirect
5. #41×C₅₄₁	C₂	semidirect
6. C₂²×D₅₄₁	C₂²	semidirect
7. Dic₁₀₈₂	C₂²	semidirect
8. D₂₁₆₄	C₂²	semidirect
9. D₂₁₆₄	C₂²	semidirect
10. C₂²⋊D₅₄₁	C₂²	semidirect
11. C₄×D₅₄₁	C₂²	semidirect
12. Q₈×C₅₄₁	C₂²	semidirect
13. D₄×C₅₄₁	C₂²	semidirect
14. C₂×C₄×C₅₄₁	C₂²	semidirect
24. C₅₄₁	H(5,𝔽₂)	semidirect
28. #41	D₅₄₁	semidirect
33. Q₈	C₂×D₅₄₁	semidirect
37. D₄	C₂×D₅₄₁	semidirect
39. C₂×C₄	C₂×D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#22575837	2
#22575836	2
C₂×D₂₁₆₄	2
D₄×D₅₄₁	2
#41×C₅₄₁	2
H(5,𝔽₂)	541