Subgroups — #46830980

Order 34624.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772674	C₂	1	2	yes	maximal, complemented
2	#31772656	C₂	1	6	yes	maximal, complemented
3	#31772650	C₂	1	3	yes	maximal, complemented
4	C₂×D₄₃₂₈	C₂	1	3	yes	maximal, complemented
5	#131×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575837	—	2	2	no	—
7	#22575836	C₂²	1	6	yes	complemented
8	C₂×D₂₁₆₄	C₂²	1	6	yes	complemented
9	D₄×D₅₄₁	—	2	6	no	—
10	Dic₂₁₆₄	C₂²	1	1	yes	characteristic, complemented
11	#22575825	C₂²	1	6	yes	complemented
12	D₄₃₂₈	C₂²	1	3	yes	complemented
13	D₄₃₂₈	C₂²	1	6	yes	complemented
14	#41×C₅₄₁	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₄₁	C₂²	1	3	yes	complemented
16	C₂×C₈×C₅₄₁	C₂²	1	3	yes	complemented
17	C₂²×D₅₄₁	—	2	6	no	—
18	Dic₁₀₈₂	C₂³	1	2	yes	—
19	D₂₁₆₄	—	2	6	no	—
20	D₂₁₆₄	C₂³	1	6	yes	—
21	C₂²⋊D₅₄₁	—	2	6	no	—
22	C₄×D₅₄₁	—	2	6	no	—
23	Q₈×C₅₄₁	D₄	1	1	yes	characteristic
24	D₄×C₅₄₁	D₄	1	3	yes	—
25	C₂×C₄×C₅₄₁	C₂³	1	3	yes	—
26	C₈×C₅₄₁	C₂³	1	1	yes	characteristic
27	C₈×C₅₄₁	C₂³	1	3	yes	—
28	C₂²×C₅₄₁	C₂×D₄	1	3	yes	—
29	C₂×D₅₄₁	—	2	6	no	—
30	C₂×D₅₄₁	—	8	6	no	—
31	C₄×C₅₄₁	C₂×D₄	1	3	yes	—
32	C₄×C₅₄₁	C₂⁴	1	1	yes	characteristic
33	Dic₅₄₁	—	2	2	no	—
34	C₂×C₅₄₁	—	2	3	no	—
35	C₂×C₅₄₁	C₂²×D₄	1	1	yes	characteristic
36	D₅₄₁	—	4	6	no	—
37	C₅₄₁	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	541	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	541	2	no	—
40	AGL(1,ℤ/8)	—	541	6	no	—
41	#135	—	541	3	no	—
42	C₂×D₈	—	541	3	no	—
43	#131	D₅₄₁	1	1	yes	characteristic, complemented
44	#41	—	541	6	no	—
45	#41	—	1082	2	no	—
46	#41	C₂×D₅₄₁	1	1	yes	characteristic
47	C₂×D₄	—	541	6	no	—
48	C₂×D₄	—	1082	6	no	—
49	Q₁₆	—	541	1	no	—
50	SD₁₆	—	541	6	no	—
51	D₈	—	541	3	no	—
52	D₈	—	541	6	no	—
53	M₁₆	C₂×D₅₄₁	1	3	yes	complemented
54	C₂×C₈	C₂×D₅₄₁	1	3	yes	complemented
55	C₂³	—	1082	6	no	—
56	Q₈	—	541	2	no	—
57	Q₈	D₂₁₆₄	1	1	yes	characteristic
58	D₄	—	541	6	no	—
59	D₄	—	1082	6	no	—
60	D₄	—	1082	6	no	—
61	D₄	D₂₁₆₄	1	3	yes	—
62	C₂×C₄	—	1082	6	no	—
63	C₂×C₄	C₂²×D₅₄₁	1	3	yes	—
64	C₈	C₂²×D₅₄₁	1	1	yes	characteristic
65	C₈	C₂²×D₅₄₁	1	3	yes	—
66	C₂²	—	1082	6	no	—
67	C₂²	—	4328	6	no	—
68	C₂²	C₂×D₂₁₆₄	1	3	yes	—
69	C₄	—	1082	2	no	—
70	C₄	C₂×D₂₁₆₄	1	3	yes	—
71	C₄	C₂³×D₅₄₁	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	2164	6	no	—
74	C₂	C₂²×D₂₁₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₁₆₄
commutator, cosocle	C₄×C₅₄₁	C₂⁴
Frattini	C₄	C₂³×D₅₄₁
Fitting	#131×C₅₄₁	C₂
socle	C₂×C₅₄₁	C₂²×D₄
hypercenter	#131	D₅₄₁
hypocenter	C₅₄₁	#576
LCS	G ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #31772674	C₂	semidirect
2. #31772656	C₂	semidirect
3. #31772650	C₂	semidirect
4. C₂×D₄₃₂₈	C₂	semidirect
5. #131×C₅₄₁	C₂	semidirect
7. #22575836	C₂²	semidirect
8. C₂×D₂₁₆₄	C₂²	semidirect
10. Dic₂₁₆₄	C₂²	semidirect
11. #22575825	C₂²	semidirect
12. D₄₃₂₈	C₂²	semidirect
13. D₄₃₂₈	C₂²	semidirect
15. M₁₆×C₅₄₁	C₂²	semidirect
16. C₂×C₈×C₅₄₁	C₂²	semidirect
37. C₅₄₁	#576	semidirect
43. #131	D₅₄₁	semidirect
53. M₁₆	C₂×D₅₄₁	semidirect
54. C₂×C₈	C₂×D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772674	2
#31772656	2
#31772650	2
C₂×D₄₃₂₈	2
#131×C₅₄₁	2
#576	541