Subgroups — #46830607

Order 34624.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772650	C₂	1	2	yes	maximal, complemented
2	Dic₄₃₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#31772573	C₂	1	2	yes	maximal, complemented
4	D₈₆₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575836	—	2	2	no	—
7	Dic₂₁₆₄	C₂²	1	2	yes	—
8	#22575825	—	2	2	no	—
9	D₄₃₂₈	C₂²	1	2	yes	—
10	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
11	C₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
12	C₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
13	Dic₁₀₈₂	—	2	2	no	—
14	D₂₁₆₄	—	2	2	no	—
15	C₂²⋊D₅₄₁	—	4	2	no	—
16	C₄×D₅₄₁	—	4	2	no	—
17	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
18	C₈×C₅₄₁	D₄	1	1	yes	characteristic
19	C₈×C₅₄₁	C₂³	1	1	yes	characteristic
20	C₂²×C₅₄₁	D₈	1	1	yes	characteristic
21	C₂×D₅₄₁	—	4	2	no	—
22	C₄×C₅₄₁	D₈	1	1	yes	characteristic
23	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
24	Dic₅₄₁	—	4	2	no	—
25	C₂×C₅₄₁	—	2	1	no	—
26	C₂×C₅₄₁	C₂×D₈	1	1	yes	characteristic
27	D₅₄₁	—	8	2	no	—
28	C₅₄₁	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	541	1	no	maximal, Sylow
30	#135	—	541	2	no	—
31	Q₃₂	—	541	1	no	—
32	SD₃₂	—	541	2	no	—
33	D₁₆	—	541	1	no	—
34	C₂×C₁₆	D₅₄₁	1	1	yes	characteristic, complemented
35	#41	—	1082	2	no	—
36	Q₁₆	—	541	2	no	—
37	SD₁₆	—	1082	2	no	—
38	D₈	—	541	2	no	—
39	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
40	C₁₆	C₂×D₅₄₁	1	1	yes	characteristic
41	C₁₆	C₂×D₅₄₁	1	1	yes	characteristic
42	Q₈	—	1082	2	no	—
43	D₄	—	1082	2	no	—
44	D₄	—	2164	2	no	—
45	C₂×C₄	—	2164	2	no	—
46	C₂×C₄	D₂₁₆₄	1	1	yes	characteristic
47	C₈	D₂₁₆₄	1	1	yes	characteristic
48	C₈	C₂²×D₅₄₁	1	1	yes	characteristic
49	C₂²	—	2164	2	no	—
50	C₂²	D₄₃₂₈	1	1	yes	characteristic
51	C₄	—	2164	2	no	—
52	C₄	D₄₃₂₈	1	1	yes	characteristic
53	C₄	C₂×D₂₁₆₄	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	4328	2	no	—
56	C₂	C₂×D₄₃₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₃₂₈
commutator, cosocle	C₈×C₅₄₁	C₂³
Frattini	C₈	C₂²×D₅₄₁
Fitting	C₂×C₁₆×C₅₄₁	C₂
socle	C₂×C₅₄₁	C₂×D₈
hypercenter	C₂×C₁₆	D₅₄₁
hypocenter	C₅₄₁	#508
LCS	G ▷ C₈×C₅₄₁ ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #31772650	C₂	semidirect
2. Dic₄₃₂₈	C₂	semidirect
3. #31772573	C₂	semidirect
4. D₈₆₅₆	C₂	semidirect
5. C₂×C₁₆×C₅₄₁	C₂	semidirect
28. C₅₄₁	#508	semidirect
34. C₂×C₁₆	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772650	2
Dic₄₃₂₈	2
#31772573	2
D₈₆₅₆	2
C₂×C₁₆×C₅₄₁	2
#508	541