Subgroups — #46830608

Order 34624.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772651	C₂	1	2	yes	maximal, complemented
2	#31772571	C₂	1	2	yes	maximal, complemented
3	D₈₆₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575837	—	2	2	no	—
7	#22575823	—	2	2	no	—
8	D₄₃₂₈	C₂²	1	2	yes	—
9	C₈×D₅₄₁	C₂²	1	1	yes	characteristic
10	#22575811	C₂²	1	1	yes	characteristic
11	Q₁₆×C₅₄₁	C₂²	1	2	yes	—
12	C₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
13	D₂₁₆₄	—	2	2	no	—
14	D₂₁₆₄	—	4	2	no	—
15	C₄×D₅₄₁	—	4	2	no	—
16	C₄×D₅₄₁	D₄	1	1	yes	characteristic
17	#16864513	D₄	1	1	yes	characteristic
18	Q₈×C₅₄₁	—	2	2	no	—
19	C₈×C₅₄₁	C₂³	1	1	yes	characteristic
20	C₂×D₅₄₁	—	4	2	no	—
21	C₂×D₅₄₁	D₈	1	1	yes	characteristic
22	C₄×C₅₄₁	—	4	2	no	—
23	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
24	Dic₅₄₁	D₈	1	1	yes	characteristic
25	C₂×C₅₄₁	C₂×D₈	1	1	yes	characteristic
26	D₅₄₁	—	2	1	no	—
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	D₅₄₁	1	1	yes	characteristic, complemented
32	SD₃₂	—	541	2	no	—
33	D₁₆	—	541	1	no	—
34	C₂×C₁₆	—	541	1	no	—
35	#41	—	1082	2	no	—
36	Q₁₆	C₂×D₅₄₁	1	2	yes	—
37	SD₁₆	—	1082	2	no	—
38	D₈	—	541	2	no	—
39	C₂×C₈	—	541	1	no	—
40	C₁₆	—	541	1	no	—
41	C₁₆	C₂×D₅₄₁	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	1082	2	no	—
44	D₄	—	2164	2	no	—
45	C₂×C₄	—	541	1	no	—
46	C₂×C₄	—	2164	2	no	—
47	C₈	—	541	1	no	—
48	C₈	C₂²×D₅₄₁	1	1	yes	characteristic
49	C₂²	—	541	1	no	—
50	C₂²	—	2164	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	541	1	no	—
53	C₄	D₄×D₅₄₁	1	1	yes	characteristic
54	C₂	—	1082	1	no	—
55	C₂	—	4328	2	no	—
56	C₂	D₈×D₅₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₅₄₁
commutator, cosocle	C₈×C₅₄₁	C₂³
Frattini	C₈	C₂²×D₅₄₁
Fitting	Q₃₂×C₅₄₁	C₂
socle	C₂×C₅₄₁	C₂×D₈
hypercenter	Q₃₂	D₅₄₁
hypocenter	C₅₄₁	#508
LCS	G ▷ C₈×C₅₄₁ ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #31772651	C₂	semidirect
2. #31772571	C₂	semidirect
3. D₈₆₅₆	C₂	semidirect
4. C₁₆×D₅₄₁	C₂	semidirect
5. Q₃₂×C₅₄₁	C₂	semidirect
28. C₅₄₁	#508	semidirect
31. Q₃₂	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772651	2
#31772571	2
D₈₆₅₆	2
C₁₆×D₅₄₁	2
Q₃₂×C₅₄₁	2
#508	541