Subgroups — #46831088

Order 34624.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₄₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#31772697	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₅₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#14524194	—	2	1	no	—
5	#22575839	—	2	1	no	—
6	Q₈×D₅₄₁	—	2	1	no	—
7	Dic₂₁₆₄	C₄	1	1	yes	characteristic, complemented
8	#22575826	—	2	1	no	—
9	C₈×D₅₄₁	C₂²	1	1	yes	characteristic
10	Q₁₆×C₅₄₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#14524194	—	4	1	no	—
12	Dic₁₀₈₂	—	2	1	no	—
13	Dic₁₀₈₂	—	4	1	no	—
14	C₄×D₅₄₁	—	4	1	no	—
15	C₄×D₅₄₁	D₄	1	1	yes	characteristic
16	#16864513	D₄	1	1	yes	characteristic
17	Q₈×C₅₄₁	—	2	1	no	—
18	C₈×C₅₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂×D₅₄₁	D₈	1	1	yes	characteristic
20	#14524194	—	8	1	no	—
21	C₄×C₅₄₁	—	4	1	no	—
22	C₄×C₅₄₁	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₅₄₁	—	4	1	no	—
24	Dic₅₄₁	SD₁₆	1	1	yes	characteristic
25	C₂×C₅₄₁	#102	1	1	yes	characteristic
26	D₅₄₁	SD₃₂	1	1	yes	characteristic
27	D₅₄₁	Q₃₂	1	1	yes	characteristic
28	C₅₄₁	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	541	1	no	maximal, Sylow
30	C₂×Q₁₆	—	541	1	no	—
31	C₂×C₁₆	—	541	1	no	—
32	#107	—	541	1	no	—
33	C₂×Q₈	—	1082	1	no	—
34	Q₁₆	—	541	1	no	—
35	Q₁₆	—	1082	1	no	—
36	Q₁₆	#14524194	1	1	yes	characteristic, complemented
37	C₂×C₈	—	541	1	no	—
38	C₄⋊C₄	—	1082	1	no	—
39	C₁₆	—	1082	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	1082	1	no	—
42	Q₈	—	2164	1	no	—
43	C₂×C₄	—	541	1	no	—
44	C₂×C₄	—	2164	1	no	—
45	C₂×C₄	—	2164	1	no	—
46	C₈	—	541	1	no	—
47	C₈	C₂×#14524194	1	1	yes	characteristic
48	C₂²	—	541	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	541	1	no	—
51	C₄	—	2164	1	no	—
52	C₄	—	4328	1	no	—
53	C₄	C₂²⋊#14524194	1	1	yes	characteristic
54	C₂	—	541	1	no	—
55	C₂	—	541	1	no	—
56	C₂	#31772690	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31772690
commutator	C₈×C₅₄₁	C₂×C₄
Frattini	C₈	C₂×#14524194
Fitting	Q₁₆×C₅₄₁	C₄
socle	C₂×C₅₄₁	#102
prime radical	C₂×D₅₄₁	D₈
cosocle	C₈×D₅₄₁	C₂²
hypercenter	Q₁₆	#14524194
hypocenter	C₅₄₁	#358
LCS	G ▷ C₈×C₅₄₁ ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
7. Dic₂₁₆₄	C₄	semidirect
10. Q₁₆×C₅₄₁	C₄	semidirect
28. C₅₄₁	#358	semidirect
36. Q₁₆	#14524194	semidirect

Maximal subgroups

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

subgroup	index
D₅₄₁⋊C₁₆	2
#31772697	2
Q₁₆×D₅₄₁	2
#358	541