Subgroups — #46829986

Order 34624.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772650	C₂	1	1	yes	maximal, characteristic
2	#31772560	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575836	—	2	1	no	—
5	Dic₂₁₆₄	C₄	1	1	yes	characteristic, complemented
6	#22575825	—	2	1	no	—
7	D₄₃₂₈	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₅₄₁	—	2	1	no	—
9	C₂×C₈×C₅₄₁	C₂²	1	1	yes	characteristic
10	C₁₆×C₅₄₁	—	2	1	no	—
11	Dic₁₀₈₂	—	2	1	no	—
12	D₂₁₆₄	—	2	1	no	—
13	C₂²⋊D₅₄₁	—	4	1	no	—
14	C₂×Dic₅₄₁	—	4	1	no	—
15	C₄×D₅₄₁	—	4	1	no	—
16	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
17	C₈×C₅₄₁	C₂×C₄	1	1	yes	characteristic
18	C₈×C₅₄₁	D₄	1	1	yes	characteristic
19	C₂²×C₅₄₁	D₈	1	1	yes	characteristic
20	C₂×D₅₄₁	—	4	1	no	—
21	C₄×C₅₄₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₅₄₁	SD₁₆	1	1	yes	characteristic
23	Dic₅₄₁	—	4	1	no	—
24	Dic₅₄₁	—	8	1	no	—
25	C₂×C₅₄₁	—	2	1	no	—
26	C₂×C₅₄₁	#102	1	1	yes	characteristic
27	D₅₄₁	—	8	1	no	—
28	C₅₄₁	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	541	1	no	maximal, Sylow
30	#135	—	541	1	no	—
31	M₃₂	D₅₄₁	1	1	yes	characteristic, complemented
32	#106	—	541	1	no	—
33	#41	—	1082	1	no	—
34	Q₁₆	—	541	1	no	—
35	SD₁₆	—	1082	1	no	—
36	D₈	—	541	1	no	—
37	C₂×C₈	C₂×D₅₄₁	1	1	yes	characteristic
38	C₄⋊C₄	—	1082	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	1082	1	no	—
41	D₄	—	1082	1	no	—
42	D₄	—	2164	1	no	—
43	C₂×C₄	—	2164	1	no	—
44	C₂×C₄	—	2164	1	no	—
45	C₂×C₄	D₂₁₆₄	1	1	yes	characteristic
46	C₈	C₄×D₅₄₁	1	1	yes	characteristic
47	C₈	C₂²⋊D₅₄₁	1	1	yes	characteristic
48	C₂²	—	2164	1	no	—
49	C₂²	D₄₃₂₈	1	1	yes	characteristic
50	C₄	—	2164	1	no	—
51	C₄	—	4328	1	no	—
52	C₄	#22575813	1	1	yes	characteristic
53	C₄	#22575825	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	4328	1	no	—
56	C₂	#31772550	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31772550
commutator	C₈×C₅₄₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₅₄₁
Fitting	M₃₂×C₅₄₁	C₂
socle	C₂×C₅₄₁	#102
prime radical	#31772650	C₂
cosocle	C₂×C₈×C₅₄₁	C₂²
hypercenter	M₃₂	D₅₄₁
hypocenter	C₅₄₁	#360
LCS	G ▷ C₈×C₅₄₁ ▷ C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #31772560	C₂	semidirect
3. M₃₂×C₅₄₁	C₂	semidirect
5. Dic₂₁₆₄	C₄	semidirect
7. D₄₃₂₈	C₄	semidirect
28. C₅₄₁	#360	semidirect
31. M₃₂	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
#31772560	2
M₃₂×C₅₄₁	2
#360	541