Subgroups — #46830617

Order 34624.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772651	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#31772574	C₂	1	1	yes	maximal, characteristic, complemented
4	#31772573	C₂	1	1	yes	maximal, characteristic, complemented
5	#31772571	C₂	1	1	yes	maximal, characteristic, complemented
6	#31772567	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#22575837	—	2	1	no	—
9	Q₈×D₅₄₁	—	2	1	no	—
10	Dic₂₁₆₄	C₂²	1	1	yes	characteristic
11	#22575826	—	2	1	no	—
12	#22575823	—	2	1	no	—
13	D₄₃₂₈	C₂²	1	1	yes	characteristic
14	C₈×D₅₄₁	C₂²	1	1	yes	characteristic
15	#22575811	C₂²	1	1	yes	characteristic
16	Q₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
17	Q₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
18	C₁₆×C₅₄₁	C₂²	1	1	yes	characteristic
19	Dic₁₀₈₂	—	2	1	no	—
20	Dic₁₀₈₂	—	4	1	no	—
21	D₂₁₆₄	—	2	1	no	—
22	D₂₁₆₄	—	4	1	no	—
23	C₄×D₅₄₁	—	4	1	no	—
24	C₄×D₅₄₁	—	4	1	no	—
25	C₄×D₅₄₁	D₄	1	1	yes	characteristic
26	#16864513	D₄	1	1	yes	characteristic
27	Q₈×C₅₄₁	—	2	1	no	—
28	Q₈×C₅₄₁	—	2	1	no	—
29	C₈×C₅₄₁	C₂³	1	1	yes	characteristic
30	C₂×D₅₄₁	—	4	1	no	—
31	C₂×D₅₄₁	D₈	1	1	yes	characteristic
32	C₄×C₅₄₁	—	4	1	no	—
33	C₄×C₅₄₁	—	4	1	no	—
34	C₄×C₅₄₁	C₂×D₄	1	1	yes	characteristic
35	Dic₅₄₁	—	4	1	no	—
36	Dic₅₄₁	D₈	1	1	yes	characteristic
37	C₂×C₅₄₁	C₂×D₈	1	1	yes	characteristic
38	D₅₄₁	—	2	1	no	—
39	D₅₄₁	—	8	1	no	—
40	C₅₄₁	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	541	1	no	maximal, Sylow
42	#135	—	541	1	no	—
43	C₂×Q₁₆	—	541	1	no	—
44	Q₃₂	—	541	1	no	—
45	Q₃₂	D₅₄₁	1	1	yes	characteristic, complemented
46	SD₃₂	—	541	1	no	—
47	SD₃₂	—	541	1	no	—
48	M₃₂	—	541	1	no	—
49	#41	—	1082	1	no	—
50	C₂×Q₈	—	1082	1	no	—
51	Q₁₆	—	541	1	no	—
52	Q₁₆	—	1082	1	no	—
53	Q₁₆	C₂×D₅₄₁	1	1	yes	characteristic
54	Q₁₆	C₂×D₅₄₁	1	1	yes	characteristic
55	SD₁₆	—	1082	1	no	—
56	D₈	—	541	1	no	—
57	C₂×C₈	—	541	1	no	—
58	C₁₆	—	541	1	no	—
59	C₁₆	C₂×D₅₄₁	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	1082	1	no	—
63	Q₈	—	2164	1	no	—
64	D₄	—	1082	1	no	—
65	D₄	—	2164	1	no	—
66	C₂×C₄	—	541	1	no	—
67	C₂×C₄	—	2164	1	no	—
68	C₂×C₄	—	2164	1	no	—
69	C₈	—	541	1	no	—
70	C₈	C₂²×D₅₄₁	1	1	yes	characteristic
71	C₂²	—	541	1	no	—
72	C₂²	—	2164	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	541	1	no	—
76	C₄	—	2164	1	no	—
77	C₄	D₄×D₅₄₁	1	1	yes	characteristic
78	C₂	—	1082	1	no	—
79	C₂	—	4328	1	no	—
80	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
prime radical	#31772651	C₂
hypercenter	Q₃₂	D₅₄₁
hypocenter	C₅₄₁	#510
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
2. Q₁₆×D₅₄₁	C₂	semidirect
3. #31772574	C₂	semidirect
4. #31772573	C₂	semidirect
5. #31772571	C₂	semidirect
6. #31772567	C₂	semidirect
7. Q₃₂×C₅₄₁	C₂	semidirect
40. C₅₄₁	#510	semidirect
45. 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
Q₁₆×D₅₄₁	2
#31772574	2
#31772573	2
#31772571	2
#31772567	2
Q₃₂×C₅₄₁	2
#510	541