Subgroups — #35212724

Order 20864.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782097	C₂	1	1	yes	maximal, characteristic
2	#23781493	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104757	—	2	1	no	—
5	Dic₁₃₀₄	C₄	1	1	yes	characteristic, complemented
6	#19104680	—	2	1	no	—
7	D₂₆₀₈	C₄	1	1	yes	characteristic, complemented
8	#19104669	—	2	1	no	—
9	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₆₃	—	2	1	no	—
11	#14579699	—	4	1	no	—
12	Dic₆₅₂	—	2	1	no	—
13	#14579688	—	4	1	no	—
14	D₁₃₀₄	—	2	1	no	—
15	C₄⋊Dic₁₆₃	—	4	1	no	—
16	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
17	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
19	Dic₃₂₆	—	4	1	no	—
20	D₆₅₂	—	4	1	no	—
21	C₂²⋊D₁₆₃	—	8	1	no	—
22	C₂×Dic₁₆₃	—	8	1	no	—
23	C₄×D₁₆₃	—	8	1	no	—
24	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
25	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
28	C₂×D₁₆₃	—	8	1	no	—
29	C₄×C₁₆₃	#102	1	1	yes	characteristic
30	C₄×C₁₆₃	SD₃₂	1	1	yes	characteristic
31	Dic₁₆₃	—	8	1	no	—
32	Dic₁₆₃	—	16	1	no	—
33	C₂×C₁₆₃	—	2	1	no	—
34	C₂×C₁₆₃	#357	1	1	yes	characteristic
35	D₁₆₃	—	16	1	no	—
36	C₁₆₃	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	163	1	no	maximal, Sylow
38	#508	—	163	1	no	—
39	M₆₄	D₁₆₃	1	1	yes	characteristic, complemented
40	#367	—	163	1	no	—
41	#135	—	326	1	no	—
42	Q₃₂	—	163	1	no	—
43	SD₃₂	—	326	1	no	—
44	D₁₆	—	163	1	no	—
45	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
46	#107	—	326	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	652	1	no	—
49	Q₁₆	—	326	1	no	—
50	SD₁₆	—	652	1	no	—
51	D₈	—	326	1	no	—
52	C₂×C₈	D₆₅₂	1	1	yes	characteristic
53	C₄⋊C₄	—	652	1	no	—
54	C₁₆	C₄×D₁₆₃	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
56	Q₈	—	652	1	no	—
57	D₄	—	1304	1	no	—
58	D₄	—	652	1	no	—
59	C₂×C₄	—	1304	1	no	—
60	C₂×C₄	—	1304	1	no	—
61	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
62	C₈	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
63	C₈	#14579688	1	1	yes	characteristic
64	C₂²	—	1304	1	no	—
65	C₂²	D₂₆₀₈	1	1	yes	characteristic
66	C₄	—	1304	1	no	—
67	C₄	—	2608	1	no	—
68	C₄	#19104657	1	1	yes	characteristic
69	C₄	#19104680	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2608	1	no	—
72	C₂	#23781467	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781467
commutator	C₁₆×C₁₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	M₆₄×C₁₆₃	C₂
socle	C₂×C₁₆₃	#357
prime radical	#23782097	C₂
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	M₆₄	D₁₆₃
hypocenter	C₁₆₃	#1418
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #23781493	C₂	semidirect
3. M₆₄×C₁₆₃	C₂	semidirect
5. Dic₁₃₀₄	C₄	semidirect
7. D₂₆₀₈	C₄	semidirect
36. C₁₆₃	#1418	semidirect
39. M₆₄	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782097	2
#23781493	2
M₆₄×C₁₆₃	2
#1418	163