Subgroups — #35212509

Order 20864.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104675	—	2	1	no	—
5	C₄×C₁₆₃⋊C₈	—	2	1	no	—
6	#135×C₁₆₃	C₄	1	1	yes	characteristic
7	#131×C₁₆₃	—	2	1	no	—
8	#108×C₁₆₃	C₄	1	1	yes	characteristic
9	#104×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₆₃⋊C₈	—	2	2	no	—
12	C₁₆₃⋊C₁₆	—	4	1	no	—
13	#41×C₁₆₃	—	2	1	no	—
14	Q₁₆×C₁₆₃	C₈	1	1	yes	characteristic, complemented
15	SD₁₆×C₁₆₃	—	2	1	no	—
16	D₈×C₁₆₃	C₈	1	1	yes	characteristic, complemented
17	M₁₆×C₁₆₃	—	2	1	no	—
18	M₁₆×C₁₆₃	—	4	1	no	—
19	C₂×C₈×C₁₆₃	—	4	1	no	—
20	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
22	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
23	C₁₆₃⋊C₈	—	2	4	no	—
24	Q₈×C₁₆₃	—	2	1	no	—
25	D₄×C₁₆₃	—	2	1	no	—
26	D₄×C₁₆₃	—	4	1	no	—
27	C₂×C₄×C₁₆₃	—	2	1	no	—
28	C₂×C₄×C₁₆₃	—	4	1	no	—
29	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₁₆₃	—	4	1	no	—
31	C₈×C₁₆₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
33	C₈×C₁₆₃	D₈	1	1	yes	characteristic
34	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₆₃	—	4	1	no	—
36	C₂²×C₁₆₃	#104	1	1	yes	characteristic
37	C₄×C₁₆₃	—	2	2	no	—
38	C₄×C₁₆₃	—	4	1	no	—
39	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₁₆₃	#102	1	1	yes	characteristic
41	C₂×C₁₆₃	—	2	1	no	—
42	C₂×C₁₆₃	—	8	1	no	—
43	C₂×C₁₆₃	D₄⋊C₈	1	1	yes	characteristic
44	C₁₆₃	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	163	1	no	maximal, Sylow
46	#443	D₁₆₃	1	1	yes	characteristic
47	#364	—	163	1	no	—
48	C₈²	—	163	1	no	—
49	#135	Dic₁₆₃	1	1	yes	characteristic
50	#131	—	2	1	no	—
51	M₃₂	—	326	1	no	—
52	#108	Dic₁₆₃	1	1	yes	characteristic
53	#104	—	2	1	no	—
54	C₄×C₈	—	326	1	no	—
55	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
56	#41	—	2	1	no	—
57	Q₁₆	C₁₆₃⋊C₈	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	C₁₆₃⋊C₈	1	1	yes	characteristic, complemented
60	M₁₆	—	2	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	326	2	no	—
64	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
66	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
67	C₁₆	—	652	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	326	4	no	—
76	C₈	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
77	C₈	#14579683	1	1	yes	characteristic
78	C₈	#14579684	1	1	yes	characteristic
79	C₈	#14579687	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#19104662	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
85	C₄	#19104656	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579684
commutator	C₈×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#443×C₁₆₃	C₂
socle	C₂×C₁₆₃	D₄⋊C₈
prime radical	#135×C₁₆₃	C₄
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#443	D₁₆₃
hypocenter	C₁₆₃	#1335
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #23781486	C₂	semidirect
2. C₈×C₁₆₃⋊C₈	C₂	semidirect
14. Q₁₆×C₁₆₃	C₈	semidirect
16. D₈×C₁₆₃	C₈	semidirect
44. C₁₆₃	#1335	semidirect
57. Q₁₆	C₁₆₃⋊C₈	semidirect
59. D₈	C₁₆₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23781486	2
C₈×C₁₆₃⋊C₈	2
#443×C₁₆₃	2
#1335	163