Subgroups — #35212594

Order 20864.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#23781418	C₂	1	1	yes	maximal, characteristic
3	#500×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₃⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₆₃⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₁₆₃	—	2	1	no	—
7	#107×C₁₆₃	—	2	1	no	—
8	#107×C₁₆₃	C₄	1	2	yes	—
9	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆₃⋊C₈	—	4	1	no	—
11	C₁₆₃⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₁₆₃	—	4	1	no	—
13	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₁₆₃	—	2	2	no	—
16	C₄⋊C₄×C₁₆₃	—	4	1	no	—
17	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
18	C₁₆₃⋊C₈	—	8	1	no	—
19	Q₈×C₁₆₃	—	8	2	no	—
20	C₂×C₄×C₁₆₃	—	4	2	no	—
21	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₆₃	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₆₃	—	2	1	no	—
25	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
26	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₃	#102	1	1	yes	characteristic
28	C₄×C₁₆₃	—	2	1	no	—
29	C₄×C₁₆₃	—	8	2	no	—
30	C₄×C₁₆₃	#101	1	1	yes	characteristic
31	C₄×C₁₆₃	#103	1	1	yes	characteristic
32	C₄×C₁₆₃	SD₃₂	1	1	yes	characteristic
33	C₄×C₁₆₃	Q₃₂	1	1	yes	characteristic
34	C₂×C₁₆₃	#332	1	1	yes	characteristic
35	C₂×C₁₆₃	#358	1	1	yes	characteristic
36	C₂×C₁₆₃	#361	1	1	yes	characteristic
37	C₁₆₃	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	163	1	no	maximal, Sylow
39	#500	D₁₆₃	1	1	yes	characteristic
40	C₄⋊C₁₆	—	163	1	no	—
41	#335	—	163	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₂×C₁₆	—	326	1	no	—
44	#107	—	2	1	no	—
45	#107	Dic₁₆₃	1	2	yes	—
46	C₄⋊C₈	—	326	1	no	—
47	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
48	C₂×Q₈	—	4	1	no	—
49	C₂×C₈	—	652	1	no	—
50	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
55	C₁₆	—	652	1	no	—
56	Q₈	—	8	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
59	C₂×C₄	#14579684	1	1	yes	characteristic
60	C₂×C₄	#14579687	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	1304	1	no	—
63	C₈	#14579686	1	1	yes	characteristic
64	C₈	#14579689	1	1	yes	characteristic
65	C₂²	#19104656	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	8	2	no	—
68	C₄	#19104654	1	1	yes	characteristic
69	C₄	#19104659	1	1	yes	characteristic
70	C₄	#19104678	1	1	yes	characteristic
71	C₄	#19104681	1	1	yes	characteristic
72	C₂	#23781411	1	1	yes	characteristic
73	C₂	#23781469	1	1	yes	characteristic
74	C₂	#23781478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19104656
commutator	C₂×C₈×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#500×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#102
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#500	D₁₆₃
hypocenter	C₁₆₃	#1365
LCS	G ▷ C₂×C₈×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
37. C₁₆₃	#1365	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₆₃⋊C₁₆)	2
#23781418	2
#500×C₁₆₃	2
#1365	163