Subgroups — #35214130

Order 20864.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782070	C₂	1	1	yes	maximal, characteristic
2	#23782061	C₂	1	2	yes	maximal
3	#23782017	C₂	1	2	yes	maximal
4	C₈×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#494×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104735	C₂²	1	2	yes	—
7	#19104669	—	2	2	no	—
8	#19104666	—	2	4	no	—
9	#19104658	—	2	4	no	—
10	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₁₆₃	—	2	2	no	—
13	C₄⋊Q₈×C₁₆₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₂₆	—	2	2	no	—
16	C₂×C₁₆₃⋊C₈	D₄	1	2	yes	—
17	C₂×C₁₆₃⋊C₈	D₄	1	2	yes	—
18	C₄⋊Dic₁₆₃	—	2	2	no	—
19	C₄⋊Dic₁₆₃	—	2	2	no	—
20	C₂×Q₈×C₁₆₃	—	2	2	no	—
21	Q₁₆×C₁₆₃	—	4	4	no	—
22	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₆₃	—	2	2	no	—
24	C₄⋊C₄×C₁₆₃	—	2	2	no	—
25	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
26	Dic₃₂₆	—	4	4	no	—
27	C₂×Dic₁₆₃	—	4	2	no	—
28	C₁₆₃⋊C₈	—	2	2	no	—
29	C₁₆₃⋊C₈	Q₁₆	1	4	yes	complemented
30	Q₈×C₁₆₃	—	4	4	no	—
31	C₂×C₄×C₁₆₃	—	4	2	no	—
32	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₆₃	SD₁₆	1	4	yes	—
36	C₂²×C₁₆₃	#127	1	1	yes	characteristic
37	C₄×C₁₆₃	—	8	2	no	—
38	C₄×C₁₆₃	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₆₃	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₆₃	#135	1	2	yes	—
41	Dic₁₆₃	—	8	2	no	—
42	C₂×C₁₆₃	#492	1	1	yes	characteristic
43	C₂×C₁₆₃	#494	1	1	yes	characteristic
44	C₂×C₁₆₃	#495	1	1	yes	characteristic
45	C₁₆₃	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	163	1	no	maximal, Sylow
47	#500	—	163	1	no	—
48	#498	—	163	2	no	—
49	#494	D₁₆₃	1	1	yes	characteristic
50	#487	—	163	2	no	—
51	C₈²	—	163	1	no	—
52	C₂×Q₁₆	—	2	2	no	—
53	C₄⋊Q₈	—	163	2	no	—
54	C₄⋊Q₈	C₂×D₁₆₃	1	2	yes	—
55	#107	—	326	2	no	—
56	#106	—	326	4	no	—
57	#103	—	326	4	no	—
58	C₄×C₈	—	163	1	no	—
59	C₄×C₈	—	163	1	no	—
60	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	326	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	163	2	no	—
65	C₂×C₈	—	163	2	no	—
66	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	326	2	no	—
70	C₄⋊C₄	—	326	2	no	—
71	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	652	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	652	2	no	—
76	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
79	C₈	—	326	2	no	—
80	C₈	—	163	4	no	—
81	C₈	#14579687	1	4	yes	—
82	C₂²	#19104731	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1304	2	no	—
85	C₄	C₂×#14579687	1	2	yes	—
86	C₄	Q₁₆×D₁₆₃	1	2	yes	—
87	C₄	#19104758	1	2	yes	—
88	C₂	#23782037	1	1	yes	characteristic
89	C₂	#23782043	1	1	yes	characteristic
90	C₂	#23782050	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19104731
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#494×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#127
hypercenter	#494	D₁₆₃
hypocenter	C₁₆₃	#1707
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
29. C₁₆₃⋊C₈	Q₁₆	semidirect
45. C₁₆₃	#1707	semidirect

Maximal subgroups

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

subgroup	index
#23782070	2
#23782061	2
#23782017	2
C₈×C₁₆₃⋊C₈	2
#494×C₁₆₃	2
#1707	163