Subgroups — #35212628

Order 20864.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104669	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₁₆₃	—	2	2	no	—
4	#19104669	—	2	2	no	—
5	#19104669	C₄	1	4	yes	complemented
6	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₆₃	—	2	2	no	—
8	C₂×C₁₆×C₁₆₃	C₄	1	2	yes	complemented
9	C₂²×Dic₁₆₃	—	4	2	no	—
10	C₄⋊Dic₁₆₃	—	2	4	no	—
11	C₄⋊Dic₁₆₃	—	4	2	no	—
12	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
16	C₂×C₈×C₁₆₃	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₆₃	—	2	2	no	—
18	C₂×Dic₁₆₃	—	4	4	no	—
19	C₂×Dic₁₆₃	—	8	2	no	—
20	C₂×Dic₁₆₃	—	8	2	no	—
21	C₂³×C₁₆₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
25	C₂×C₄×C₁₆₃	Q₁₆	1	1	yes	characteristic
26	C₈×C₁₆₃	C₄²	1	1	yes	characteristic
27	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₁₆₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₆₃	#102	1	2	yes	—
30	C₂²×C₁₆₃	#107	1	1	yes	characteristic
31	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
32	C₂²×C₁₆₃	SD₃₂	1	2	yes	—
33	C₂²×C₁₆₃	Q₃₂	1	1	yes	characteristic
34	C₄×C₁₆₃	#95	1	1	yes	characteristic
35	C₄×C₁₆₃	#103	1	2	yes	—
36	C₄×C₁₆₃	#106	1	1	yes	characteristic
37	Dic₁₆₃	—	8	4	no	—
38	C₂×C₁₆₃	#340	1	1	yes	characteristic
39	C₂×C₁₆₃	#357	1	2	yes	—
40	C₂×C₁₆₃	#358	1	2	yes	—
41	C₂×C₁₆₃	#366	1	1	yes	characteristic
42	C₂×C₁₆₃	#367	1	1	yes	characteristic
43	C₁₆₃	#1380	1	1	yes	characteristic, complemented, Sylow
44	#1380	—	163	1	no	maximal, Sylow
45	C₂²×C₁₆	D₁₆₃	1	1	yes	characteristic
46	C₂×#107	—	163	2	no	—
47	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	326	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₁₆₃	1	2	yes	complemented
51	#107	—	326	2	no	—
52	#107	—	163	4	no	—
53	C₂²×C₄	—	652	2	no	—
54	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
55	C₂×C₈	C₄×D₁₆₃	1	2	yes	—
56	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
58	C₂×C₈	Dic₃₂₆	1	1	yes	characteristic
59	C₄⋊C₄	—	326	4	no	—
60	C₄⋊C₄	—	652	2	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₁₃₀₄	1	1	yes	characteristic
63	C₂×C₄	—	1304	2	no	—
64	C₂×C₄	—	1304	2	no	—
65	C₂×C₄	—	652	4	no	—
66	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₁₆₃	1	1	yes	characteristic
68	C₂×C₄	#14579688	1	2	yes	—
69	C₂×C₄	Dic₆₅₂	1	1	yes	characteristic
70	C₈	C₄×Dic₁₆₃	1	1	yes	characteristic
71	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
72	C₈	Dic₁₆₃⋊C₄	1	2	yes	—
73	C₂²	#19104657	1	2	yes	—
74	C₂²	#19104669	1	1	yes	characteristic
75	C₂²	D₂₆₀₈	1	1	yes	characteristic
76	C₂²	#19104680	1	2	yes	—
77	C₂²	Dic₁₃₀₄	1	1	yes	characteristic
78	C₄	—	1304	4	no	—
79	C₄	C₁₆₃⋊#95	1	1	yes	characteristic
80	C₄	#19104660	1	2	yes	—
81	C₄	#19104667	1	1	yes	characteristic
82	C₂	#23781428	1	1	yes	characteristic
83	C₂	#23781467	1	2	yes	—
84	C₂	#23781470	1	2	yes	—
85	C₂	#23781491	1	1	yes	characteristic
86	C₂	#23781493	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₃₀₄
commutator	C₈×C₁₆₃	C₄²
Frattini	C₂²×C₈	C₂×D₁₆₃
Fitting	C₂²×C₁₆×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	D₈
cosocle	C₂²×C₈×C₁₆₃	C₂²
hypercenter	C₂²×C₁₆	D₁₆₃
hypocenter	C₁₆₃	#1380
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #19104669	C₄	semidirect
8. C₂×C₁₆×C₁₆₃	C₄	semidirect
43. C₁₆₃	#1380	semidirect
50. C₂×C₁₆	Dic₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104669	2
C₂²×C₁₆×C₁₆₃	2
#1380	163