Subgroups — #35215938

Order 20864.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781713	C₂	1	3	yes	maximal
2	C₂×#19104653	C₂	1	3	yes	maximal
3	#338×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
5	C₂×#14579683	C₂²	1	3	yes	—
6	#19104670	—	2	6	no	—
7	#19104653	—	2	6	no	—
8	C₂×M₁₆×C₁₆₃	C₂²	1	3	yes	—
9	C₂×Dic₃₂₆	—	2	3	no	—
10	C₂×Dic₃₂₆	—	2	3	no	—
11	C₂²×Dic₁₆₃	Q₈	1	2	yes	—
12	#14579683	—	2	3	no	—
13	#14579683	D₄	1	6	yes	—
14	C₂×C₁₆₃⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₆₃	—	2	3	no	—
17	M₁₆×C₁₆₃	—	2	3	no	—
18	C₂×C₈×C₁₆₃	—	2	3	no	—
19	Dic₃₂₆	—	4	3	no	—
20	Dic₃₂₆	—	4	1	no	—
21	C₂×Dic₁₆₃	—	2	6	no	—
22	C₁₆₃⋊C₈	—	2	6	no	—
23	C₂³×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₆₃	#41	1	3	yes	—
26	C₈×C₁₆₃	—	4	3	no	—
27	C₂²×C₁₆₃	—	4	1	no	—
28	C₂²×C₁₆₃	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₁₆₃	#121	1	3	yes	—
30	C₄×C₁₆₃	#126	1	1	yes	characteristic
31	Dic₁₆₃	—	4	2	no	—
32	C₂×C₁₆₃	—	2	3	no	—
33	C₂×C₁₆₃	#396	1	1	yes	characteristic
34	C₁₆₃	#2070	1	1	yes	characteristic, complemented, Sylow
35	#2070	—	163	1	no	maximal, Sylow
36	#430	—	163	3	no	—
37	C₂×#101	—	163	3	no	—
38	#338	D₁₆₃	1	1	yes	characteristic
39	C₂²×Q₈	—	163	1	no	—
40	C₂×M₁₆	—	163	3	no	—
41	C₂×M₁₆	C₂×D₁₆₃	1	3	yes	—
42	#108	—	326	6	no	—
43	#101	—	326	6	no	—
44	C₂×Q₈	—	326	3	no	—
45	C₂×Q₈	—	326	3	no	—
46	C₂²×C₄	—	163	2	no	—
47	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	326	3	no	—
51	M₁₆	—	163	6	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	326	3	no	—
54	C₂³	Q₈×D₁₆₃	1	1	yes	characteristic
55	Q₈	—	652	3	no	—
56	Q₈	—	652	1	no	—
57	C₂×C₄	—	326	6	no	—
58	C₂×C₄	D₄×D₁₆₃	1	3	yes	—
59	C₂×C₄	#14579699	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	326	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#19104713	1	3	yes	—
64	C₄	—	652	2	no	—
65	C₄	#19104708	1	3	yes	—
66	C₄	#19104729	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#23781589	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781589
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#338×C₁₆₃	C₂
socle	C₂×C₁₆₃	#396
prime radical	C₂³×C₁₆₃	C₂×Q₈
hypercenter	#338	D₁₆₃
hypocenter	C₁₆₃	#2070
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
34. C₁₆₃	#2070	semidirect

Maximal subgroups

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

subgroup	index
#23781713	2
C₂×#19104653	2
#338×C₁₆₃	2
#2070	163