Subgroups — #35214688

Order 20864.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₆₃⋊C₈)	C₂	1	6	yes	maximal
2	C₄³×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆₃⋊C₈	C₂²	1	4	yes	—
4	C₄⋊(C₁₆₃⋊C₈)	—	2	12	no	—
5	C₂×C₄²×C₁₆₃	C₄	1	3	yes	—
6	C₂×C₄²×C₁₆₃	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₁₆₃	C₂²	1	3	yes	—
8	C₂×C₁₆₃⋊C₈	—	2	12	no	—
9	C₂²×C₄×C₁₆₃	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₁₆₃	C₂×C₄	1	3	yes	—
11	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
12	C₄²×C₁₆₃	—	2	6	no	—
13	C₄²×C₁₆₃	C₈	1	4	yes	complemented
14	C₄²×C₁₆₃	D₄	1	6	yes	—
15	C₄²×C₁₆₃	Q₈	1	6	yes	—
16	C₁₆₃⋊C₈	—	4	4	no	—
17	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₆₃	—	2	6	no	—
19	C₂×C₄×C₁₆₃	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₁₆₃	C₄⋊C₄	1	6	yes	—
21	C₂×C₄×C₁₆₃	C₂×C₈	1	6	yes	—
22	C₂×C₄×C₁₆₃	M₁₆	1	6	yes	—
23	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
24	C₂×C₄×C₁₆₃	C₂×Q₈	1	3	yes	—
25	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	3	yes	—
26	C₂²×C₁₆₃	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₁₆₃	C₂×M₁₆	1	3	yes	—
28	C₄×C₁₆₃	—	2	6	no	—
29	C₄×C₁₆₃	C₄⋊C₈	1	12	yes	complemented
30	C₄×C₁₆₃	#127	1	1	yes	characteristic
31	C₄×C₁₆₃	C₄⋊Q₈	1	3	yes	—
32	C₂×C₁₆₃	#384	1	1	yes	characteristic
33	C₂×C₁₆₃	C₂×C₄⋊C₈	1	3	yes	—
34	C₂×C₁₆₃	#423	1	3	yes	—
35	C₁₆₃	#1842	1	1	yes	characteristic, complemented, Sylow
36	#1842	—	163	1	no	maximal, Sylow
37	C₂×C₄⋊C₈	—	163	6	no	—
38	C₄³	D₁₆₃	1	1	yes	characteristic
39	C₂²×C₈	—	163	4	no	—
40	C₂×C₄²	Dic₁₆₃	1	3	yes	—
41	C₂×C₄²	Dic₁₆₃	1	1	yes	characteristic
42	C₂×C₄²	C₂×D₁₆₃	1	3	yes	—
43	C₄⋊C₈	—	326	12	no	—
44	C₂²×C₄	C₂×Dic₁₆₃	1	3	yes	—
45	C₂²×C₄	C₂×Dic₁₆₃	1	3	yes	—
46	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
47	C₂×C₈	—	326	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	C₁₆₃⋊C₈	1	4	yes	complemented
50	C₄²	D₆₅₂	1	6	yes	—
51	C₄²	Dic₃₂₆	1	6	yes	—
52	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₄⋊Dic₁₆₃	1	6	yes	—
55	C₂×C₄	C₄⋊Dic₁₆₃	1	6	yes	—
56	C₂×C₄	C₂×C₁₆₃⋊C₈	1	6	yes	—
57	C₂×C₄	#14579683	1	6	yes	—
58	C₂×C₄	C₂×D₆₅₂	1	3	yes	—
59	C₂×C₄	C₂×Dic₃₂₆	1	3	yes	—
60	C₈	—	652	4	no	—
61	C₂²	C₂×C₄⋊Dic₁₆₃	1	3	yes	—
62	C₂²	C₂²×C₁₆₃⋊C₈	1	1	yes	characteristic
63	C₂²	C₂×#14579683	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊(C₁₆₃⋊C₈)	1	12	yes	complemented
66	C₄	#19104732	1	1	yes	characteristic
67	C₄	#19104735	1	3	yes	—
68	C₂	#23781535	1	1	yes	characteristic
69	C₂	C₂×C₄⋊(C₁₆₃⋊C₈)	1	3	yes	—
70	C₂	#23781689	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator	C₂²×C₁₆₃	C₂²×C₈
Fitting	C₄³×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₄³	D₁₆₃
hypocenter	C₁₆₃	#1842
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
13. C₄²×C₁₆₃	C₈	semidirect
29. C₄×C₁₆₃	C₄⋊C₈	semidirect
35. C₁₆₃	#1842	semidirect
49. C₄²	C₁₆₃⋊C₈	semidirect
65. C₄	C₄⋊(C₁₆₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₆₃⋊C₈)	2
C₄³×C₁₆₃	2
#1842	163