Subgroups — #35214148

Order 20864.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782016	C₂	1	3	yes	maximal, complemented
2	#23782012	C₂	1	3	yes	maximal
3	C₈²×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104735	C₂²	1	3	yes	—
5	#19104732	C₂²	1	1	yes	characteristic
6	#19104660	—	2	6	no	—
7	#19104657	—	2	6	no	—
8	C₄×C₈×C₁₆₃	C₂²	1	3	yes	—
9	C₂×Dic₃₂₆	—	2	3	no	—
10	C₂×D₆₅₂	—	2	3	no	—
11	C₄⋊Dic₁₆₃	—	2	3	no	—
12	C₄⋊Dic₁₆₃	—	2	3	no	—
13	C₂×C₈×C₁₆₃	D₄	1	6	yes	—
14	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
15	C₂²×D₁₆₃	—	4	1	no	—
16	Dic₃₂₆	—	4	6	no	—
17	D₆₅₂	—	4	6	no	—
18	C₂×Dic₁₆₃	—	4	3	no	—
19	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
20	C₈×C₁₆₃	—	2	6	no	—
21	C₂²×C₁₆₃	#127	1	1	yes	characteristic
22	C₂×D₁₆₃	—	8	3	no	—
23	C₄×C₁₆₃	#135	1	6	yes	—
24	Dic₁₆₃	—	8	3	no	—
25	C₂×C₁₆₃	#495	1	3	yes	—
26	D₁₆₃	—	16	1	no	—
27	C₁₆₃	#1711	1	1	yes	characteristic, complemented, Sylow
28	#1711	—	163	1	no	maximal, Sylow
29	#487	—	163	3	no	—
30	#486	—	163	3	no	—
31	C₈²	D₁₆₃	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	163	3	no	—
33	#127	—	163	1	no	—
34	#103	—	326	6	no	—
35	#102	—	326	6	no	—
36	C₄×C₈	C₂×D₁₆₃	1	3	yes	—
37	C₂×Q₈	—	326	3	no	—
38	C₂×D₄	—	326	3	no	—
39	C₂×C₈	D₆₅₂	1	6	yes	—
40	C₄⋊C₄	—	326	3	no	—
41	C₄⋊C₄	—	326	3	no	—
42	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
43	C₂³	—	652	1	no	—
44	Q₈	—	652	6	no	—
45	D₄	—	652	6	no	—
46	C₂×C₄	—	652	3	no	—
47	C₂×C₄	C₂×D₆₅₂	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	1304	3	no	—
50	C₂²	#19104732	1	1	yes	characteristic
51	C₄	—	1304	3	no	—
52	C₄	#19104757	1	6	yes	—
53	C₂	—	2608	1	no	—
54	C₂	#23782049	1	3	yes	—

	subgroup	G/H
center	C₂²	#19104732
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	C₈²×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#127
prime radical	#19104732	C₂²
hypercenter	C₈²	D₁₆₃
hypocenter	C₁₆₃	#1711
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #23782016	C₂	semidirect
3. C₈²×C₁₆₃	C₂	semidirect
27. C₁₆₃	#1711	semidirect
31. C₈²	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782016	2
#23782012	2
C₈²×C₁₆₃	2
#1711	163