Subgroups — #35212676

Order 20864.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781499	C₂	1	2	yes	maximal, complemented
2	C₂×M₃₂×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₁₆₃⋊C₃₂	—	2	2	no	—
4	C₂²×C₈×C₁₆₃	C₄	1	1	yes	characteristic
5	M₃₂×C₁₆₃	—	2	2	no	—
6	C₂×C₁₆×C₁₆₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₁₆₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₁₆₃	—	2	1	no	—
10	C₂×C₈×C₁₆₃	—	2	1	no	—
11	C₂×C₈×C₁₆₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₆₃	—	2	1	no	—
14	C₁₆×C₁₆₃	D₄	1	2	yes	—
15	C₂³×C₁₆₃	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₁₆₃	—	2	1	no	—
17	C₂×C₄×C₁₆₃	—	2	1	no	—
18	C₂×C₄×C₁₆₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₆₃	—	2	1	no	—
21	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
23	C₂²×C₁₆₃	—	2	1	no	—
24	C₂²×C₁₆₃	—	4	1	no	—
25	C₂²×C₁₆₃	C₂×C₁₆	1	1	yes	characteristic
26	C₄×C₁₆₃	—	2	1	no	—
27	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₆₃	M₃₂	1	1	yes	characteristic
29	C₂×C₁₆₃	—	2	1	no	—
30	C₂×C₁₆₃	—	4	1	no	—
31	C₂×C₁₆₃	C₂²⋊C₁₆	1	1	yes	characteristic
32	C₁₆₃	#1400	1	1	yes	characteristic, complemented, Sylow
33	#1400	—	163	1	no	maximal, Sylow
34	C₂×M₃₂	D₁₆₃	1	1	yes	characteristic
35	M₆₄	—	163	2	no	—
36	C₂²×C₈	Dic₁₆₃	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	Dic₁₆₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
40	C₃₂	—	326	2	no	—
41	C₂²×C₄	C₁₆₃⋊C₈	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	C₁₆₃⋊C₈	1	1	yes	characteristic
45	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
46	C₁₆	—	2	1	no	—
47	C₁₆	C₂²⋊D₁₆₃	1	2	yes	—
48	C₂³	C₁₆₃⋊C₁₆	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₁₆₃⋊C₁₆	1	1	yes	characteristic
52	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
55	C₈	#14579683	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×C₁₆₃⋊C₁₆	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
61	C₄	#19104675	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₂²⋊(C₁₆₃⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₁₆₃
commutator	C₂²×C₁₆₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	C₂×M₃₂×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂²⋊C₁₆
prime radical	C₂³×C₁₆₃	C₁₆
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	C₂×M₃₂	D₁₆₃
hypocenter	C₁₆₃	#1400
LCS	G ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #23781499	C₂	semidirect
32. C₁₆₃	#1400	semidirect

Maximal subgroups

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

subgroup	index
#23781499	2
C₂×M₃₂×C₁₆₃	2
#1400	163