Subgroups — #35212489

Order 20864.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781447	C₂	1	2	yes	maximal
2	#424×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#19104675	—	2	2	no	—
4	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₁₆₃	C₄	1	1	yes	characteristic
6	#117×C₁₆₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₆₃	—	2	2	no	—
8	C₁₆₃⋊C₁₆	—	4	2	no	—
9	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
10	M₁₆×C₁₆₃	—	4	1	no	—
11	C₂×C₈×C₁₆₃	—	2	1	no	—
12	C₂×C₈×C₁₆₃	—	4	1	no	—
13	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
14	C₄⋊C₄×C₁₆₃	—	2	1	no	—
15	C₂²⋊C₄×C₁₆₃	C₈	1	2	yes	—
16	C₄²×C₁₆₃	—	2	1	no	—
17	C₂³×C₁₆₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₁₆₃	—	2	1	no	—
19	C₂×C₄×C₁₆₃	—	2	1	no	—
20	C₂×C₄×C₁₆₃	—	2	2	no	—
21	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	M₁₆	1	1	yes	characteristic
23	C₈×C₁₆₃	—	2	2	no	—
24	C₈×C₁₆₃	—	4	1	no	—
25	C₂²×C₁₆₃	—	2	1	no	—
26	C₂²×C₁₆₃	—	4	1	no	—
27	C₂²×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₆₃	—	2	1	no	—
29	C₄×C₁₆₃	—	4	2	no	—
30	C₄×C₁₆₃	#99	1	1	yes	characteristic
31	C₄×C₁₆₃	#101	1	1	yes	characteristic
32	C₂×C₁₆₃	—	2	1	no	—
33	C₂×C₁₆₃	—	4	1	no	—
34	C₂×C₁₆₃	#324	1	1	yes	characteristic
35	C₁₆₃	#1328	1	1	yes	characteristic, complemented, Sylow
36	#1328	—	163	1	no	maximal, Sylow
37	#424	D₁₆₃	1	1	yes	characteristic
38	#349	—	163	2	no	—
39	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
40	C₂²×C₈	Dic₁₆₃	1	1	yes	characteristic
41	#117	Dic₁₆₃	1	1	yes	characteristic
42	M₃₂	—	326	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	C₁₆₃⋊C₈	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	652	2	no	—
53	C₂³	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
58	C₂×C₄	#14579683	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#19104650	1	1	yes	characteristic
67	C₄	#19104654	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#23781391	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19104650
commutator, prime radical	C₂³×C₁₆₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₁₆₃
Fitting	#424×C₁₆₃	C₂
socle	C₂×C₁₆₃	#324
cosocle	C₂×M₁₆×C₁₆₃	C₂²
hypercenter	#424	D₁₆₃
hypocenter	C₁₆₃	#1328
LCS	G ▷ C₂³×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #424

Complements

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

N	H ≅ G/N	type
35. C₁₆₃	#1328	semidirect

Maximal subgroups

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

subgroup	index
#23781447	2
#424×C₁₆₃	2
#1328	163