Subgroups — #35212519

Order 20864.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781795	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₆₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₃₂₆	C₄	1	1	yes	characteristic
5	C₂×C₁₆₃⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₁₆₃⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₁₆₃⋊C₈	—	2	1	no	—
8	#19104645	—	2	1	no	—
9	C₂×C₁₆×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₂₆	C₈	1	1	yes	characteristic
12	C₂×C₁₆₃⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₆₃	C₈	1	1	yes	characteristic
14	Dic₁₆₃⋊C₄	—	2	1	no	—
15	C₄×Dic₁₆₃	—	2	1	no	—
16	C₁₆₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₆₃	—	2	2	no	—
21	Dic₃₂₆	—	4	1	no	—
22	C₂×Dic₁₆₃	—	2	1	no	—
23	C₂×Dic₁₆₃	—	2	1	no	—
24	C₁₆₃⋊C₈	—	4	1	no	—
25	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₆₃	M₁₆	1	1	yes	characteristic
28	C₈×C₁₆₃	—	2	1	no	—
29	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₆₃	—	2	1	no	—
33	C₄×C₁₆₃	—	2	1	no	—
34	C₄×C₁₆₃	#103	1	1	yes	characteristic
35	C₄×C₁₆₃	#104	1	1	yes	characteristic
36	Dic₁₆₃	—	4	1	no	—
37	Dic₁₆₃	—	4	1	no	—
38	C₂×C₁₆₃	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₁₆₃	#349	1	1	yes	characteristic
40	C₂×C₁₆₃	#350	1	1	yes	characteristic
41	C₁₆₃	#1338	1	1	yes	characteristic, complemented, Sylow
42	#1338	—	163	1	no	maximal, Sylow
43	#446	—	163	1	no	—
44	C₄⋊C₁₆	—	163	1	no	—
45	#346	D₁₆₃	1	1	yes	characteristic
46	C₄×Q₈	—	163	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	326	1	no	—
49	C₄⋊C₈	—	326	1	no	—
50	C₄⋊C₈	—	163	1	no	—
51	#97	—	326	1	no	—
52	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
53	C₂×Q₈	—	163	1	no	—
54	C₂×C₈	—	326	1	no	—
55	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
56	C₂×C₈	D₆₅₂	1	1	yes	characteristic
57	C₄⋊C₄	—	326	1	no	—
58	C₄⋊C₄	—	163	1	no	—
59	C₄²	—	326	1	no	—
60	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	652	1	no	—
63	Q₈	—	652	1	no	—
64	C₂×C₄	—	326	1	no	—
65	C₂×C₄	—	326	1	no	—
66	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
67	C₂×C₄	C₈×D₁₆₃	1	1	yes	characteristic
68	C₂×C₄	#14579682	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	652	1	no	—
71	C₈	#14579688	1	1	yes	characteristic
72	C₈	Dic₆₅₂	1	1	yes	characteristic
73	C₂²	(C₂×D₁₆₃)⋊C₈	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	652	1	no	—
77	C₄	—	652	1	no	—
78	C₄	#19104660	1	1	yes	characteristic
79	C₄	#19104663	1	1	yes	characteristic
80	C₂	#23781397	1	1	yes	characteristic
81	C₂	#23781446	1	1	yes	characteristic
82	C₂	#23781450	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₆₃)⋊C₄
commutator	C₂×C₄×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#346×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#346	D₁₆₃
hypocenter	C₁₆₃	#1338
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
41. C₁₆₃	#1338	semidirect

Maximal subgroups

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

subgroup	index
#23781795	2
C₄⋊(C₁₆₃⋊C₁₆)	2
#346×C₁₆₃	2
#1338	163