Subgroups — #35212504

Order 20864.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃₀₄	C₂	1	1	yes	maximal, characteristic
2	#23781416	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₀₄	C₄	1	1	yes	characteristic
5	C₄×D₆₅₂	—	2	1	no	—
6	#19104669	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₆₃⋊C₈)	—	2	1	no	—
8	#19104657	—	2	1	no	—
9	C₂×C₁₆×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×D₆₅₂	—	2	1	no	—
12	C₂×C₄×D₁₆₃	—	4	1	no	—
13	D₁₃₀₄	—	2	1	no	—
14	D₁₃₀₄	C₈	1	2	yes	complemented
15	C₂×C₁₆₃⋊C₈	—	4	1	no	—
16	C₄⋊Dic₁₆₃	—	2	1	no	—
17	(C₂×D₁₆₃)⋊C₄	—	4	1	no	—
18	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
21	C₁₆×C₁₆₃	—	2	2	no	—
22	C₂²×D₁₆₃	—	4	1	no	—
23	D₆₅₂	—	2	2	no	—
24	D₆₅₂	—	4	1	no	—
25	C₂×Dic₁₆₃	—	4	1	no	—
26	C₄×D₁₆₃	—	8	2	no	—
27	C₁₆₃⋊C₈	—	8	1	no	—
28	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₃	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₆₃	—	2	1	no	—
32	C₈×C₁₆₃	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
34	C₂²×C₁₆₃	#102	1	1	yes	characteristic
35	C₂×D₁₆₃	—	4	2	no	—
36	C₂×D₁₆₃	—	8	1	no	—
37	C₂×D₁₆₃	—	8	1	no	—
38	C₄×C₁₆₃	—	2	1	no	—
39	C₄×C₁₆₃	—	2	1	no	—
40	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₁₆₃	#104	1	1	yes	characteristic
42	Dic₁₆₃	—	8	1	no	—
43	C₂×C₁₆₃	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₁₆₃	#360	1	1	yes	characteristic
45	C₂×C₁₆₃	#361	1	1	yes	characteristic
46	D₁₆₃	—	8	2	no	—
47	C₁₆₃	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	163	1	no	maximal, Sylow
49	C₄×D₈	—	163	1	no	—
50	#346	D₁₆₃	1	1	yes	characteristic, complemented
51	#334	—	163	1	no	—
52	C₂×D₈	—	163	1	no	—
53	C₄×D₄	—	326	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	163	1	no	—
56	C₄⋊C₈	—	326	1	no	—
57	#102	—	326	1	no	—
58	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
59	C₂×D₄	—	326	1	no	—
60	C₂²×C₄	—	652	1	no	—
61	D₈	—	326	1	no	—
62	D₈	—	163	2	no	—
63	C₂×C₈	—	652	1	no	—
64	C₂×C₈	C₄×D₁₆₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
66	C₄⋊C₄	—	326	1	no	—
67	C₂²⋊C₄	—	652	1	no	—
68	C₄²	D₆₅₂	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	652	1	no	—
71	D₄	—	326	2	no	—
72	D₄	—	652	1	no	—
73	C₂×C₄	—	1304	2	no	—
74	C₂×C₄	—	652	1	no	—
75	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
77	C₂×C₄	#14579688	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1304	1	no	—
80	C₈	C₈×D₁₆₃	1	1	yes	characteristic
81	C₈	#14579682	1	1	yes	characteristic
82	C₂²	—	1304	1	no	—
83	C₂²	—	1304	1	no	—
84	C₂²	—	652	2	no	—
85	C₂²	#19104657	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	1304	1	no	—
89	C₄	(C₂×D₁₆₃)⋊C₈	1	1	yes	characteristic
90	C₄	#19104663	1	1	yes	characteristic
91	C₂	—	1304	2	no	—
92	C₂	#23781394	1	1	yes	characteristic
93	C₂	#23781476	1	1	yes	characteristic
94	C₂	#23781479	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19104657
commutator	C₈×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#346×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#102
prime radical	C₂×D₁₃₀₄	C₄
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#346	D₁₆₃
hypocenter	C₁₆₃	#1333
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #23781416	C₂	semidirect
3. #346×C₁₆₃	C₂	semidirect
14. D₁₃₀₄	C₈	semidirect
47. C₁₆₃	#1333	semidirect
50. #346	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₃₀₄	2
#23781416	2
#346×C₁₆₃	2
#1333	163