Subgroups — #35212576

Order 20864.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782038	C₂	1	1	yes	maximal, characteristic
2	#23781486	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579688	—	2	1	no	—
5	C₂×#14579688	—	2	1	no	—
6	#19104735	C₄	1	1	yes	characteristic
7	#19104732	C₄	1	1	yes	characteristic
8	#19104675	—	2	1	no	—
9	M₃₂×C₁₆₃	—	2	1	no	—
10	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₂₆	—	2	1	no	—
12	C₂×D₆₅₂	—	2	1	no	—
13	C₂×D₆₅₂	—	4	1	no	—
14	#14579688	—	4	1	no	—
15	#14579688	—	4	1	no	—
16	#14579688	—	4	1	no	—
17	#14579688	—	4	1	no	—
18	C₄⋊Dic₁₆₃	—	4	1	no	—
19	C₁₆₃⋊C₁₆	—	4	1	no	—
20	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
22	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₆₃	—	4	1	no	—
24	C₂²×D₁₆₃	—	4	1	no	—
25	Dic₃₂₆	—	4	1	no	—
26	Dic₃₂₆	—	4	1	no	—
27	D₆₅₂	—	4	1	no	—
28	D₆₅₂	—	4	1	no	—
29	D₆₅₂	—	8	2	no	—
30	C₂×Dic₁₆₃	—	4	1	no	—
31	C₂×C₄×C₁₆₃	—	2	1	no	—
32	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₆₃	D₈	1	1	yes	characteristic
34	C₈×C₁₆₃	D₈	1	1	yes	characteristic
35	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₁₆₃	#100	1	1	yes	characteristic
38	C₂×D₁₆₃	—	8	2	no	—
39	C₂×D₁₆₃	—	8	1	no	—
40	C₄×C₁₆₃	—	2	2	no	—
41	C₄×C₁₆₃	#102	1	1	yes	characteristic
42	C₄×C₁₆₃	#102	1	1	yes	characteristic
43	Dic₁₆₃	—	8	1	no	—
44	C₂×C₁₆₃	—	2	1	no	—
45	C₂×C₁₆₃	#331	1	1	yes	characteristic
46	D₁₆₃	—	16	1	no	—
47	C₁₆₃	#1359	1	1	yes	characteristic, complemented, Sylow
48	#1359	—	163	1	no	maximal, Sylow
49	#492	—	163	1	no	—
50	#364	—	163	1	no	—
51	#364	D₁₆₃	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	326	1	no	—
53	C₂×SD₁₆	—	326	1	no	—
54	C₄⋊Q₈	—	163	1	no	—
55	#127	—	163	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	326	1	no	—
58	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
59	C₂×Q₈	—	326	1	no	—
60	C₂×D₄	—	326	1	no	—
61	C₂×D₄	—	652	1	no	—
62	SD₁₆	—	652	1	no	—
63	SD₁₆	—	652	1	no	—
64	SD₁₆	—	652	1	no	—
65	SD₁₆	—	652	1	no	—
66	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
67	C₂×C₈	D₆₅₂	1	1	yes	characteristic
68	C₄⋊C₄	—	652	1	no	—
69	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	652	1	no	—
72	C₂³	—	652	1	no	—
73	Q₈	—	652	1	no	—
74	Q₈	—	652	1	no	—
75	D₄	—	1304	2	no	—
76	D₄	—	652	1	no	—
77	D₄	—	652	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	652	1	no	—
80	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
81	C₈	#14579684	1	1	yes	characteristic
82	C₈	D₁₃₀₄	1	1	yes	characteristic
83	C₈	#14579686	1	1	yes	characteristic
84	C₈	#14579688	1	1	yes	characteristic
85	C₂²	—	1304	2	no	—
86	C₂²	—	1304	1	no	—
87	C₂²	#19104651	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	1304	1	no	—
90	C₄	#19104655	1	1	yes	characteristic
91	C₄	#19104657	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2608	1	no	—
94	C₂	#23781408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781408
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#364×C₁₆₃	C₂
socle	C₂×C₁₆₃	#331
prime radical	#19104732	C₄
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#364	D₁₆₃
hypocenter	C₁₆₃	#1359
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #23781486	C₂	semidirect
3. #364×C₁₆₃	C₂	semidirect
47. C₁₆₃	#1359	semidirect
51. #364	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782038	2
#23781486	2
#364×C₁₆₃	2
#1359	163