Subgroups — #35212572

Order 20864.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782042	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×D₁₃₀₄	—	2	1	no	—
5	C₂×D₁₃₀₄	—	2	1	no	—
6	#19104732	C₄	1	2	yes	—
7	#19104675	—	2	1	no	—
8	M₃₂×C₁₆₃	—	2	1	no	—
9	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₂×D₆₅₂	—	2	2	no	—
11	C₂×D₆₅₂	—	4	2	no	—
12	D₁₃₀₄	—	4	1	no	—
13	D₁₃₀₄	—	4	1	no	—
14	D₁₃₀₄	—	4	1	no	—
15	D₁₃₀₄	—	4	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	—	4	1	no	—
21	C₂²×D₁₆₃	—	4	2	no	—
22	D₆₅₂	—	4	2	no	—
23	D₆₅₂	—	4	2	no	—
24	D₆₅₂	—	8	4	no	—
25	C₂×C₄×C₁₆₃	—	2	1	no	—
26	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₆₃	D₈	1	1	yes	characteristic
28	C₈×C₁₆₃	D₈	1	1	yes	characteristic
29	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₆₃	#100	1	1	yes	characteristic
32	C₂×D₁₆₃	—	8	4	no	—
33	C₂×D₁₆₃	—	8	2	no	—
34	C₄×C₁₆₃	—	2	2	no	—
35	C₄×C₁₆₃	#102	1	1	yes	characteristic
36	C₄×C₁₆₃	#102	1	1	yes	characteristic
37	C₂×C₁₆₃	—	2	1	no	—
38	C₂×C₁₆₃	#331	1	1	yes	characteristic
39	D₁₆₃	—	16	2	no	—
40	C₁₆₃	#1357	1	1	yes	characteristic, complemented, Sylow
41	#1357	—	163	1	no	maximal, Sylow
42	#493	—	163	1	no	—
43	#364	—	163	1	no	—
44	#364	D₁₆₃	1	1	yes	characteristic, complemented
45	C₂×D₈	—	326	1	no	—
46	C₂×D₈	—	326	1	no	—
47	#127	—	163	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	326	1	no	—
50	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
51	C₂×D₄	—	326	2	no	—
52	C₂×D₄	—	652	2	no	—
53	D₈	—	652	1	no	—
54	D₈	—	652	1	no	—
55	D₈	—	652	1	no	—
56	D₈	—	652	1	no	—
57	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
58	C₂×C₈	D₆₅₂	1	1	yes	characteristic
59	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	652	1	no	—
62	C₂³	—	652	2	no	—
63	D₄	—	1304	4	no	—
64	D₄	—	652	2	no	—
65	D₄	—	652	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
68	C₈	#14579684	1	1	yes	characteristic
69	C₈	D₁₃₀₄	1	1	yes	characteristic
70	C₈	#14579686	1	1	yes	characteristic
71	C₈	#14579688	1	1	yes	characteristic
72	C₂²	—	1304	4	no	—
73	C₂²	—	1304	2	no	—
74	C₂²	#19104651	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#19104655	1	1	yes	characteristic
77	C₄	#19104657	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	2608	2	no	—
80	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	#23782042	C₂
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#364	D₁₆₃
hypocenter	C₁₆₃	#1357
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
40. C₁₆₃	#1357	semidirect
44. #364	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782042	2
#23781486	2
#364×C₁₆₃	2
#1357	163