Subgroups — #35215641

Order 20864.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782465	C₂	1	3	yes	maximal, complemented
2	#23781633	C₂	1	3	yes	maximal, complemented
3	#338×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14579699	C₂²	1	3	yes	—
5	#19104768	—	2	6	no	—
6	#19104763	—	2	6	no	—
7	#19104757	—	2	6	no	—
8	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
9	(C₂×D₁₆₃)⋊C₈	—	2	6	no	—
10	C₂×M₁₆×C₁₆₃	C₂²	1	3	yes	—
11	#14579699	—	2	6	no	—
12	#14579699	—	2	6	no	—
13	C₂×Dic₃₂₆	D₄	1	3	yes	—
14	C₂×D₆₅₂	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₆₃	—	2	3	no	—
16	C₂×C₄×D₁₆₃	—	2	3	no	—
17	Dic₆₅₂	—	2	6	no	—
18	#14579688	—	2	12	no	—
19	D₁₃₀₄	—	2	6	no	—
20	C₂×C₁₆₃⋊C₈	—	2	3	no	—
21	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₆₃	—	2	6	no	—
23	C₂×C₈×C₁₆₃	D₄	1	6	yes	—
24	C₂²×D₁₆₃	—	2	3	no	—
25	Dic₃₂₆	—	2	6	no	—
26	D₆₅₂	—	2	6	no	—
27	C₂²⋊D₁₆₃	—	4	6	no	—
28	C₂×Dic₁₆₃	—	2	3	no	—
29	C₄×D₁₆₃	—	4	3	no	—
30	C₄×D₁₆₃	—	4	3	no	—
31	C₁₆₃⋊C₈	—	4	1	no	—
32	C₂³×C₁₆₃	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
35	C₈×C₁₆₃	—	2	6	no	—
36	C₂²×C₁₆₃	—	4	1	no	—
37	C₂²×C₁₆₃	#121	1	3	yes	—
38	C₂×D₁₆₃	—	4	6	no	—
39	C₂×D₁₆₃	—	4	3	no	—
40	C₄×C₁₆₃	#120	1	3	yes	—
41	C₄×C₁₆₃	#127	1	1	yes	characteristic
42	Dic₁₆₃	—	4	3	no	—
43	C₂×C₁₆₃	—	2	3	no	—
44	C₂×C₁₆₃	#392	1	1	yes	characteristic
45	D₁₆₃	—	8	3	no	—
46	C₁₆₃	#2017	1	1	yes	characteristic, complemented, Sylow
47	#2017	—	163	1	no	maximal, Sylow
48	#575	—	163	3	no	—
49	#408	—	163	3	no	—
50	#338	D₁₆₃	1	1	yes	characteristic, complemented
51	C₂×#41	—	163	3	no	—
52	#137	—	326	6	no	—
53	AGL(1,ℤ/8)	—	326	6	no	—
54	#135	—	326	6	no	—
55	C₂×M₁₆	C₂×D₁₆₃	1	3	yes	—
56	C₂²×C₈	—	163	1	no	—
57	C₂²⋊C₈	—	326	6	no	—
58	#41	—	326	6	no	—
59	#41	—	326	6	no	—
60	C₂×Q₈	—	163	3	no	—
61	C₂×D₄	—	326	3	no	—
62	C₂×D₄	—	163	3	no	—
63	C₂²×C₄	—	326	3	no	—
64	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
65	Q₁₆	—	326	6	no	—
66	SD₁₆	—	326	12	no	—
67	D₈	—	326	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	326	3	no	—
70	C₂×C₈	D₆₅₂	1	6	yes	—
71	C₂³	—	326	3	no	—
72	C₂³	#14579700	1	1	yes	characteristic
73	Q₈	—	326	6	no	—
74	D₄	—	326	6	no	—
75	D₄	—	652	6	no	—
76	C₂×C₄	—	326	3	no	—
77	C₂×C₄	—	652	3	no	—
78	C₂×C₄	—	652	3	no	—
79	C₂×C₄	D₄×D₁₆₃	1	3	yes	—
80	C₂×C₄	C₂×D₆₅₂	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	652	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	652	6	no	—
85	C₂²	—	652	3	no	—
86	C₂²	#19104710	1	3	yes	—
87	C₄	—	652	3	no	—
88	C₄	#19104705	1	3	yes	—
89	C₄	#19104732	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	1304	3	no	—
92	C₂	#23781575	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19104732
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#338×C₁₆₃	C₂
socle	C₂×C₁₆₃	#392
hypercenter	#338	D₁₆₃
hypocenter	C₁₆₃	#2017
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #23782465	C₂	semidirect
2. #23781633	C₂	semidirect
3. #338×C₁₆₃	C₂	semidirect
46. C₁₆₃	#2017	semidirect
50. #338	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782465	2
#23781633	2
#338×C₁₆₃	2
#2017	163