Subgroups — #35212423

Order 20864.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₆₃⋊C₈)	C₂	1	2	yes	maximal
2	#407×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆₃⋊C₈	—	2	2	no	—
4	C₂²⋊(C₁₆₃⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₁₆₃⋊C₈)	C₄	1	4	yes	—
6	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₆₃	C₄	1	2	yes	—
8	C₂²⋊C₈×C₁₆₃	—	2	2	no	—
9	C₂×C₁₆₃⋊C₈	—	2	4	no	—
10	C₂×C₁₆₃⋊C₈	—	4	4	no	—
11	C₂⁴×C₁₆₃	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₁₆₃	—	4	2	no	—
13	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
16	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
17	M₁₆×C₁₆₃	—	4	2	no	—
18	C₂×C₈×C₁₆₃	—	2	2	no	—
19	C₁₆₃⋊C₈	—	4	4	no	—
20	C₂³×C₁₆₃	—	2	4	no	—
21	C₂³×C₁₆₃	—	4	1	no	—
22	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
23	C₂³×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	—	2	2	no	—
25	C₂×C₄×C₁₆₃	—	2	4	no	—
26	C₂×C₄×C₁₆₃	—	4	2	no	—
27	C₂×C₄×C₁₆₃	—	4	1	no	—
28	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₆₃	—	4	2	no	—
32	C₂²×C₁₆₃	—	2	2	no	—
33	C₂²×C₁₆₃	—	2	2	no	—
34	C₂²×C₁₆₃	—	2	2	no	—
35	C₂²×C₁₆₃	—	2	1	no	—
36	C₂²×C₁₆₃	—	4	4	no	—
37	C₂²×C₁₆₃	#95	1	1	yes	characteristic
38	C₂²×C₁₆₃	#108	1	4	yes	—
39	C₄×C₁₆₃	—	2	1	no	—
40	C₄×C₁₆₃	—	2	2	no	—
41	C₄×C₁₆₃	#99	1	2	yes	—
42	C₂×C₁₆₃	—	2	4	no	—
43	C₂×C₁₆₃	—	4	1	no	—
44	C₂×C₁₆₃	#341	1	2	yes	—
45	C₂×C₁₆₃	#342	1	1	yes	characteristic
46	C₁₆₃	#1298	1	1	yes	characteristic, complemented, Sylow
47	#1298	—	163	1	no	maximal, Sylow
48	#407	D₁₆₃	1	1	yes	characteristic
49	C₂×C₂²⋊C₈	—	163	2	no	—
50	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
51	C₂×M₁₆	Dic₁₆₃	1	2	yes	—
52	C₂²×C₈	—	326	2	no	—
53	C₂²⋊C₈	—	2	2	no	—
54	C₂²⋊C₈	—	326	2	no	—
55	C₂²⋊C₈	—	163	4	no	—
56	C₂⁴	Dic₃₂₆	1	1	yes	characteristic
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
59	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
60	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
61	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
62	M₁₆	—	4	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	326	4	no	—
65	C₂×C₈	—	652	4	no	—
66	C₂³	—	2	4	no	—
67	C₂³	—	4	1	no	—
68	C₂³	Dic₁₆₃⋊C₄	1	2	yes	—
69	C₂³	C₄⋊Dic₁₆₃	1	1	yes	characteristic
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
75	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
76	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
77	C₈	—	4	2	no	—
78	C₈	—	652	4	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	4	4	no	—
84	C₂²	C₁₆₃⋊#95	1	1	yes	characteristic
85	C₂²	#19104670	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#19104650	1	2	yes	—
89	C₂	—	2	4	no	—
90	C₂	—	4	1	no	—
91	C₂	#23781429	1	2	yes	—
92	C₂	#23781432	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₆₃
commutator	C₂×C₄×C₁₆₃	C₄²
Frattini	C₂³×C₄	C₂×D₁₆₃
Fitting	#407×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#95
prime radical	C₂⁴×C₁₆₃	Q₈
cosocle	C₂³×C₄×C₁₆₃	C₂²
hypercenter	#407	D₁₆₃
hypocenter	C₁₆₃	#1298
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
46. C₁₆₃	#1298	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₆₃⋊C₈)	2
#407×C₁₆₃	2
#1298	163