Subgroups — #35214679

Order 20864.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781420	C₂	1	6	yes	maximal
2	C₄³×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆₃⋊C₈	C₂²	1	4	yes	—
4	C₂×C₄²×C₁₆₃	C₄	1	3	yes	—
5	C₂×C₄²×C₁₆₃	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₁₆₃	C₂²	1	3	yes	—
7	C₂×C₁₆₃⋊C₈	—	2	12	no	—
8	C₂²×C₄×C₁₆₃	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₁₆₃	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
11	C₄²×C₁₆₃	—	2	6	no	—
12	C₄²×C₁₆₃	—	2	6	no	—
13	C₄²×C₁₆₃	C₈	1	4	yes	complemented
14	C₁₆₃⋊C₈	—	4	4	no	—
15	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₆₃	—	2	3	no	—
17	C₂×C₄×C₁₆₃	—	2	3	no	—
18	C₂×C₄×C₁₆₃	—	2	3	no	—
19	C₂×C₄×C₁₆₃	—	2	3	no	—
20	C₂×C₄×C₁₆₃	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₁₆₃	M₁₆	1	6	yes	—
22	C₂×C₄×C₁₆₃	#41	1	6	yes	—
23	C₂²×C₁₆₃	#117	1	3	yes	—
24	C₂²×C₁₆₃	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₁₆₃	C₂×M₁₆	1	3	yes	—
26	C₄×C₁₆₃	—	2	6	no	—
27	C₄×C₁₆₃	—	2	6	no	—
28	C₄×C₁₆₃	#126	1	4	yes	—
29	C₂×C₁₆₃	#383	1	1	yes	characteristic
30	C₂×C₁₆₃	#431	1	3	yes	—
31	C₂×C₁₆₃	#432	1	3	yes	—
32	C₁₆₃	#1839	1	1	yes	characteristic, complemented, Sylow
33	#1839	—	163	1	no	maximal, Sylow
34	C₄³	D₁₆₃	1	1	yes	characteristic
35	#336	—	163	6	no	—
36	C₂²×C₈	—	163	4	no	—
37	C₂×C₄²	Dic₁₆₃	1	3	yes	—
38	C₂×C₄²	Dic₁₆₃	1	1	yes	characteristic
39	C₂×C₄²	C₂×D₁₆₃	1	3	yes	—
40	C₂²×C₄	C₂×Dic₁₆₃	1	3	yes	—
41	C₂²×C₄	C₂×Dic₁₆₃	1	3	yes	—
42	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
43	C₂×C₈	—	326	12	no	—
44	C₄²	—	2	6	no	—
45	C₄²	—	2	6	no	—
46	C₄²	C₁₆₃⋊C₈	1	4	yes	complemented
47	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	C₂×C₁₆₃⋊C₈	1	6	yes	—
53	C₂×C₄	#14579683	1	6	yes	—
54	C₂×C₄	#14579699	1	6	yes	—
55	C₈	—	652	4	no	—
56	C₂²	#19104694	1	3	yes	—
57	C₂²	C₂²×C₁₆₃⋊C₈	1	1	yes	characteristic
58	C₂²	C₂×#14579683	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	2	6	no	—
61	C₄	#19104729	1	4	yes	—
62	C₂	#23781533	1	1	yes	characteristic
63	C₂	#23781718	1	3	yes	—
64	C₂	#23781722	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₃
commutator	C₂²×C₁₆₃	C₂²×C₈
Fitting	C₄³×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₄³	D₁₆₃
hypocenter	C₁₆₃	#1839
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
13. C₄²×C₁₆₃	C₈	semidirect
32. C₁₆₃	#1839	semidirect
46. C₄²	C₁₆₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23781420	2
C₄³×C₁₆₃	2
#1839	163