Subgroups — #35212369

Order 20864.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781718	C₂	1	2	yes	maximal
2	C₄³×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₆₃⋊C₈)	C₄	1	4	yes	complemented
4	C₂²⋊(C₁₆₃⋊C₈)	—	2	2	no	—
5	C₄×C₁₆₃⋊C₈	—	2	2	no	—
6	C₂×C₄²×C₁₆₃	—	2	2	no	—
7	C₂×C₄²×C₁₆₃	C₄	1	2	yes	—
8	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆₃⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₁₆₃	—	2	2	no	—
11	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
13	C₄²×C₁₆₃	—	2	8	no	—
14	C₄²×C₁₆₃	—	2	2	no	—
15	C₄²×C₁₆₃	C₈	1	4	yes	complemented
16	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
17	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
18	C₄²×C₁₆₃	Q₈	1	1	yes	characteristic
19	C₁₆₃⋊C₈	—	4	4	no	—
20	C₂³×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₆₃	—	2	8	no	—
22	C₂×C₄×C₁₆₃	—	2	4	no	—
23	C₂×C₄×C₁₆₃	—	2	4	no	—
24	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₆₃	M₁₆	1	2	yes	—
30	C₂²×C₁₆₃	—	2	2	no	—
31	C₂²×C₁₆₃	#95	1	1	yes	characteristic
32	C₂²×C₁₆₃	C₂²⋊C₈	1	2	yes	—
33	C₄×C₁₆₃	—	2	8	no	—
34	C₄×C₁₆₃	—	2	2	no	—
35	C₄×C₁₆₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₁₆₃	#97	1	1	yes	characteristic
37	C₄×C₁₆₃	#104	1	4	yes	—
38	C₄×C₁₆₃	C₄⋊C₈	1	2	yes	—
39	C₂×C₁₆₃	—	2	2	no	—
40	C₂×C₁₆₃	#336	1	1	yes	characteristic
41	C₂×C₁₆₃	#339	1	2	yes	—
42	C₁₆₃	#1276	1	1	yes	characteristic, complemented, Sylow
43	#1276	—	163	1	no	maximal, Sylow
44	#431	—	163	2	no	—
45	C₄³	D₁₆₃	1	1	yes	characteristic
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	Dic₁₆₃	1	2	yes	—
48	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
49	C₄⋊C₈	—	163	4	no	—
50	C₂²⋊C₈	—	326	2	no	—
51	C₄×C₈	—	326	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
55	C₂×C₈	—	326	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	C₁₆₃⋊C₈	1	4	yes	complemented
59	C₄²	C₄×D₁₆₃	1	2	yes	—
60	C₄²	D₆₅₂	1	1	yes	characteristic
61	C₄²	Dic₃₂₆	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
67	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
68	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
69	C₂×C₄	C₄⋊Dic₁₆₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₁₆₃⋊C₈	1	2	yes	—
71	C₂×C₄	#14579683	1	2	yes	—
72	C₈	—	652	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₁₆₃⋊#95	1	1	yes	characteristic
75	C₂²	C₂²⋊(C₁₆₃⋊C₈)	1	2	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×C₁₆₃⋊C₈	1	1	yes	characteristic
79	C₄	#19104646	1	1	yes	characteristic
80	C₄	#19104663	1	4	yes	—
81	C₄	C₄⋊(C₁₆₃⋊C₈)	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	#23781420	1	1	yes	characteristic
84	C₂	#23781425	1	2	yes	—

	subgroup	G/H
center	C₄²	D₆₅₂
commutator	C₄×C₁₆₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₆₃
Fitting	C₄³×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#95
prime radical	C₂³×C₁₆₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₆₃	C₂²
hypercenter	C₄³	D₁₆₃
hypocenter	C₁₆₃	#1276
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₁₆₃⋊C₈)	C₄	semidirect
15. C₄²×C₁₆₃	C₈	semidirect
42. C₁₆₃	#1276	semidirect
58. 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
#23781718	2
C₄³×C₁₆₃	2
#1276	163