Subgroups — #35212570

Order 20864.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782034	C₂	1	1	yes	maximal, characteristic
2	#23781443	C₂	1	1	yes	maximal, characteristic
3	#347×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104735	C₄	1	1	yes	characteristic
5	#19104727	C₄	1	1	yes	characteristic
6	#19104675	—	2	1	no	—
7	#19104660	—	2	1	no	—
8	#19104660	—	2	1	no	—
9	M₃₂×C₁₆₃	—	2	1	no	—
10	#97×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₂₆	—	2	1	no	—
12	C₄⋊Dic₁₆₃	—	2	1	no	—
13	C₄⋊Dic₁₆₃	—	4	1	no	—
14	Dic₁₆₃⋊C₄	—	4	1	no	—
15	C₁₆₃⋊C₁₆	—	2	2	no	—
16	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
18	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₆₃	—	2	2	no	—
20	Dic₃₂₆	—	4	1	no	—
21	Dic₃₂₆	—	4	1	no	—
22	C₂×Dic₁₆₃	—	4	1	no	—
23	C₂×Dic₁₆₃	—	4	1	no	—
24	C₂×C₄×C₁₆₃	—	2	1	no	—
25	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₃	—	2	1	no	—
27	C₈×C₁₆₃	—	2	1	no	—
28	C₂²×C₁₆₃	#100	1	1	yes	characteristic
29	C₄×C₁₆₃	—	4	1	no	—
30	C₄×C₁₆₃	#104	1	1	yes	characteristic
31	C₄×C₁₆₃	#104	1	1	yes	characteristic
32	Dic₁₆₃	—	8	1	no	—
33	Dic₁₆₃	—	8	1	no	—
34	C₂×C₁₆₃	—	2	1	no	—
35	C₂×C₁₆₃	#329	1	1	yes	characteristic
36	C₁₆₃	#1356	1	1	yes	characteristic, complemented, Sylow
37	#1356	—	163	1	no	maximal, Sylow
38	#491	—	163	1	no	—
39	#347	—	163	1	no	—
40	#347	D₁₆₃	1	1	yes	characteristic
41	C₄⋊Q₈	—	163	1	no	—
42	#125	—	163	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	326	1	no	—
45	#103	—	326	1	no	—
46	#103	—	326	1	no	—
47	#97	C₂×D₁₆₃	1	1	yes	characteristic
48	C₂×Q₈	—	326	1	no	—
49	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
50	C₂×C₈	D₆₅₂	1	1	yes	characteristic
51	C₄⋊C₄	—	326	1	no	—
52	C₄⋊C₄	—	652	1	no	—
53	C₄⋊C₄	—	652	1	no	—
54	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	326	2	no	—
57	Q₈	—	652	1	no	—
58	Q₈	—	652	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	652	1	no	—
61	C₂×C₄	—	652	1	no	—
62	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#19104651	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	1304	1	no	—
68	C₄	—	1304	1	no	—
69	C₄	#19104661	1	1	yes	characteristic
70	C₄	#19104663	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#23781404	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781404
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	#97	C₂×D₁₆₃
Fitting	#347×C₁₆₃	C₂
socle	C₂×C₁₆₃	#329
prime radical	C₂²×C₁₆₃	#100
cosocle	#97×C₁₆₃	C₂²
hypercenter	#347	D₁₆₃
hypocenter	C₁₆₃	#1356
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
36. C₁₆₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#23782034	2
#23781443	2
#347×C₁₆₃	2
#1356	163